Introduction to Quantum Computing for Business

G oenland, Koen
Book
In oduc ion o Quan um Compu ing o Business
P o ided in Coope a ion wi h:
Ams e dam Uni e si y P ess (AUP)
Sugges ed Ci a ion: G oenland, Koen (2025) : In oduc ion o Quan um Compu ing o Business, ISBN
978-90-4856-899-4, Ams e dam Uni e si y P ess, Ams e dam,
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koen g oenland
INTRODUCTION TO
QUANTUM COMPUTING
FOR BUSINESS
ow will businesses use quan um echnology in he u u e?
Wha p oblems will a quan um compu e sol e? How long
will i ake be o e hese de ices become comme cially ele an ?
Wi h he i s gene a ion o quan um compu e s on he ho izon,
unde s anding hei impac is mo e ele an han e e . Luckily,
you don’ need a physics deg ee o unde s and he unc ionali y
o hese compu e s – jus like you don’ need o know how a
ansis o wo ks o excel in con en ional i .
This book is he pe ec in oduc ion o he oppo uni ies and h ea s
o quan um echnologies. I equips you wi h he necessa y knowledge
o join cu ing-edge discussions and make s a egic decisions.
koen g oenland is a heo e ical physicis wi h a PhD in he
nea - e m applica ions o quan um compu e s. He wo ks as an
inno a ion o ice a he Uni e si y o Ams e dam, whe e he is
esponsible o se ing up esea ch collabo a ions and de eloping
li elong lea ning educa ion o p o essionals. He is one o he
d i ing o ces behind Quan um. Ams e dam, he inno a ion hub
ha d i es he comme cialisa ion o quan um echnologies
a ound he Du ch capi al.
“Easy o ead and ull o insigh s, a mus - ead o anyone looking
o unde s and he eal-wo ld impac o quan um compu ing.”
– Diede ick C oese, Di ec o o Cen e o Quan um and Socie y
“This book o e s a well- ounded, scien i ically accu a e o e iew
o quan um echnology, highligh ing i s signi ican po en ial o
inno a ion.” – Ch is ian Scha ne , P o esso in Theo e ical Compu e
Science, Di ec o o QuSo
H
koen g oenland INTRODUCTION TO QUANTUM COMPUTING FOR BUSINESS
 

In oduc ion o Quan um Compu ing o Business.indd 1In oduc ion o Quan um Compu ing o Business.indd 1 11-02-2025 12:0511-02-2025 12:05
In oduc ion o Quan um Compu ing o Business
In oduc ion o Quan um Compu ing
o Business
Koen G oenland
Ams e dam Uni e si y P ess

Co e illus a ion: © Dada a
Co e design: Mijke Wonde gem
Lay-ou : C ius G oup, Hulshou
Illus a ions: © Dada a
isbn 978 90 4856 898 7
e-isbn 978 90 4856 899 4 (pd )
doi 10.5117/9789048568987
nu 120
C ea i e Commons License CC-BY NC ND (h p://c ea i ecommons.o g/licenses/by-nc-nd/4.0)
K. G oenland / Ams e dam Uni e si y P ess B.V., Ams e dam 2025
Some igh s ese ed. Wi hou limi ing he igh s unde copy igh ese ed abo e, any pa o
his book may be ep oduced, s o ed in o in oduced in o a e ie al sys em, o ansmi ed,
in any o m o by any means (elec onic, mechanical, pho ocopying, eco ding o o he wise).
Table o Con en s
Pa 1 The essen ials
P e ace: Why his book? 11
1 An in oduc ion o he quan um wo ld 15
1.1 Wha is quan um? 15
1.2 Fou su p ising phenomena 16
1.3 Wha does a quan um compu e look like? 22
1.4 Fu he eading 26
2 The backg ound: Why a e we so en husias ic abou quan um
echnology? 27
2.1 Wha is quan um echnology? 27
2.2 The impo ance o high-pe o mance compu ing 28
2.3 Why can quan um compu e s ha e an ad an age? 29
2.4 F om algo i hm o so wa e 34
2.5 Fu he eading 35
2.6 No es 35
3 The applica ions: Wha p oblems will we sol e wi h quan um
compu e s? 37
3.1 Wha applica ions o e a quan um speedup? 38
3.2 How can we compa e di e en ypes o speedups? 44
3.3 Whe e is he kille applica ion? 47
3.4 Fu he eading 51
3.5 No es 52
4 Timelines: When can we expec a use ul quan um compu e ? 55
4.1 Wha pa ame e s a e ele an ? 55
4.2 How many qubi s a e needed? 58
4.3 How long un il we ha e million-qubi machines? 63
4.4 Pu ing i all oge he 67
4.5 Fu he eading 69
4.6 No es 69
5 Fou my hs abou quan um compu ing 71
5.1 My h 1: Quan um compu e s ind all solu ions a once 71
5.2 My h 2: Qubi s can s o e much mo e da a han he same
numbe o classical bi s 72
5.3 My h 3: En anglemen allows you o send in o ma ion as e
han ligh o o in luence objec s a a dis ance 73
5.4 My h 4: Quan um compu e s a e always en yea s away. 75
5.5 Fu he eading 76
5.6 No es 77
Pa 2 Mo e abou he applica ions
6 Applica ions in chemis y and ma e ial science 81
6.1 Wha p oblems in chemis y and ma e ial science will we sol e? 81
6.2 Algo i hms o quan um chemis y 83
6.3 A hype a ound quan um compu ing o clima e change 85
6.4 A case s udy o a po en ial kille applica ion: FeMoco 86
6.5 Fu he eading 88
6.6 No es 89
7 The impac on cybe secu i y 91
7.1 C yp og aphy is much mo e han jus sec ecy 91
7.2 The quan um h ea is mainly o public key c yp og aphy 93
7.3 Wha solu ions exis ? 97
7.4 Conclusion 100
7.5 Fu he eading 100
7.6 No e 101
8 Applica ions o quan um ne wo ks 103
8.1 The p omises o he quan um in e ne 103
8.2 How use ul is he quan um in e ne in p ac ice? 104
8.3 The case o QKD 105
8.4 Conclusion 107
8.5 Fu he eading 107
9 Op imisa ion and AI: Wha a e companies doing oday? 109
9.1 Compa ing Algo i hms and O anges 109
9.2 Whe e should we look o a new kille applica ion? 113
9.3 Examples o esul s in di e en sec o s 114
9.4 Fu he eading 122
9.5 No es 123
Pa 3 The ha dwa e and s a egic ac ions
10 Quan um ha dwa e 127
10.1 Di e en unc ionali ies 127
10.2 Di e en building blocks 131
10.3 Fu he eading 132
10.4 No e 132
11 E o co ec ion 133
11.1 Wha is e o co ec ion? 134
11.2 Longe compu a ions need mo e qubi s 138
11.3 Wha is he cu en s a e-o - he-a ? 141
11.4 Conclusion 143
11.5 Fu he eading 144
12 Wha s eps should you o ganisa ion ake? 145
12.1 Common i s s eps 145
12.2 P epa e o use quan um applica ions 146
12.3 Mig a ing o pos -quan um c yp og aphy 149
12.4 Fu he eading 153
12.5 No e 153
Pa 4 The inal bi s
13 Fu he eading 157
13.1 I wan o lea n he echnical de ails 157
13.2 I wan o lea n o p og am a quan um compu e 159
13.3 I wan o s ay up o da e wi h he la es de elopmen s 160
13.4 I wan o lea n mo e abou business implica ions 161
14 O e iew o quan um compu e s a ailable oday 163
15 Quan um Hype Bingo 165
16 Acknowledgemen s 167
17 Bibliog aphy 169
18 Index 173

1 An in oduc ion o he quan um wo ld
A a glance
you don’ need o unde s and quan um mechanics o unde s and he
unc ionali y o quan um compu e s. Bu i you insis , quan um mechanics
desc ibes he beha iou o he smalles pa icles. I leads o many coun e -
in ui i e phenomena: compu e memo y can s o e mul iple pieces o da a
simul aneously, bu , when measu ed, na u e selec s jus a single piece and
h ows away all he o he s.
I you wan o d i e a ca , do you need o unde s and how i s engine wo ks?
O cou se, you don’ ! In a simila ein, you don’ need o know he de ails o
quan um physics o ead he es o his book. So, eel ee o skip his chap e .
Ne e heless, we know ha mos people wan o ha e some concep ual in-
ui ion abou wha quan um mechanics eally is. I is no na u al o lea e one
o he mos used wo ds in his book as an abs ac concep , and i migh be
ha d o he human b ain o p oceed wi hou a leas seeing some examples.
He e is my bes a emp o explain quan um mechanics in accessible e ms.
P oceed wi h cau ion, as hings will almos ce ainly ge con using om he e.
1.1 Wha is quan um?
Quan um physics o quan um mechanicsis he heo y ha desc ibes
he inies pa icles, such as elec ons, a oms, and small molecules. The
heo y is mean o desc ibe he undamen al laws o na u e using a se o
ma hema ical equa ions, allowing us o p edic cause and e ec a he scale
o nanome es. I answe s ques ions like ‘Wha happens when I b ing wo
elec ons close oge he ?’ o ‘Will hese wo subs ances unde go a chemical
eac ion?’. You can con as quan um mechanics o New on’s classical
physics, which we lea ned in high school. The classical heo y wo ks g ea
o objec s he size o a building o a oo ball bu becomes inaccu a e a much
smalle scales. Quan um is, in a sense, a e inemen o classical physics: he
heo ies a e e ec i ely iden ical when applied o a co ee mug, bu he mo e
di icul quan um heo y is needed o desc ibe e y small hings.
Some examples o sys ems whe e quan um could play a ole a e:
– A oms and he elec ons ha o bi a ound hem.
– Flows o elec ici y in mic oscopic (nano-scale) wi es and chips.
– Pho ons, he pa icles ou o which ligh is made.
16 In oduC Ion o Quan um Compu Ing o BusIness
We a e going o need some physics ja gon o p oceed. We like o use he
wo d ‘s a e’, which is a comple e desc ip ion o all he physical p ope ies
o he wo ld a one ins ance: he loca ions o all he di e en pa icles, hei
eloci ies, how much hey o a e, e c. Usually, he en i e uni e se is oo big
o s udy, so we o en simpli y ou wo ld o a single, isola ed pa icle o o
a limi ed piece o compu e memo y. Le ’s imagine a ba e pa icle in an
o he wise emp y wo ld. We may be in e es ed in i s loca ion, which we’ll
call x . Fo example, he wo ld migh look some hing like he image below,
which can be desc ibed by a e y simple s a e: x = 5 ( he ule is jus i ual).
In he spi i o compu ing, we migh look a a ‘bi ’ ha s o es in o ma ion.
Think o i as a iny magne ha can ei he poin ‘up’ (1) o ‘down’ (0). The
s a e o a piece o memo y is easy o desc ibe, simply by exp essing he bi
alues one by one. Fo example: 11010.
Impo an ly, he s a e o he wo ld can change o e ime. We will o en
ca e abou he s a e o he wo ld a a ce ain momen , o example, a he
beginning o a compu a ion o a he end o i .
1.2 Fou su p ising phenomena
The mos iconic quan um phenomenon is supe posi ion. Think abou
any p ope y ha we can (classically) measu e, such as he posi ion o a
pa icle o he alue o a bi on a ha d d i e (0 o 1). In quan um mechanics,
many di e en measu emen ou comes can be somewha ‘ ue’ a he same
ime: a pa icle can be in mul iple posi ions a once, o a bi could be 0 and
1 simul aneously. When we say ‘a he same ime’ we mean ha , o p edic
an In oduC Ion o he Quan um Wo ld 17
any cause and e ec , we need o keep ack o all hese possibili ies. To
illus a e a supe posi ion, I some imes pic u e a quan um pa icle spli ing
in o many opaque copies o i sel , sp ead ou o e space, whe e he deg ee
o anspa ency de e mines how likely he pa icle is o be ound he e: he
da ke i is, he mo e p esence i has a ha loca ion.
To h ow in some mo e examples o supe posi ions: an elec on can mo e
a a eloci y o 10m/s and 100m/s a he same ime (which ob iously also
leads o a supe posi ion in i s loca ion). Mo e ele an o us: a compu e
memo y migh s o e he numbe s 5 and 11 ‘simul aneously’ o e en 46 di -
e en Mic oso Excel sp eadshee s ‘a once’. An impo an building block
o make his all wo k is he qubi , which is any kind o ha dwa e ha can
s o e bi alues 0 and 1, and any possible supe posi ion o hese wo. I we
ha e a bunch o qubi s oge he , we’ll call i a quan um memo y.
Le us illus a e he wei dness o supe posi ions wi h an example whe e
he 46 sp eadshee s each ake 1 megabi (Mb) o s o e. A egula , classical
ha d d i e would alloca e he i s Mb o a i s sp eadshee , hen ano he
Mb o s o e he second, and so o h. In o al, i would use 46 Mb. The
quan um memo y has an addi ional op ion o s o e he sp eadshee s in
supe posi ion: using he qubi -equi alen o jus 1Mb (one million qubi s) i
would encode all he da a in jus ha limi ed amoun o memo y. Whe eas
1 Mb o classical memo y can i jus one sp eadshee , a quan um memo y
o 1 Mb can ep esen se e al o hem, all hanks o he unique p ope ies
o quan um physics. Howe e , as we’ll see la e , he e is a ca ch o s o ing
all ha da a so compac ly.
How can you possibly desc ibe a wo ld whe e pa icles and compu e
memo ies a e in supe posi ion? Fo now, le ’s ocus on an isola ed pa icle.
We speci y i s s a e using a leng hy lis , whe e o each possible posi ion,
we s o e a numbe called he ampli ude, which is ela ed o how likely he
pa icle is o be ound a ha loca ion. In o he wo ds, he s a e desc ibes
p ecisely o wha ex en a pa icle is a posi ion x = 0 , o wha ex en a
posi ion x = 1 , and so o h, o e e y possible loca ion ha he pa icle
can be a . And indeed, his lis could be in ini ely long! Luckily, when
dealing wi h compu e s, we wo k wi h simple objec s. A quan um bi
18 In oduC Ion o Quan um Compu Ing o BusIness
needs jus wo ampli udes, which deno e he ex en o which he bi is ‘0’
o ‘1’, espec i ely.
The ampli udes used o desc ibe quan um s a es eel somewha analo-
gous o p obabili ies, which can simila ly ell us he likelihood ha , o
example, a pa icle can be ound a a pa icula loca ion. Howe e , he e
is a undamen al di e ence. P obabili ies in he classical wo ld help us
deal wi h in o ma ion we don’ ha e: su ely, he pa icle is al eady a some
loca ion, bu pe haps we jus don’ know which loca ion ye . Quan um
mechanics is di e en . E en i we know e e y iny de ail abou he loca ion
o a pa icle, we s ill need o desc ibe i as a supe posi ion. Fundamen ally,
he loca ion is no de e mined ye . Hence, he e is li e ally no be e way
o desc ibe he pa icle han by acking his con olu ed supe posi ion.
Ampli udes a e also mo e inicky o deal wi h han p obabili ies because
hese numbe s can become nega i e (and o ma h expe s, hey can e en
be complex numbe s).
The second wei d phenomenon is how quan um measu emen s wo k.
Why do we ne e obse e an elec on a wo places a he same ime? Why
do I ne e ind a ca bo h mo ing and s anding s ill? In quan um mechanics,
as soon as we measu e he loca ion o a pa icle, i ins an ly jumps o a single
loca ion a andom – making i s loca ion ully de e mined. Simila ly, when
we measu e a qubi , i jumps o ei he ‘0’ o ‘1’. When we measu e he da a
in a quan um memo y, we may ind any one o he 46 sp eadshee s ha
we e s o ed. A measu emen essen ially changes a sys em in o a no mal,
classical s a e.
The e ec o a measu emen is in insically andom (and hence, ou wo ld
is no de e minis ic!). Bu his doesn’ imply ha we canno unde s and
quan um mechanics. We can calcula e he p obabili ies o measu emen
ou comes wi h inc edible p ecision as long as we know he s a e be o e he
measu emen .
I is impo an o no e ha we canno lea n any hing abou he wo ld
wi hou measu ing – i is ou only way o ob ain da a abou physical objec s.
Any obse a ion, e en a sligh peek a ou sys em, is a measu emen in
quan um mechanics. Addi ionally, measu emen s a e des uc i e in he
sense ha hey change he s a e o he wo ld. We undamen ally canno
‘look’ a a pa icle wi hou dis u bing i . In ac , measu emen s dele e all he
ich da a encoded in a supe posi ion! I a pa icle was ini ially a posi ion
x = 0 ,  x = 3 and x = 10, all simul aneously, hen upon measu emen ,
i jumps o one o hese h ee op ions. To gi e you a bi o ja gon, we call
his ins an aneous change a ‘collapse.’ F om ha momen , i is 100% a a
ixed loca ion: i , a i s , we measu e he pa icle o be a x = 3 , hen any
an In oduC Ion o he Quan um Wo ld 19
subsequen measu emen will gi e he same esul , un il some o he o ce
mo es i again. In he con ex o a quan um compu a ion, his means ha
we should ca e ully choose when we pe o m any measu emen s – we
canno jus peek a he da a a any momen we like, o we isk dis u bing
a supe posi ion.
This also means ha a single piece o quan um memo y canno s o e an
immense numbe o sp eadshee s a he same ime – a leas , you wouldn’
be able o e ie e each o hem. To s o e 15 Mb wo h o classical da a, we
need 15 Mb wo h o qubi s. Hence, quan um compu e s a e no pa icula ly
use ul o s o ing classical da a.
The ac ha a measu emen changes he s a e o he wo ld poses a
se ious p oblem o he enginee s who a e building quan um compu e s.
No ma e wha ma e ial we cons uc ou qubi s om, hey will su ely
in e ac wi h o he nea by pa icles, and some o hese in e ac ions could
ac like des uc i e measu emen s. We call his e ec decohe ence, and,
as we will see la e , his o ms one o he co e challenges o la ge-scale
quan um compu a ion.
A his poin , quan um da a doesn’ seem pa icula ly use ul. Why would
we wan o deal wi h supe posi ions i hey lead o all his unce ain y? The
impo an ad an age s ems om he way in which a quan um compu e can
p ocess quan um da a. Using quan um mechanics, a de ice can manipula e
da a in ways ha a classical compu e could ne e do.
Tha leads us o he hi d unique phenomenon. A quan um compu e
can manipula e he da a i s o es using so-called quan um ga es, o simply
‘ga es’ o sho . These a e apid bu s s o some physical o ces ha change
he s a e o one o mo e qubi s. They can u n a classical-looking s a e in o
a quan um supe posi ion o ice e sa. They can ac like logical ope a ions,
like he AND and OR ga es ha a e used in classical elec onics, bu also
like new quan um logic ha has no classical coun e pa .
F om a unc ional pe spec i e, a quan um ga e akes one o mo e
qubi s as inpu , changes hei in e nal s a e, and hen ou pu s he same
numbe o qubi s (wi h hei al e ed s a es). In o he wo ds, he numbe
o physical objec s emains unchanged, bu he o e all s a e changes. As
an example, you may hink o ou p o o ypical magne ha was ini ially
poin ing ‘up’, bu a quan um ga e migh lip his o ‘down’. The e a e many
such ga es possible, each ha ing a di e en e ec on hei inpu . We like
o gi e hem names in capi al le e s, such as X, Z, H, and CX. Impo an ly,
a quan um ga e is de e minis ic, meaning ha i s inpu -ou pu beha iou
is always he same, as opposed o he quan um measu emen s we saw
ea lie .

20 In oduC Ion o Quan um Compu Ing o BusIness
The canonical way o desc ibe a quan um compu e p og am is by de ining
a sequence o quan um ga es, whe e o each ga e, we also indica e wha
qubi s a e supposed o be he ga e’s inpu . A he end o he compu a ion,
we measu e all qubi s. An example o such a p og am, using he s anda d
Quan um Assembly (QASM) language, is gi en below.
Toge he , hese s eps can be g aphically displayed in a quan um ci cui ,
as shown he e on he igh . Quan um ci cui s ep esen each qubi wi h
a ho izon al line and indica e ime lowing om le o igh . Whene e a
box wi h a le e is displayed o e a qubi line, hen he co esponding ga e
should be applied. This isn’ unlike he way we ead shee music! You may
no ice ha some imes, wo o mo e ga es can be pe o med in pa allel as
long as hey ac on di e en qubi s.
When we un a ci cui on an ac ual quan um compu e , he inal measu e-
men s lead o p obabilis ic ou comes. We ge o see a bunch o ones and ze oes:
one classical bi o each qubi . I he ci cui is a good quan um algo i hm,
hen, wi h high p obabili y, hese classical bi s will ell us he answe we
a e looking o . Bu e en hen, we migh need o edo he compu a ion a ew
imes and ake ( o example) he mos common esul as ou inal answe .
I you a e comple ely con used a his poin , you a e no alone. The whole
business o quan um supe posi ion and quan um ope a ions is inc edibly
complex and is no some hing you could possibly mas e a e eading a
ew pages. Scien is s who ha e s udied he subjec o many yea s a e s ill
an In oduC Ion o he Quan um Wo ld 21
equen ly ba led by decep i e pa adoxes and coun e -in ui i e phenomena.
On he o he hand, we hope ha he unc ionali y o quan um ci cui s
makes some sense: we de ine a lis o ins uc ions and eed hem in o a
machine ha can execu e hem. We don’ ha e o know p ecisely wha ’s
going on unde he hood!
The e is one emaining quan um phenomenon o co e – one ha comes
wi h a mys e ious lai su ounding i . We’ e alking abou quan um en-
anglemen , which we’ll desc ibe using he ollowing example.
Imagine ha we ha e wo qubi s, which we can anspo independen ly
om each o he wi hou dis u bing he da a hey s o e. Toge he , he qubi s
can ep esen he s a es 00, 01, 10, o 11, o any supe posi ion o hese. Ac-
co ding o quan um mechanics, we can c ea e a e y speci ic s a e whe e
he pai o qubi s is simul aneously 00 and 11. Now, imagine ha compu e
scien is Alice g abs one o he qubi s, akes i on he ocke ship, and lies
i all he way o he dwa plane Plu o. The o he qubi emains on Ea h
in he hands o physicis Bob. Upon a i ing on Plu o, Alice measu es he
qubi and inds ou come ‘1’. A deep ques ion is: wha do we now know abou
Bob’s qubi ?
Since he only possible measu emen ou comes we e 00 and 11, he o he
qubi can only be measu ed as ‘1’ om now onwa ds. I essen ially collapses
o be 100% in he s a e ‘1’. Bu how could he Ea h-based qubi possibly know
ha a measu emen occu ed on Plu o? Wha mechanism made i collapse?
Acco ding o Eins ein’s heo y o ela i i y, in o ma ion canno a el as e
han he speed o ligh , which ansla es in o a ew hou s be ween Ea h
and Plu o. Ne e heless, measu ing he qubi s in wo a away loca ions will
always gi e a consis en esul , e en when he wo qubi s a e measu ed a
exac ly he same ime.
This pa adox e eals, once again, how con using quan um mechanics
can be. Howe e , he s o y abo e is pe ec ly consis en wi h bo h quan-
um mechanics and he heo y o ela i i y. The co e p inciple is ha no
in o ma ion can be sen as e han ligh be ween Alice and Bob. Fo example,
can you see why Bob has no way o de ec ing when Alice pe o ms he
measu emen jus by looking a his en angled qubi ? In he mos common
in e p e a ion o quan um mechanics, he Ea h qubi does indeed change
i s s a e ins an aneously when Alice measu es he qubi , al hough he e is
no way o exploi his e ec o as messaging.
Mo e gene ally, en anglemen is he phenomenon whe e wo o mo e a a-
way qubi s can ha e co ela ed measu emen ou comes ha a e classically
impossible. The e is a ascina ing u he discussion abou he philosophy
behind en anglemen , bu we’ll lea e ha o o he sou ces. Wha ma e s
22 In oduC Ion o Quan um Compu Ing o BusIness
o us is ha en anglemen leads o new unc ionali ies ha we can exploi .
We will disco e wha hese a e in he chap e on quan um ne wo ks.
So, he e you ha e i : ou su p ising phenomena you may hea equen ly
in quan um echnology con e sa ions. To summa ise:
– Supe posi ion: he phenomenon whe e a qubi is bo h 0 and 1 a he same
ime.
– Quan um measu emen : measu ing a quan um memo y des oys supe
-
posi ion. The esul we ob ain is p obabilis ic.
– Quan um ga es: de e minis ic changes o he s a e o qubi s, which gen-
e alise classical logic ga es like OR, AND, NOT. A lis o se e al quan um
ga es ( oge he wi h he qubi s hey ac on) o ms a quan um ci cui .
– En anglemen : qubi s sepa a ed o e a long dis ance can s ill sha e unique
p ope ies.
1.3 Wha does a quan um compu e look like?
Mos la ge-scale compu ing oday happens in da a cen es, whe e we don’
ca e much abou he speci ics o he de ices ha do ou calcula ions. We
also expec ha u u e quan um compu e s will mos ly be ucked away in
he ‘cloud’, making hei appea ance and inne wo kings la gely i ele an
o mos use s. Howe e , o his op ional chap e , we can ake he oppo -
uni y o iew wha oday’s cu ing-edge ha dwa e looks like. The e a e
many di e en ways o build a quan um compu e , each based on dis inc
physical sys ems and p inciples. He e, we desc ibe he example o so-called
supe conduc ing qubi s, a ela i ely ma u e pla o m used by companies like
IBM, Google, and Rige i and se e al academic ins i u es. Resea ch ins i u e
QuTech in Del , he Ne he lands, was kind enough o p o ide pho os ha
allow us o look inside hei labs. We will see ha only a iny pa o he
compu e is ac ually ‘quan um’, whe eas mos o he machine consis s o
classical machine y ha ’s equi ed o keep he compu e wo king.
The eal quan um magic happens on a chip, no unlike he compu e
chips used in you lap op o phone. The qubi s a e o med by iny elec onic
ci cui s whe e he low o elec ical cu en is es ic ed o jus one ou
o wo s a es: he ‘bi ’ s a es 0 and 1. Since his is a quan um sys em, he
cu en can also be in a supe posi ion – pic u e all he elec ons in he wi e
pa icipa ing bo h in low ‘0’ and low ‘1’ simul aneously! This only wo ks
when he chip is cooled down o unimaginably low empe a u es, down
o a ound 10 millikel in – a hund ed h o a deg ee abo e absolu e ze o. A
hese empe a u es, he elec onic ci cui s become supe conduc ing, such
an In oduC Ion o he Quan um Wo ld 23
ha an ini ial cu en can low inde ini ely. This is impo an because any
damping o he cu en would cause unwan ed dis u bance o he qubi s a e.
The empe a u e cons ain is why he quan um chip is placed in a
massi e dilu ion e ige a o , a cylinde o abou hal a me e in diame e
and o e a me e all, which specialises in keeping he quan um chip cool.
In he u u e, la ge quan um compu e s may need e en bigge idges o
combine se e al o hese close oge he . Deepe pa s o he idge ha e
inc easingly low empe a u es, allowing us o cool in s ages. An example
could be o cool a i s en i onmen o 35 Kel in (-283 °Celsius o -396.7
°Fah enhei ), ollowed by subsequen s ages o ~3K, 900mK, 100mK, un il
he inal s age o ~10mK is eached.
Enginee s ypically suspend he idge on he ceiling so ha he highe
empe a u es a e on op, and he ul acold quan um chip is placed a he
e y bo om. The in e nals a e shaped acco dingly: se e al laye s o gold
disks a e hung below one ano he , one disk o each empe a u e zone. A
la ge numbe o wi es un be ween he disks, anspo ing signals be ween
he ceiling and he lowe mos a eas. The whole s uc u e o ms he iconic
me al chandelie ha you o en see in images, al hough i would all be
co e ed by a bo ing me al case when he idge is in ope a ion.
To make he qubi s do some hing use ul, like execu ing a quan um ga e
o pe o ming a measu emen , we need o send signals in o he chip. Jus
like wi h classical compu e s, a ‘signal’ is a ol age di e ence be ween
a quan um chip. pho o c edi s: ma c Blommae o Qu ech.
30 In oduC Ion o Quan um Compu Ing o BusIness
meaning ha numbe s up o 18,446,744,073,709,551,615 can be p ocessed.
Each o hese elemen a y s eps can be some hing like addi ion, mul iplica-
ion, a compa ison, e c., and we ha e powe ul ools o wea e hese basic
ope a ions oge he o o m e icien so wa e.
Now, quan um compu e s a e supposed o be e en as e , igh ? Well, i ’s
no ha d o ind suppo o ha claim:
news heade s by ech ada 2 and I lscience3.
You may be disappoin ed o hea ha , as o 2024, quan um compu e s canno
e en add o mul iply numbe s o mo e han 3 o 4 bi s. And e en i hey
could, hei a e o ope a ion would by no means each se e al GHz, bu mo e
likely se e al MHz (a ew million ope a ions pe second) a bes . In o he
wo ds, hey’ e mo e han a housand imesslowe .To make hings wo se,
he in o ma ion in quan um compu e s is ex emely agile and needs o
be cons an ly checked and co ec ed using so-callede o co ec ion.This
is a o m o o e head ha could make quan um compu e s ano he se e al
o de s o magni ude slowe . E en in he a u u e, when quan um compu e s
a e mo e ma u e and mo e eliable, we s ill expec hem o be much slowe
han he classical chips a ha ime.
How does his hyme wi h he news abou e e - as e quan um compu -
e s? And why a e we s ill in e es ed in hese slow machines? As we claimed
be o e, we hope o do ce ain compu a ions in a undamen ally di e en
way. Le ’s look a a beau i ul analogy ha Andy Ma uschak and Michael
Nielsen b ing up in hei online cou se Quan um Coun y4.

he BaCkg ound: Why a e We so en husIas IC aBou Quan um eChnology? 31
Imagine ha you’d like o a el om Mo occo o Spain, which a e sepa a ed
by a small piece o sea called he S ai o Gib al a . I you echnology does
no allow you o c oss he sea, hen you’d need o ake a la ge de ou , all he
way h ough No h A ica, pas he A abian Peninsula, and h ough Eu ope,
be o e you can each you des ina ion. This ep esen s he s eps aken by a
classical compu e . In he same analogy, a quan um compu e g an s you
he abili y o a e se bo h land and sea (much like a ho e c a ) so ha
you can ake a much mo e di ec ou e.
The beau y o quan um compu a ion is ha we ha e a undamen ally
di e en way o a el (do compu a ions), which can some imes b ing us o
ou des ina ion using a sho e ou e (doing ewe compu a ional s eps). E en
wi h a much slowe ehicle (compu e ), one may a i e a he des ina ion
soone . In ac , he quan um ad an age o en g ows as p oblems become
la ge and mo e complica ed.
The analogy also shows ha quan um compu e s do no always ha e
an ad an age: you would no wan o a el om Ams e dam o Be lin
by ho e c a . Un o una ely, in many cases, we don’ ye know wha he
as es means o anspo a ion is. I is s ill an ac i e a ea o esea ch o
comple ely map ou he landscape o e which quan um and classical
compu e s can a el and o de e mine which p oblems allow a speedup,
and which don’ .
Fo his eason, we don’ expec ha classical compu e s will be
eplaced any ime soon. Ins ead, classical and quan um p ocesso s will
li e side by side, and p og amme s will pick whiche e ool is be e
sui ed o sol e a ce ain p oblem. The si ua ion could be simila o how
we use g aphical p ocessing uni s (GPUs) oday, which o e emendous
he BaCkg ound: Why a e We so en husIas IC aBou Quan um eChnology? 33
speedups o he aining o a i icial in elligence models bu a e no made
o eplace egula classical p ocesso s (CPUs). Pe haps we should e en
gi e quan um compu e s a simila abb e ia ion, like ‘QPU’ o Quan um
P ocessing Uni .
In he analogy wi h he S ai o Gib al a , he p ecise ou e ha you a el
deno es he chosenalgo i hm.In he ield o compu e science, an algo i hm
is as ep-by-s ep lis o ins uc ions ha desc ibes how a compu a ional
p oblem should be sol ed. The‘s eps’he e should be su icien ly simple so
ha i is comple ely unambiguous how o do hem. They could be ope a ions
such as adding, mul iplying, o compa ing wo numbe s. Needless o say,
he ewe s eps he algo i hm equi es, he be e .
By exploi ing quan um mechanics, a quan um compu e in oduces
new basic s eps ha a e impossible o pe o m on a classical compu e .
Fo example, he p e ious chap e in oduced quan um logic ga es ha
gene alise ope a ions like AND and OR. Using hese building blocks, we
can o mula e quan um algo i hms ha ake much ewe s eps han he
bes classical algo i hm e e could!
In he end, he ime needed o sol e a p oblem can be e y oughly calcula ed
as:
“Time o sol e a p oblem” = “ ime pe s ep” × “numbe o s eps equi ed”
The ‘ ime pe s ep’ is a p ope y o he ha dwa e ha you use. Clea ly, a as e
CPU will lead o as e solu ions. The ‘numbe o s eps equi ed’ is dic a ed
by he algo i hm. The la e is p ecisely how quan um compu e s can o e
spec acula speedups. As long as he imp o emen in he ‘numbe o s eps
equi ed’ compensa es o he disad an age in ‘ ime pe s ep’, a quan um
compu e can help us sol e p oblems in less ime!
A ecu ing heme in his book is he sea ch o indus ially ele an
quan um algo i hms. This u ns ou o be mo e challenging han i seems a
i s sigh . Quan um algo i hms a e buil on deep and complex ma hema ics,
ely on coun e -in ui i e quan um phenomena, and equi e in en i e new
me hods o ackle a p oblem. Simple weaks o exis ing classical algo i hms
a e a ely su icien . In ac , o mos p oblems, no quan um speedups ha e
been iden i ied a all, despi e he bes a emp s by scien is s wo ldwide. We
migh go as a as o say ha , e en i we had a la ge-scale quan um compu e
oday, i s alue would be limi ed. Fo his eason, he ongoing de elopmen
o no el algo i hms is exceedingly impo an .
34 In oduC Ion o Quan um Compu Ing o BusIness
2.4 F om algo i hm o so wa e
In he end, simply inding a good algo i hm is no enough: i has o be u ned
in o so wa e, a piece o language ha explici ly ells a compu e how o
execu e he s ep-by-s ep ins uc ions.
The di e ence be ween ‘algo i hms’ and ‘so wa e’ is sub le. An algo i hm
is a pu ely ma hema ical desc ip ion ha desc ibes p ecisely how numbe s
should be manipula ed. I could ell which wo numbe s mus be mul iplied,
wha unc ion mus be e alua ed, o how an image mus be ans o med.
Howe e , di e en compu e s can use di e en ypes o p ocesso s and
memo y, and an algo i hm does no desc ibe how hese ope a ions a e done
on a speci ic compu e . This is whe e so wa e comes in o play. I desc ibes
p ecisely wha ha dwa e ope a ion mus be called, whe e each numbe is
s o ed in memo y, and how an image is ep esen ed in bina y.
As an analogy, you may hink o he algo i hm as a ecipe o bake he
pe ec chocola e cookie. The algo i hm should unambiguously desc ibe
wha should happen o he ing edien s: in wha o de hey should be mixed,
how long hey should be hea ed a wha empe a u e, e c. Howe e , o build
a ac o y ha p oduces hese cookies, you need o be e en mo e speci ic:
Whe e is he suga s o ed? Ou o wha pipe does he dough low? How a e
cookies laid nex o each o he in he o en?
Fundamen ally, co e scien i ic b eak h oughs come om inding new
algo i hms. Once a new algo i hm is ound, i can be e-used many di e en
imes on any capable machine (assuming a good so wa e de elope will
u n i in o app op ia e code!).
In his book, we ca e less abou quan um so wa e and mo e abou quan-
um algo i hms. Fi s ly, he algo i hms ell us p ecisely he unc ionali y ha
quan um compu e s can o e . Mo eo e , we don’ ye know how a ma u e
quan um compu e will be p og ammed o how quan um ha dwa e and
so wa e will change in he ollowing yea s. On he o he hand, once a new
algo i hm is ound, i can be che ished o e e .
Now ha we ha e come o app ecia e algo i hms, i is na u al o ask which
quan um algo i hms we know o . Wha p oblems do quan um compu -
e s sol e well? And how do hese algo i hms compa e o hei classical
equi alen s? This will be he opic o he nex chap e .
he BaCkg ound: Why a e We so en husIas IC aBou Quan um eChnology? 35
2.5 Fu he eading
The Map o Quan um Compu ing (YouTube)– a 30-minu e o e iew ideo
by domain o science ha o ms a g ea supplemen o his book.
Ch is e ie’s book Wha You Shouldn’ Know Abou Quan um Compu e s
debunks se e al my hs abou quan um compu e s, p esen ed in an
accessible way.
a e you looking o a much mo e ex ensi e and echnical sou ce ha
co e s p e y much e e y hing he e is o know abou quan um compu -
e s? ench consul an oli ie ez a y has w i en a 1500+ page book,
Unde s anding Quan um Technologies.
2.6 No es
1. See e.g. h ps://www.ma ke sandma ke s.com/Ma ke -Repo s/Quan um-High-
Pe o mance-Compu ing-Ma ke -631.h ml and h ps://www.mo do in elligence.com/
indus y- epo s/cloud-high-pe o mance-compu ing-hpc-ma ke .
2. Wyciślik-Wilson, S.E. (2019) ‘Google c ea es quan um chip millions o imes as e han
he as es supe compu e ’, TechRada . h ps://www. ech ada .com/news/google-
c ea es-quan um-chip-millions-o - imes- as e - han- he- as es -supe compu e .
3. Dunhill, J. (2021) ‘Chinese Scien is s C ea e Quan um P ocesso 60,000 Times Fas e
Than Cu en Supe compu e s’, IFLScience. h ps://www.i lscience.com/chinese-
scien is s-c ea e-quan um-p ocesso -60000- imes- as e - han-cu en -supe compu -
e s-61475.
4. Ma uschak, A. and Nielsen, M. (2019) ‘Quan um Coun y’. h ps://quan um.coun y.

3 The applica ions: Wha p oblems will
we sol e wi h quan um compu e s?
A a glance
he mos impo an applica ion a eas a e:
1. he simula ion o ma e ial p ope ies and chemical p ocesses;
2. c acking c yp og aphy;
3. using quan um ne wo ks o dis ibu e c yp og aphic keys; and
4. sol ing la ge-scale op imisa ion and aI p oblems.
ge ing u ili y ou o a quan um compu e is no s aigh o wa d. I e-
qui es an algo i hm ha bea s all o he known me hods (e en hose ha
un on e y as classical compu e s), and i mus ackle a p oblem wi h
eal-wo ld ele ance. especially in op imisa ion and aI, we ha e no ound
a con incing ‘kille applica ion’ ye .
In he p e ious chap e , we saw ha quan um algo i hms can sol e ce ain
p oblems in ewe s eps, allowing a la ge-scale quan um compu e o com
-
ple e speci ic asks much as e han any classical compu e could. Howe e ,
he p ecise speedup depends s ongly on he ask a hand. The e o e, he
mos impo an ques ion in his ield is: o which p oblems do quan um
compu e s o e a meaning ul ad an age?
The Quan um Algo i hm Zoo
1
lis s p e y much all known quan um
algo i hms. I has become an imp essi e lis ha ci es o e 400 pape s.
Un o una ely, upon close inspec ion, i ’s ha d o ex ac p ecisely he
use ul business applica ions, o a ew easons. Some algo i hms sol e
highly a i icial p oblems o which no eal business use cases a e known.
O he s may make un ealis ic assump ions o may only o e a speedup
when dealing wi h an ou ageously la ge amoun o da a ( ha we ne e
encoun e in he eal wo ld). Ne e heless, sc olling h ough i is de ini ely
ecommended.
Fo his book, we ake a di e en app oach. We ocus speci ically on
algo i hms wi h plausible business applica ions. To assess hei ad an age,
we spli ou main ques ion in o wo pa s:
– Wha applica ions o e a quan um speedup?
– How la ge is his speedup in p ac ice?
38 In oduC Ion o Quan um Compu Ing o BusIness
3.1 Wha applica ions o e a quan um speedup?
We o esee ou majo amilies o use cases whe e quan um compu ing
can make a eal impac on socie y. We b ie ly discuss each o hem he e.
Fo mo e de ails, we dedica e a mo e in-dep h chap e o each applica ion
amily in Pa 2.
1. Simula ion o o he quan um sys ems: Molecules, ma e ials, and
chemical p ocesses
Mos ma e ials can be accu a ely simula ed on classical compu e s. Howe e ,
in some speci ic si ua ions, he loca ions o a oms and elec ons become
no o iously ha d o desc ibe, some imes equi ing quan um mechanics o
make use ul p edic ions. Such p oblems a e he p o o ypical examples o
whe e a quan um compu e can o e a g ea ad an age. Realis ic applica-
ions could be in designing new chemical p ocesses (leading o cheape and
mo e ene gy-e icien ac o ies), es ima ing he e ec s o new medicine, o
wo king owa ds ma e ials wi h desi able p ope ies (like supe conduc o s
o semiconduc o s). O cou se, scien is s will also be exci ed o simula e
he physics ha occu in exo ic ci cums ances, like a he La ge Had on
Collide o in black holes.
Simula ion is, howe e , no a sil e bulle , and quan um compu e s
will no be spi ing ou ecipes o new pha maceu icals by hemsel es.
B eak h oughs in chemis y and ma e ial science will s ill equi e a mix
o heo y, lab es ing, compu a ion, and, mos o all, he ha d wo k o sma
scien is s and enginee s. F om his pe spec i e, quan um compu e s ha e
he po en ial o become a alued new ool o R&D depa men s.
2. C acking a ce ain ype o c yp og aphy
The secu i y o oday’s in e ne communica ion elies hea ily on a c yp-
og aphic p o ocol in en ed by Ri es , Shami , and Adleman (RSA) in
he la e 70s. The p o ocol helps dis ibu e sec e enc yp ion keys (so ha
nobody else can ead messages in ansi ) and gua an ees he o igin o iles
and webpages (so ha you know ha he la es Windows upda e ac ually
came om Mic oso , and no om some e il cybe c iminal). RSA wo ks
hanks o an ingenious ma hema ical ick: hones use s can se up hei
enc yp ion using ela i ely ew compu a ional s eps, whe eas ‘spying’ on
o he s would equi e one o sol e an ex emely ha d p oblem. Fo he RSA
c yp osys em, ha p oblem isp ime ac o isa ion,whe e he goal is o
he applICa Ions: Wha p oBlems WIll We sol e WI h Quan um Compu e s? 39
decompose a e y la ge numbe ( o illus a ion pu poses, le ’s hink o 15)
in o i s p ime ac o s (he e: 3 and 5). As a as we know, o su icien ly la ge
numbe s, his ask akes such an inc edibly long ime ha nobody would
e e succeed in b eaking a ele an code – a leas on a classical compu e .
This all changed in 1994 when compu e scien is Pe e Sho disco e ed ha
quan um compu e s happen o be qui e good a ac o ing.
The quan um algo i hm by Sho can c ack RSA (and also i s cousin
calledellip ic cu e c yp og aphy, abb e ia ed o ECC) in a ela i ely e -
icien way using a quan um compu e . To be mo e conc e e, acco ding
oa ecen pape ,2 a plausible quan um compu e could ac o he equi ed
2048-bi numbe in oughly eigh hou s (and using app oxima ely wen y
million impe ec qubi s). No e ha u u e b eak h oughs may u he
educe he s a ed ime and qubi equi emen s.
Fo una ely, no all c yp og aphy is b oken as easily by a quan um com-
pu e . RSA and ECC all in o he ca ego y o public key c yp og aphy,which
deli e s a ce ain ange o unc ionali ies. A di e en class o p o ocols
issymme ic key c yp og aphy,which is easonably sa e agains quan um
compu e s bu doesn’ p o ide he same ich unc ionali y aspublic
keyc yp o. The mos sensible app oach is eplacing RSA and ECC wi h
so-calledpos -quan um c yp og aphy(PQC): public key c yp osys ems
esilien o a acke s wi h a la ge-scale quan um compu e . In e es ingly,
PQC doesno  equi e hones use s ( ha ’s you) o ha e a quan um compu e :
i will wo k pe ec ly ine on oday’s PCs, lap ops, and se e s.
A he ime o w i ing, a complex mig a ion lies ahead o p e y much
e e y la ge o ganisa ion in he wo ld, which comes in addi ion o many
exis ing cybe secu i y h ea s. The ounda ions ha e been laid: hanks
o he Ame ican Na ional Ins i u e o S anda ds and Technology (NIST),
c yp og aphe s om a ound he globe came oge he o selec he bes
quan um-sa e al e na i es, culmina ing in he publica ion o he i s
s anda ds in Augus 2024. These a e he new algo i hms ha he as
majo i y o use s will adop .
Un o una ely, many go e nmen s and en e p ises un a g ea amoun o
legacy so wa e ha is ha d o upda e, making his a complex IT mig a ion
ha could easily ake 5–15 yea s, depending on he o ganisa ion. The e’s a
se ious h ea ha quan um compu e s will be able o un Sho ’s algo i hm
wi hin such a ime ame, so o ganisa ions a e encou aged o s a mig a ing
as ea ly as possible.
A new ype o c yp og aphy comes wi h i s own addi ional isks: he new
s anda ds ha e no ye been es ed as ho oughly as he nea ly i y-yea -old
RSA algo i hm. Ideally, new implemen a ions will behyb id, meaning ha
46 In oduC Ion o Quan um Compu Ing o BusIness
He e is a ough o e iew o quan um speedups as we unde s and hem
oday, ca ego ised by hei ype o asymp o ic speedup:
C acking RSA / ECC (Sho ’s algo i hm)
Some chemis y and ma e ial science
B u e- o ce sea ch (G o e ’s algo i hm)
Di e enal equaons, Lasso, …
NP-comple e p oblems
So ng
Loading a la ge amoun o da a om
a d i e
Exponenal
Polynomial
No speedup
Heu isc (unknown)
Annealing (opmizaon)
Va iaonal Quan um Ci cui s
Some chemis y and ma e ial science
Bina y opmizaon
Neu al Ne wo ks
Suppo Vec o Machines
– The ‘exponen ial’ box is he mos in e es ing one, ea u ing applica ions
whe e quan um compu e s seem o ha e a g oundb eaking bene i o e
classical compu e s. I con ainsSho ’s algo i hm o ac o ing, explaining
he owe ing ad an age ha quan um compu e s ha e in codeb eaking.
We also belie e i con ains some applica ions inchemis y and ma e ial
science, especially hose ela ing o dynamics (s udying how molecules
and ma e ials change o e ime).
– The’polynomial’box is s ill in e es ing, bu i s applicabili y is unclea . Re-
call ha a quan um compu e would need much mo e imepe s ep– and,
mo eo e , i will ha e conside able o e head due oe o co ec ion. Does
a polynomial educ ion in he numbe o s eps o e come his slowness?
Acco ding o a ecen pape ,10 small polynomial speedups (as achie ed
byG o e ’s algo i hm) will no cu i , a leas no in he o eseeable u u e.
– Fo some compu a ions, a quan um compu e o e sno speedup.Exam-
ples include so ing a lis o loading la ge amoun s o da a.
I his we e he comple e s o y, hen mos people would ag ee ha
quan um compu ing is a bi disappoin ing. I would be a niche p oduc
o hacke s and a iny communi y o physicis s and chemis s who s udy
quan um mechanics i sel .
– Fo una ely, he e is ye ano he ca ego y: many o he mos exci ing claims
come om heheu is icalgo i hms. This e m is used when an algo i hm
migh gi e a subop imal solu ion (which could s ill be use ul) o when we
canno igo ously quan i y he un ime. Such algo i hms a e common on
classical compu e s: neu al ne wo ks all in his ca ego y, and hese caused
a signi ican e olu ion in AI. Un o una ely, i is unclea wha he impac
o cu en ly known heu is ic quan um algo i hms will be.

he applICa Ions: Wha p oBlems WIll We sol e WI h Quan um Compu e s? 47
In summa y, he po en ial o economic alue a ies g ea ly ac oss quan um
algo i hms. The case o ac o ing has a clea and con incing speedup, bu is
only use ul o codeb eaking (whe e we hope ha impac is limi ed hanks o
he adop ion o quan um-sa e c yp og aphy). In con as , machine lea ning
and op imisa ion do ackle a b oad pale e o ele an p oblems, bu he
speed ad an age o a quan um compu e emains unce ain in his ield.
The applica ions o chemis y and ma e ial science all somewhe e in he
middle, wi h some ele an a eas o applicabili y and conc e e indica ions
o a p ac ical speed ad an age.
3.3 Whe e is he kille applica ion?
Is he e hope ha we’ll ind new quan um algo i hms wi h a la ge com-
me cial o socie al alue? Fo a quan um algo i hm o be uly impac ul,
we equi e wo p ope ies:
1. [Use ul] The algo i hm sol es a p oblem wi h eal-wo ld signi icance ( o
example, because o ganisa ions can wo k mo e e icien ly o because i
helps answe a scien i ic ques ion).
2. [Be e / as e ] Using his pa icula algo i hm is he mos sensible* choice
om a echnical pe spec i e,** e en when compa ed o all o he possible
me hods.
Th oughou his book, we will use he e m quan um u ili y when bo h
p ope ies a e con incingly sa is ied.
The p ecise de ini ion can be a bi inicky, so be o e we s a sea ching
o u ili y, we need o ge some echnical de ails ou o he way.
* Wha is ‘sensible’ (2) depends s ongly on he con ex o he eal-wo ld
p oblem (1). In mos cases, we ca e abou how as a p oblem is sol ed, bu
one should also ake in o accoun he o al cos o de eloping he so wa e,
he cos o leasing he ha dwa e, he ene gy consump ion, he p obabili y o
e o s, and so o h. Fo example, a high- equency ade migh be happy
wi h a 2% as e algo i hm e en i he cos s a e sky-high and he e’s a decen
chance o ailu e, whe eas a hospi al could dismiss a 200x as e quan um
app oach i he cos s don’ ou weigh he bene i s. Indeed, wha is ‘sensible’ is
highly subjec i e. In p ac ice, we can elax his equi emen somewha and
ocus p ima ily on speed, which is a su icien ly complex igu e o me i on
i s own. Ideally, he quan um algo i hm should enjoy anexponen ialspeedup
o a leas a la ge polynomial speedup.
48 In oduC Ion o Quan um Compu Ing o BusIness
** We explici ly look o echnical pe spec i es. O he wise, one migh
also say ha using a quan um algo i hm is comme cially he bes op ion
because i c ea es good PR o because i keeps he wo k o ce exci ed. Then,
pe haps, he i s u ili y has al eady been eached! Howe e , his is no he
compu a ional e olu ion ha we’ e looking o , so we explici ly exclude
such non- echnical easons in p ope y (2). Simila ly, we don’ wan o
wo y oo much abou legal issues (‘i doesn’ comply wi h egula ions’)
because i eels somewha a i icial o dismiss a quan um algo i hm o
such easons.
Sup emacy, ad an age, u ili y
A ound 2019 and 2020, he e ms quan um sup emacy and quan um
ad an age we e popula ly used when quan um compu e s did, o he
i s ime, bea he bes supe compu e s in e ms o speed (p ope y 2).11, 12
This in ol ed an algo i hm ha was che y-picked o pe o m well on a
ela i ely small and noisy quan um compu e whils being as challenging
as possible o a con en ional supe compu e . Quan um ad an age was
mos ly a man-on- he-moon- ype scien i ic achie emen , showcasing he
apid p og ess in ha dwa e enginee ing and silencing he scep ics who s ill
hough quan um compu ing wouldn’ wo k. The e was no a emp o ha e
any p ac ical alue (1).
As a na u al nex s ep, he ace is on o be he i s o un some hing
use ul whils lea ing classical supe compu e s in he dus . This led IBM o
coin he e m quan um u ili y,
13
which we adap ed abo e. In he ollowing
yea s, we can expec he leading ha dwa e and so wa e manu ac u e s o
maximise he amoun o ‘u ili y’ ha hey could possibly squeeze ou o
medium-sized quan um compu e s, whils compe i o s will use hei bes
classical simula ions o dispu e hese claims. The i s ba les ha e al eady
been ough : in June2023, IBM claimed o simula e ce ain ma e ial science
models be e han classically possible,14 quickly ollowed by wo scien i ic
esponses ha showed how easy i was o simula e he same expe imen
on a lap op.1516
I seems o us ha such heal hy compe i ion is good o he ield o e all.
I should lead o inc easingly con incing and igo ous quan um u ili y, om
which he end-use s will e en ually p o i !
In pa allel, he e is a apidly expanding numbe o p ess eleases by
s a ups and en e p ises ha claim o c ea e business alue by sol ing
indus ial p oblems on oday’s ha dwa e, o en wi hou sha ing many
de ails. These app oaches ypically s a wi h a ele an p oblem in mind
and hence sco e well on use ulness (1). Howe e , i is ques ionable whe he
he applICa Ions: Wha p oBlems WIll We sol e WI h Quan um Compu e s? 49
quan um algo i hms we e indeed he bes op ion (2), and mos epo s
we’ e seen ha dly bo he o show any a gumen a ion in his di ec ion.
Such claims should only be aken se iously i a igo ous benchma k agains
s a e-o - he-a classical echniques is included.
Do known algo i hms p o ide u ili y?
Wi h he quan um u ili y c i e ia in mind, we can e isi he algo i hms
ha we e discussed be o e.
(1) use ul (2) Be e han
classical
op imisa ion: igo ous bu slow algo i hms ✓?
op imisa ion: as algo i hms in sea ch o a use cases ? ✓
op imisa ion: heu is ic algo i hms ✓?
simula ion o molecules and ma e ials ✓?
B eaking sa ✓ ✓
Se e al ‘ igo ous bu slow’ algo i hms, mos no ably G o e ’s algo i hm,
ha e an ex ensi e ange o indus ial applicabili y. Howe e , i seems ha ,
in p ac ice, o he (classical) app oaches sol e such p oblems as e . The
quad a ic speedup will be insu icien in he nea e m, and i ’s unclea i
i will be in he u u e.
Then, we ha e se e al exponen ial speedups, like he algo i hm o
opological da a analysis, o which no p ac ical uses ha e been ound
(despi e many scien i ic and indus ial e o s).
Mos op imis ic ou looks ocus on heu is ic algo i hms, o which he
speed ad an age will become clea wi h ma u ing ha dwa e. Ne e heless,
we judge ha no op imisa ion algo i hms can ick bo h boxes o quan um
u ili y ye .
E en o simula ion o molecules and ma e ials, i is no s aigh owa d
o pinpoin p ecisely whe e we can ind u ili y. Classical compu e s a e
al eady inc edibly as , and excellen classical algo i hmic echniques ha e
been de eloped. Scien is Ga ne Chan e en gi es alks ha a e sugges i ely
i led ‘Is The e E idence o Exponen ial Quan um Ad an age in Quan um
Chemis y?’.17The case o quan um simula ion is sub le, and we elabo a e on
his ma e in he chap e on applica ions in chemis y and ma e ial science.
To he bes o ou knowledge, codeb eaking (Sho ’s algo i hm) is he only
impac ul algo i hm ha has li le compe i ion om classical me hods.
Hope ully, mos c i ical c yp og aphy will be upda ed well be o e a quan-
um compu e a i es, making la ge-scale deploymen o Sho ’s algo i hm
50 In oduC Ion o Quan um Compu Ing o BusIness
ela i ely unin e es ing. Ei he way, he applica ion o codeb eaking is no
qui e he posi i e inno a ion ha quan um en husias s a e looking o .
Could he na u e o quan um mechanics be such ha exponen ial
speedups a e only ound in codeb eaking, chemis y, and a bunch o highly
a i icial oy p oblems, bu nowhe e else in he b oad spec um o p ac i-
cally ele an challenges? Mos people would a gue ha such a scena io
is unlikely. The e a e s ill high hopes ha ei he some o he ca ea s wi h
exis ing algo i hms will be add essed o ha new b eak h ough algo i hms
will be disco e ed.
How op imis ic you a e abou quan um compu ing should depend on
(a leas ) he ollowing ques ions:
– How impac ul will heu is ic and o-be-disco e ed algo i hms be com-
pa ed o classical algo i hms? In o he wo ds, wha is he algo i hmic
po en ial o quan um compu ing?
– How will quan um ha dwa e de elop ela i e o classical ha dwa e?
Ul ima ely, he comme cial success o quan um compu e s depends s ongly
on hese ques ions. I we allow ou sel es o do some mo e hypo he ical
d eaming, we imagine ha he ollowing u u e scena ios could be possible,
on a spec um o op imism e sus pessimism:
S a ing on he pessimis ic side, i one belie es ha op imisa ion algo i hms
u n ou o be lacklus e, hen quan um compu ing migh emain a niche
o academics. Howe e , depending on he u ili y o mo e widely applicable
algo i hms, one migh p edic ha quan um compu e s will be ins alled in
special-pu pose compu ing acili ies o , e en mo e op imis ically, ha hey
he applICa Ions: Wha p oBlems WIll We sol e WI h Quan um Compu e s? 51
become inc easingly common addi ions o da a cen es (much like GPUs
oday). Whe e would you place you sel in his igu e?
3.4 Fu he eading
‘The Po en ial Impac o Quan um Compu e s on Socie y’18 ( onald de Wol ,
2017) is an accessible o e iew o known algo i hms, oge he wi h an
assessmen o how we can ensu e a mos ly posi i e ne e ec on socie y.
‘Quan um Algo i hms: An O e iew’19 (ashley mon ana o, 2016) is a
mo e echnical o e iew pape ha desc ibes a selec ion o impac ul
algo i hms in g ea e de ail.
p o esso sco aa onson wa ns us o ‘Read The Fine
P in ’ o op imisa ion algo i hms.[appea ed inNa u e
physics, wi h paywall]
p o esso sanke das sa ma wa ns o hype wi hin he ield o quan um
op imisa ion and machine lea ning.
( echnical) a quan i a i e analysis o g o e ’s un ime compa ed o oday’s
supe compu e s.
(scien i ic pape ) amazon esea che s lay ou a comp ehensi e lis o
end- o-end complexi ies o nea ly e e y known quan um algo i hm.

52 In oduC Ion o Quan um Compu Ing o BusIness
3.5 No es
1. Jo dan, S. (2024) Quan um Algo i hm Zoo. h ps://quan umalgo i hmzoo.o g.
2. Gidney, C. and M. Eke å (2021) ‘How o Fac o 2048 Bi RSA In ege s in 8 Hou s Using
20 Million Noisy Qubi s’, Quan um, 5, p.433. h ps://doi.o g/10.22331/q-2021-04-15-433.
3. Ha ow, A am W, A ina an Hassidim, and Se h Lloyd (2008). ‘Quan um Algo i hm
o Linea Sys ems o Equa ions’. Physical Re iew Le e s, 103 (15) 150502. h ps://doi.
o g/10.1103/PhysRe Le .103.150502
4. Aa onson, S. (2015). Read he ine p in . Na u e Physics, 11(4), 291–293. h ps://doi.
o g/10.1038/nphys3272. Open access: h ps://www.sco aa onson.com/pape s/qml.pd .
5. Liu, Y., A unachalam, S., & Temme, K. (2021). A igo ous and obus quan um speed-
up in supe ised machine lea ning. Na u e Physics, 17(9), 1013–1017. h ps://doi.
o g/10.1038/s41567-021-01287-z
6. Qiski . ‘How Ewin Tang’s Dequan ized Algo i hms A e Helping Quan um Algo i hm
Resea che s’. Qiski (blog), 15Ma ch2023. h ps://medium.com/qiski /how-ewin-
angs-dequan ized-algo i hms-a e-helping-quan um-algo i hm- esea che s-
3821d3e29c65.
7. Wi h he symbol ~ we mean ‘ oughly p opo ional o’. I allows us o w i e down
an app oxima ion o a unc ion, making hem easie o ead, h owing away some
de ails a e no impo an he e.
8. You may ind e en sou ces s a ing ha Sho ’s algo i hm akes a ime p opo ional
o n2log(n). Such scaling is heo e ically possible bu elies onasymp o ic op imisa-
ions ha a e unlikely o be used in p ac ice.
9. Technically, he bes algo i hms o ac o ing, like he gene al numbe ield sie e,
ha e a scaling beha iou ha lies be ween polynomial and exponen ial. Hence,
he speedup o Sho ’s algo i hm is echnically a bi less han ‘exponen ial’ – a mo e
co ec e m would be ‘supe polynomial’. S ill, his book (and many o he sou ces)
con inue o use he e m ‘exponen ial speedup’ o emphasise he eno mous scaling
ad an age o e polynomial speedups.
10. Babbush, R. e al. (2021) ‘Focus beyond Quad a ic Speedups o E o -Co ec ed
Quan um Ad an age’, PRX Quan um, 2(1), p.010103. h ps://doi.o g/10.1103/PRXQuan-
um.2.010103.
11. Zhong, H.-S. e al. (2020) ‘Quan um compu a ional ad an age using pho ons’, Science,
370(6523), pp.1460–1463. h ps://doi.o g/10.1126/science.abe8770.
12. A u e, F. e al. (2019) ‘Quan um sup emacy using a p og ammable supe conduc ing
p ocesso ’, Na u e, 574(7779), pp.505–510. h ps://doi.o g/10.1038/s41586-019-1666-5.
13. Technically, IBM has a sub ly di e en in e p e a ion. In a blog pos (see h ps://
www.ibm.com/quan um/blog/wha -is-quan um-u li y), hey de ine ‘u ili y’ as: ‘Quan-
um compu a ion ha p o ides eliable, accu a e solu ions o p oblems ha a e beyond
he each o b u e o ce classical compu ing me hods, and which a e o he wise only
accessible o classical app oxima ion me hods’. In o he wo ds: a quan um compu e
doesn’ ha e o ou pe o m any classical algo i hm, i me ely has o compe e wi h
he silly app oach o b u e- o ce sea ch – which is almos ne e he bes algo i hm
in p ac ise. This de ini ion seems hea ily ocused on claiming u ili y as soon as pos-
sible. Ne e heless, i we look a he big pic u e, we seem o ha e a simila no ion o
‘ad an age o end-use s’ in mind, so I’m happy o adop he e m ‘u ili y’ anyway.
14. Kim, Y. e al. (2023) ‘E idence o he u ili y o quan um compu ing be o e aul ole -
ance’, Na u e, 618(7965), pp.500–505. h ps://doi.o g/10.1038/s41586-023-06096-3.
15. Begušić, T. and Chan, G.K.-L. (2023) ‘Fas classical simula ion o e idence o he
u ili y o quan um compu ing be o e aul ole ance’. a Xi . h ps://doi.o g/10.48550/
a Xi .2306.16372.
he applICa Ions: Wha p oBlems WIll We sol e WI h Quan um Compu e s? 53
16. Tindall, J. e al. (2024) ‘E icien Tenso Ne wo k Simula ion o IBM’s Eagle Kicked
Ising Expe imen ’, PRX Quan um, 5(1), p.010308. h ps://doi.o g/10.1103/PRXQuan-
um.5.010308.
17. Chan, G. (2022) ‘Is The e E idence o Exponen ial Quan um Ad an age in Quan um
Chemis y?’ Be keley Quan um Colloquium, 12Ap il. h ps://www.you ube.com/
wa ch? =DZPH7ENcRLU.
18. De Wol , R. (2017) ‘The Po en ial Impac o Quan um Compu e s on Socie y’, E hics
and In o ma ion Technology, 19(4), pp.271–276. h ps://doi.o g/10.1007/s10676-017-
9439-z (open access: h ps://a xi .o g/abs/1712.05380).
19. Mon ana o, A. (2016) ‘Quan um Algo i hms: An O e iew’, npj Quan um In o ma ion,
2(1), pp.1–8. h ps://doi.o g/10.1038/npjqi.2015.23.
4 Timelines: When can we expec a
use ul quan um compu e ?
A a glance
he ea lies comme cial quan um applica ions will need se e al million
qubi s, acco ding o he mos igo ous s udies.
assuming an exponen ial g ow h simila o moo e’s law, we p edic ha
he i s applica ions could be wi hin each a ound 2035–2040.
The billion-dolla ques ion in ou ield is:
When will quan um compu e s ou pe o m con en ional compu e s on
ele an p oblems?
In he p e ious chap e , we de ined he equi emen s mo e p ecisely and
coined he e m ‘u ili y’ o such an achie emen .
Un o una ely, nobody can con iden ly answe his ques ion oday, and
pas p edic ions o en p o ed inaccu a e. Mo eo e , a ele an quan um
compu e won’ jus appea om one day o he nex : he e’s a con inuous
e olu ion whe e hese de ices will become inc easingly capable. In his
chap e , we will show how we can make a ough p edic ion abou u u e
imelines and discuss wha will happen on he pa h owa ds la ge-scale
quan um compu a ion.
No e
as an impo an disclaime , his chap e is highly subjec i e. I ’s no ha d o
a i e a di e en conclusions simply by choosing o he sou ces and making
di e en assump ions. We did ou u mos bes o ely on he mos up- o-da e
in o ma ion, combining he iews o he mos widely accep ed pape s, and
making assump ions ha align wi h he iew o mos expe s o p esen a bal-
anced pe spec i e.
4.1 Wha pa ame e s a e ele an ?
Compa ed o cu en ly a ailable echnology, we’d equi e a undamen al
imp o emen o hese speci ica ions:
– Numbe o qubi s
62 In oduC Ion o Quan um Compu Ing o BusIness
such iny machines would b ing an exponen ial ad an age o e eno mous
supe compu e s. Now ha he ield is coming o age, many a e becoming
mo e ca e ul. To illus a e, when looking back a a 2021 epo , consul ancy
i m BCG chi al ously admi s:8
Ou assump ions o nea - e m alue c ea ion in he NISQ e a, howe e ,
ha e p o ed op imis ic and mus be e ised.
The mos se ious ecen claim abou NISQ u ili y comes om he IBM eam
in a pape i led ‘E idence o he u ili y o quan um compu ing be o e
aul - ole ance,’9 in which a quan um simula ion o a speci ic physical
sys em was pe o med using 127 noisy qubi s. Howe e , hei a gumen s
we e quickly e u ed by u he s udies ha simula ed IBM’s imp essi e
quan um expe imen on a con en ional lap op.10
Ma yland-based p o esso Sanka Das Da ma exp esses he iew o many
academics in his opinion a icle ‘Quan um compu ing has a hype p oblem’.
11
He s esses ha ‘ he comme cialisa ion po en ial [o NISQ] is a om clea ’,
poin ing ou ha claims o speedups in inance, machine lea ning and d ug
disco e y ha e so a come wi h highly unsa is ying e idence.
Tha ce ainly doesn’ mean ha NISQ u ili y is uled ou . Mos expe s seem
o keep an eye on he de elopmen s o NISQ applica ions bu will ag ee ha ,
as ye , no u ili y o NISQ machines has been ound. To illus a e, an o e iew
a icle abou pha maceu ical applica ions
12
has a ca e ul bu sugges i e message:
Mos NISQ algo i hms […] ely hea ily on classical op imisa ion heu is ics,
and he ac ual un ime is di icul o es ima e. Fu he mo e, ecen
esul s sugges ha in NISQ app oaches, he numbe o measu emen s
equi ed o achie e a gi en e o scales exponen ially wi h he dep h o
he ci cui . Fo hese easons, he e we ocus ou discussion exclusi ely
on aul - ole an quan um compu e s.
Simila ly, a ecen o e iew13 o quan um chemis y seems o emain agnos-
ic wi h ega d o NISQ ad an age while poin ing ou ha aul - ole ance
has a highe chance o succeeding:
[I] is di icul o p edic when o i algo i hms on nea - e m noisy
in e media e-scale quan um de ices will ou pe o m classical compu e s
o use ul asks. Bu i is likely ha , a some poin , he achie emen o
la ge-scale quan um e o co ec ion will enable he deploymen o a
hos o so-called e o -co ec ed quan um algo i hms.

ImelInes: When Can We expeC a use ul Quan um Compu e ? 63
In his book, we choose o ollow he iew o mos scien is s and s ick o
he well-unde s ood use cases o ea ly aul - ole an quan um compu e s
ha we discussed p e iously. Nobody can ule ou new b eak h oughs
ha allow NISQ u ili y, bu i seems unwise o coun on hese. A po en ial
scien i ic leap could comple ely s i up ou agile p edic ion – bu so would
unexpec ed backlashes in ha dwa e de elopmen o e en un o eseen
unding s ops.
4.3 How long un il we ha e million-qubi machines?
Now ha we’ e se ou a ge o oughly a million qubi s, we’d like o es ima e
when such ha dwa e will be a ailable. We highligh he ollowing sou ces:
1. Road maps and claims o ha dwa e manu ac u e s;
2. Su eys o expe s;
3. Ex apola ion o Moo e’s law.
Wha do manu ac u e s say?
Below, we see he qubi numbe s ha se e al manu ac u e s ha e al eady
ealised (solid disks) and wha hey will p oduce in he u u e acco ding
o hei public oad maps (opaque plusses). No e ha he e ical axis is
loga i hmic, displaying a b oad ange om a ound 10 o 10,000 qubi s. A
lowe numbe o qubi s by no means indica es ha hese compu e s a e
wo se. In ac , he machines wi h he lowe numbe s o qubi s on his g aph
ha e an impo an edge in o he pa ame e s, such as ga e accu acies and
qubi connec i i y.
Besides hei oad maps, companies some imes make mo e da ing
claims in media in e iews o a p esen a ions a la ge e en s. Based
on he applica ion a ge s abo e, i should come as no su p ise ha
manu ac u e s aim o a ound a million qubi s as a ‘moonsho ’ accom-
plishmen . Back in 2020, IBM claimed ha i would each he 1 million
qubi a ge by 2030.14 A ound he same ime, jou nalis s in e p e ed
Google’s p onouncemen s as meaning ha i would do his e en as e
(a ound 202915). The s a -up PsiQuan um, which made wa es hanks
o eco d-high in es men s o o e a billion dolla s o hei pho onic
quan um chips, wen as a as claiming ha i would ha e a million
qubi s by 2025.16, 17
I seems ha hese claims we e oo ambi ious. In 2024, wi h only a yea
o go and no publicly p esen ed p oduc p og ession, PsiQuan um shi ed
i s 1 million qubi oad map o 2027.18 IBM ook an e en mo e conse a i e
64 In oduC Ion o Quan um Compu Ing o BusIness
s ep, and i ’s now claiming ha i will ha e jus 100,000 qubi s in 203319
(al hough his machine should mee he e o co ec ion capabili ies ha
we assumed in he p e ious sec ions). Al hough his delay sounds disap-
poin ing, ha dwa e manu ac u e s a e s ill making imp essi e p og ess, no
leas because he numbe o a ailable qubi s g ows as e han one would
p edic acco ding o Moo e’s law o classical chips!
T apped-ion machines end o ha e ewe qubi s bu highe ga e accu a-
cies. Pe haps his is why IonQdisplays i s oad map in a di e en o ma :
hey aim o achie e 1024so-called algo i hmic qubi sby 2028.
20
This means
ha IonQ will ha ea leas  his numbe o qubi s, bu i also gua an ees
su icien ga e accu acy o un easonably long ci cui s. I ’s unclea whe he
e o co ec ion will be used o his. Compe i o Quan inuum ecen ly
announced a mo e conc e e oad map,21 p edic ing a ound 100 logical qubi s
in 2027. These should b ing he e ec i e ga e e o s down by oughly a ac o
o 10. Looking ahead o 2029, Quan inuum p ojec s housands o physical
qubi s ha o m hund eds o logical qubi s. This migh no be enough o
un he algo i hms discussed ea lie , bu i ’s no oo a o ei he .
he la ges numbe o qubi s demons a ed by a selec ion o ha dwa e manu ac u e s, shown
o di e en yea s. opaque plusses indica e manu ac u e s’ oad maps. da a aken om publicly
a ailable sou ces up un il augus 2024.
ImelInes: When Can We expeC a use ul Quan um Compu e ? 65
Wha does Moo e’s law say?
One could assume ha quan um compu e s will ‘g ow’ a a simila a e
as classical compu e s. Moo e’s law s a es ha he numbe o ansis o s
in a dense in eg a ed ci cui g ows exponen ially: he numbe doubles
oughly e e y wo yea s. This has been a su p isingly accu a e p edic o o
he de elopmen o classical IT. I we apply Moo e’s law o quan um, hen
boos ing qubi numbe s om a ound a housand o one million would ake
a ound wen y yea s – p edic ing ha he one million qubi ma k won’
be passed un il 2044. Clea ly, mos ha dwa e manu ac u e s a e mo e
op imis ic. I we assume he numbe o qubi s doubles each yea , hen
one would p edic ha one million qubi s will be a ailable in en yea s.
While doubling a quan um compu e ’s size each yea is al eady a daun ing
challenge, companies like IBM, Pasqal, and QuE a se he ba e en highe
o hemsel es, hoping o double e e y 7–9mon hs.
Wha do expe s say?
The Global Risk Ins i u e conduc s annualsu eys asking expe s o s a e
helikelihood ha quan um compu e s will pose a signi ican h ea o
public key c yp og aphy 5 yea s om now. Simila ly, esponden s also
es ima e he likeliness 10, 15, 20, and 30 yea s away.
66 In oduC Ion o Quan um Compu Ing o BusIness
This essen ially boils down o he ques ion: when will a quan um compu e
un Sho ’s algo i hm o c ack RSA-2048?We p e iously saw ha a ound 20
million qubi s would be needed o his (al hough expe s may ake in o
accoun ha his numbe can s ill be lowe ed).
We conside his an impo an sou ce because many impo an au ho i ies
in he ield (like p o esso s and co po a e leade s) ake pa in his s udy. The
esul s om Decembe 2023,
22
ga he ed om 37 pa icipan s, a e displayed
below.
esul s o he decembe 2023 expe su ey by global isk Ins i u e. igu e c edi s: m. mosca,
mpiani, www.global iskins i u e.o g.
How o ead his g aph?
le ’s look a he column labelled ‘5 yea s’. a o al o 24 co esponden s indi-
ca e ha he e is less han 1% p obabili y ha quan um compu e s pose a
secu i y h ea in he nex i e yea s. a single pe son is qui e pessimis ic and
assigns a >70% chance ha his will happen. on a e age, expe s say ha
he e’s a ai ly small likelihood ha quan um compu e s will pose a h ea
o c yp og aphy in he nex i e yea s.
u he o he igh , he a ios shi . looking a 20 yea s om now, he
majo i y o expe s belie e ha quan um compu e s pose a se ious h ea ,
wi h o e hal o hem assigning a likelihood o 70% o mo e.
ImelInes: When Can We expeC a use ul Quan um Compu e ? 67
I appea s ha he majo i y o expe s belie e ha he ipping poin is
be ween 10–20 yea s om now. Somewhe e be ween 15 and 20 yea s away,
he e’s a poin whe e he median pa icipan assigned oughly 50% chance o
see a quan um compu e capable o b eaking c yp og aphic codes.Howe e ,
we should ake in o accoun a signi ican unce ain y: e en expe s make
wildly a ying es ima es, so he e’s no ob ious conclusion om his da a.
These expe s a e almos ce ainly awa e o ha dwa e manu ac u e ’s
oad maps, as we shall see below.
4.4 Pu ing i all oge he
The g aph on he nex page sums up ou ea lie indings.
Assuming ha qubi numbe s will g ow exponen ially (and ha all o he
pa ame e s will keep up acco dingly), we can conside se e al scena ios. A
pessimis ic scena io would be ha he numbe o qubi s ‘me ely’ ollows
he classical e sion o Moo e’s law, and qubi numbe s double only once
e e y wo yea s (do ed line). Then, we would ha e o wai un il well pas
2040 o each 100,000 qubi s. An e en wo se scena io would be i we canno
achie e exponen ial g ow h, which would s e ch he imelines e en u he .
An ex emely op imis ic ou look would ollow he blue dashed line (which
ex apola es he p og ess by IBM, doubling hei qubi s e e y ~9mon hs).
I one also belie es in p ac ical applica ions wi h much less han a million
qubi s, hen hese could be a ailable by 2030.
An in e media e pe spec i e is o assume ha he numbe o qubi s
doubles annually. In e es ingly, his seems o app oxima ely align wi h IBM’s
la es claims and he ypical expe opinion. Depending on he applica ion,
i would mean ha quan um chemis y simula ion and codeb eaking can
be wi hin each be ween ~2033 and 2040.
To conclude, ou es ima es s ongly depend on he assump ions ha you’ e
willing o accep (who would’ e hough !). Do you belie e ha imp o ing
algo i hms and e o co ec ion echniques will allow o applica ions
wi h much less han a million qubi s? How quickly do you belie e ha he
ha dwa e will imp o e? I you we e o o ce me o make a p edic ion, I’d
say he i s applica ions will a ise a ound 2035, wi h he unde s anding
ha he e’s a conside able ma gin o e o .
As a inal ema k, a ull u ili y-scale quan um compu e equi es much
mo e han jus some numbe o qubi s. To each he i s use ul applica ions,
we likely equi e simul aneous p og ess in algo i hmics, so wa e, ga e
accu acies, e o co ec ion echniques, idges, lase s, and many o he

68 In oduC Ion o Quan um Compu Ing o BusIness
ImelInes: When Can We expeC a use ul Quan um Compu e ? 69
impo an sub ields o quan um compu ing. Hope ully, all hese disciplines
will ind he equi ed b eak h oughs ha will sus ain he exponen ial
g ow h o quan um compu ing ha dwa e.
4.5 Fu he eading
scien is samuel Jaques (Wa e loo) makes insigh ul g aphs ha combine
he numbe o qubi s and he e o a es, and pu s hem in he pe spec i e
o applica ions equi emen s.
4.6 No es
1. Technically, quan um ga es a e con inuous ope a ions, so numbe s like ideli y a e
de ined sligh ly di e en ly. S ill, he pic u e o disc e e bi lips is no oo a o and
will lead o he same conclusions, so we p e e his mo e accessible explana ion.
2. Gidney, C. and Eke å, M. (2021) ‘How o ac o 2048 bi RSA in ege s in 8 hou s using
20 million noisy qubi s’, Quan um, 5, p.433. h ps://doi.o g/10.22331/q-2021-04-15-433.
3. Lee, J. e al. (2021) ‘E en Mo e E icien Quan um Compu a ions o Chemis y
Th ough Tenso Hype con ac ion’, PRX Quan um, 2(3), p.030305. h ps://doi.
o g/10.1103/PRXQuan um.2.030305.
4. Goings, J.J. e al. (2022) ‘Reliably assessing he elec onic s uc u e o cy och ome
P450 on oday’s classical compu e s and omo ow’s quan um compu e s’, P o-
ceedings o he Na ional Academy o Sciences, 119(38), p. e2203533119. h ps://doi.
o g/10.1073/pnas.2203533119.
5. Be e land, M.E. e al. (2022) ‘Assessing Requi emen s o Scale o P ac ical Quan um
Ad an age’. a Xi . h ps://doi.o g/10.48550/a Xi .2211.07629.
6. See h ps://www.you ube.com/wa ch? =-U dExQW0cs& =1024s, s a ing a 17:04.
7. McKinsey Digi al (2024) ‘Quan um Technology Moni o ’. h ps://www.mckinsey.
com/capabili ies/mckinsey-digi al/ou -insigh s/s eady-p og ess-in-app oaching- he-
quan um-ad an age.
8. Bobie , J.-F. e al. (2024) The Long-Te m Fo ecas o Quan um Compu ing S ill Looks
B igh , BCG Global. h ps://www.bcg.com/publica ions/2024/long- e m- o ecas - o -
quan um-compu ing-s ill-looks-b igh .
9. Kim, Y. e al. (2023) ‘E idence o he u ili y o quan um compu ing be o e aul ole -
ance’, Na u e, 618(7965), pp.500–505. h ps://doi.o g/10.1038/s41586-023-06096-3.
10. Begušić, T. and Chan, G.K.-L. (2023) ‘Fas classical simula ion o e idence o he
u ili y o quan um compu ing be o e aul ole ance’. a Xi . h ps://doi.o g/10.48550/
a Xi .2306.16372.
11. Das Sa ma, S. (2022) ‘Quan um compu ing has a hype p oblem’. h ps://www. ech-
nology e iew.com/2022/03/28/1048355/quan um-compu ing-has-a-hype-p oblem/.
70 In oduC Ion o Quan um Compu Ing o BusIness
12. San aga i, R. e al. (2024) ‘D ug design on quan um compu e s’, Na u e Physics, 20(4),
pp.549–557. h ps://doi.o g/10.1038/s41567-024-02411-5.
13. Cao, Y. e al. (2019) ‘Quan um Chemis y in he Age o Quan um Compu ing’, Chemi-
cal Re iews, 119(19), pp.10856–10915. h ps://doi.o g/10.1021/acs.chem e .8b00803.
14. Hacke , R. (2020) IBM plans a huge leap in supe as quan um compu ing by 2023,
Fo une. h ps:// o une.com/2020/09/15/ibm-quan um-compu e -1-million-qubi s-
by-2030/.
15. Finke, D. (2020) ‘Google Goal: Build an E o Co ec ed Compu e wi h 1 Million
Physical Qubi s by he End o he Decade’, Quan um Compu ing Repo , 5Sep embe .
h ps://quan umcompu ing epo .com/google-goal-e o -co ec ed-compu e -wi h-
1-million-physical-qubi s-by- he-end-o - he-decade/.
16. Wang, B. (2020) ‘PsiQuan um Ta ge s Million Silicon Pho onic Qubi s by 2025’,
23Ap il. h ps://www.nex big u u e.com/2020/04/psiquan um- a ge s-million-sili-
con-pho onic-qubi s-by-2025.h ml.
17. Wha will million-qubi compu e s look like in a ew yea s? (2022) ICV TAnK-ic . h ps://
www.ic ank.com/newsin o/629365.h ml.
18. Finke, D. (2024) ‘PsiQuan um Recei es $940 Million AUD ($620M USD) o Ins all a
1 Million Qubi Machine in Aus alia by 2027’, Quan um Compu ing Repo , 30Ap il.
h ps://quan umcompu ing epo .com/psiquan um- ecei es-940-million-aud-620m-
usd- o-ins all-a-1-million-qubi -machine-in-aus alia-by-2027/.
19. Bake , B. (2023) IBM De ails Road o 100,000 Qubi s by 2033, IoT Wo ld Today. h ps://
www.io wo ld oday.com/indus y/ibm-de ails- oad- o-100-000-qubi s-by-2033.
20. Chapman, P. (2020) ‘Scaling IonQ’s Quan um Compu e s: The Roadmap’, IonQ,
9Decembe . h ps://ionq.com/pos s/decembe -09-2020-scaling-quan um-compu e -
oadmap.
21. Quan inuum accele a es he pa h o Uni e sal Fully Faul -Tole an Quan um
Compu ing (2024) Quan inuum. h ps://www.quan inuum.com/blog/quan inu-
um-accele a es- he-pa h- o-uni e sal- aul - ole an -quan um-compu ing-sup-
po s-mic oso s-ai-and-quan um-powe ed-compu e-pla o m-and- he-pa h- o-a-
quan um-supe compu e .
22. Mosca, M. and Piani, M. (2023) Quan um Th ea Timeline Repo 2023. h ps://global-
iskins i u e.o g/publica ion/2023-quan um- h ea - imeline- epo /.
5 Fou my hs abou quan um compu ing
This chap e elies on a bi o quan um physics ja gon. See he chap e ‘An
in oduc ion o he quan um wo ld’ o a quick in oduc ion.
5.1 My h 1: Quan um compu e s ind all solu ions a once
This my h is likely he mos echnical, and builds on a misin e p e a ion o he
concep o supe posi ion. A single qubi can be in wo s a es a he same ime
(0 and 1), wo qubi s can ep esen ou s a es (00, 01, 10, 11), and h ee qubi s
a e po en ially in eigh unique con igu a ions simul aneously. As we inc ease
he numbe o qubi s, his numbe o coexis ing s a es scales exponen ially!
This means ha a me e 1000 qubi s can e ec i ely ‘s o e’ 2
1000 unique
alues, all a he same ime. Tha ’s an incomp ehensibly la ge numbe ,
much mo e han he e a e a oms in he isible uni e se. E en he as es
compu e s in he wo ld couldn’ loop h ough all hese s a es in a li e ime.
Each o hese s a es can be in e p e ed like a ile on a compu e , be i an
Excel sp eadshee , a web page, a CAD d awing, o wha e e kind o da a
we choose o wo k wi h.
A sma compu e scien is can also de ise a way o make 1000 bi s ep-
esen ‘solu ions’ o a p oblem. Fo example, imagine ha we wan o ind
an op imal ae oplane wing ha gene a es inc edible li while equi ing as
ew ma e ials as possible. Using quan um supe posi ion, we migh ep esen
2
1000 such wings simul aneously.
We picked he example o ae oplane wings because simula ing hei
ae odynamic p ope ies equi es a p e y he y compu a ion. Le ’s assume
ha we ha e w i en such a compu e p og am ha accu a ely simula es any
wing. Le ’s call ha p og am  . I will ou pu 1 i he wing wo ks well (acco ding
o wha e e me ic), and 0 o he wise. Su ely, he p og am akes a e y la ge
numbe o compu a ion s eps, which we’ll call T. The p og am will need
some inpu , deno ed by x , which is a 1000-bi desc ip ion o all he ele an
p ope ies o a hypo he ical ae oplane wing. In o he wo ds, he compu e
p og am compu es
(
x
)
= 1 i x is a an as ic wing, and
(
x
)
= 0 i i ’s ubbish.
Now, a quan um compu e should be able o execu e any classical unc-
ion, igh ? We should be able o un   on a quan um compu e , bu now we
ha e he unique ea u e ha he 1000-qubi inpu can ep esen a humongous
numbe o po en ial ae oplane wings a he same ime. By doing a me e T
compu a ional s eps, we can check he p ope ies o 2
1000 solu ions!

Pa 2
Mo e abou he
applica ions
6 Applica ions in chemis y and
ma e ial science
Pe haps he mos c edible applica ion o quan um compu e s is o s udy
quan um physics i sel . This deepens ou unde s anding o mic oscopic
sys ems like molecules, a oms, o e en sub-a omic pa icles, ul ima ely
leading o he disco e y o new d ugs, ma e ials, and chemical p oduc ion
me hods. A i s sigh , he e seems o be a signi ican ad an age compa ed
o con en ional compu e s, which s uggle o s o e he complex quan um
s a e o sys ems wi h many pa icles. As a back as 1981, physicis Richa d
Feynman ended a con e ence alk wi h a amous quo e, hin ing a he
oppo uni ies o quan um compu ing:1
I’m no happy wi h all he analyses ha go wi h jus he classical heo y,
because na u e isn’ classical, dammi , and i you wan o make a simula
-
ion o na u e, you’d be e make i quan um mechanical.
Since hen, scien is s ha e become inc easingly adep a accu a ely con-
olling quan um sys ems. Today, uni e si ies boas a wide spec um o
analogue quan um expe imen s ha help us unde s and na u e unde exo ic
ci cums ances. We’ e now lining up ou ools o ake hese simula ions o
he nex le el: s udying na u e wi h digi al quan um machines.
In his chap e , we will assess how quan um compu e s can impac he
ields o chemis y and ma e ial science. Tha makes his chap e mo e
echnical, and we’ll assume some ( e y) basic backg ound in chemis y and
physics. We discuss he mos ele an algo i hms, e alua e claims abou
quan um compu ing’s bene i s in he igh agains clima e change, and
analyse why he ni ogenase enzyme ecei es such widesp ead a en ion.
6.1 Wha p oblems in chemis y and ma e ial science will we
sol e?
The compu a ional p oblems ha chemis s ca e abou ypically come in
wo la ou s: s a ic and dynamic p oblems. The mos s udied p oblem is he
s a ic a ian , whe e he goal is o ind he a angemen (s) o pa icles wi h
he lowes possible ene gy. We call such an a angemen he g ound s a e.
These s a es a e ele an because we usually ind sys ems in (o close o) hei
82 In oduC Ion o Quan um Compu Ing o BusIness
lowes ene gy s a es in na u e. In he con ex o molecules, he a omic nuclei
a e ela i ely hea y, while he ligh weigh elec ons mo e much as e and
a e mo e p one o be en angled o in a quan um supe posi ion. The e o e,
chemis s end o make app oxima ions ha allow hem o ocus p ima ily
on he posi ions and spins o he elec ons: he elec onic s uc u e p oblem.
The o he main p oblem is abou dynamics: gi en some ini ial con igu a-
ion o pa icles, how do hey econ igu e hemsel es a e a ce ain amoun
o ime? This is o en e e ed o as a sys em’s ( ime) e olu ion. Bo h p oblems
a e in o mally e e ed o as quan um simula ion.
We o en ecei e he ques ion o why i ’s so ha d o simula e quan um
mechanics on a classical compu e . In ui i ely, his ha dness a ises when we
deal wi h many pa icles ha exhibi la ge amoun s o supe posi ion and
en anglemen , such ha he loca ion o one pa icle is hea ily dependen on
he (undecided) posi ion o many o he pa icles. We call such s a es s ongly
co ela ed. Classical compu e s s uggle because hey need o keep ack o
all he possible loca ions ha pa icle A can be, bu also all he loca ions
o pa icle B, and he same o pa icle C, e c. As he numbe o pa icles
g ows, he numbe o possible con igu a ions o hese pa icles inc eases
exponen ially. This means ha he numbe o ele an ampli udes (see he
chap e on quan um physics) ha a classical compu e needs o p ocess
g ows e y quickly. E en wi h a me e one hund ed pa icles, b u e- o ce
simula ion is a beyond he capabili ies o he wo ld’s bes supe compu e s.
I is a common misconcep ion ha quan um compu e s s aigh o wa dly
o e an exponen ial ad an age compa ed o classical compu e s o all
chemis y p oblems. An in luen ial ecen pape epo s2:
[W]e conclude ha e idence o such an exponen ial ad an age ac oss
chemical space has ye o be ound. While quan um compu e s may s ill
p o e use ul o g ound-s a e quan um chemis y h ough polynomial
speedups, i may be p uden o assume exponen ial speedups a e no
gene ically a ailable o his p oblem.
No e ha his commen is speci ically abou inding g ound s a es, which,
a guably, emains he mos ele an p oblem in chemis y. The e is s ill
ample e idence ha quan um compu e s o e an exponen ial speedup
o ime e olu ions.
The e is mo e bad news o quan um compu e s. O e he yea s,
compu a ional chemis s ha e ound b illian app oxima ions, hacks, and
op imisa ions o wo k a ound he classical compu e ’s bo lenecks, aising a
high ba be o e a quan um compu e can meaning ully compe e. Fo nea ly
applICa Ions In ChemIs y and ma e Ial sCIenCe 83
e e y p oblem in chemis y, he e appea s o be a cle e ick o sol e i
somewha e icien ly on a classical machine.
Fo a kille applica ion, we likely need o sea ch in a ai ly speci ic niche,
igh a he swee spo whe e classical me hods s uggle while a quan um
compu e excels. I is no en i ely clea how la ge his niche is, and i is an
ac i e esea ch a ea o iden i y mo e sys ems whe e classical me hods all
sho . One p omising a ea in ol es mul i-me al sys ems, whe e mul iple
me al ions a e close oge he . Such sys ems a e p esen in biologically
ele an enzymes such as P450 and FeMoco.3 Ano he is in he e ogeneous
ca alysis, whe e he ca alys and eagen s/p oduc s a e in a di e en phase
o ma e .4
The i s p ac ical use s o quan um simula ion algo i hms will mos likely
be scien is s who s udy he undamen als o quan um sys ems. Physicis s a e
al eady employing de ices ha a e simila o ea ly quan um compu e s o
mimic ce ain classes o ma e ials. We wouldn’ call hese de ices compu e s
ye , bu a he analogue simula o s. One o he i s ac ual applica ions o
a ully digi al quan um compu e could be o analyse heo e ical models
o quan um ma e ials, such as he amous Hubba d model.5
The i s e o -co ec ed quan um compu e s will hope ully ind hei
place in indus ial R&D se ings. One o he i s applica ion a eas could
be o be e unde s and hea o emen ioned mul i-me al sys ems, which
a e ele an in hecalcula ions o ligand binding a ini ies in d ugs and in
unde s anding he mechanism behind he biological p oduc ion o ammonia.
We add ess he la e example a he end o his chap e . Ano he exci ing
a ea could be o explo e he mechanism behind Type-II supe conduc i i y
and o sea ch o ma e ials ha become supe conduc ing a e en highe
empe a u es.6 I is ha d o say wha he impac o quan um compu e s will
be beyond such niche a eas, as his will depend s ongly on he use ulness o
small polynomial speedups and unp edic able b eak h oughs in quan um
algo i hms. We see a b oad pale e o o he impac ul applica ions ha ha e
been p oposed, such as pho oca aly ic eac ions ( o example, e icien ly
spli ing wa e o p oduce hyd ogen uel),
7
ca bon cap u e mechanisms,
8
he s udy o e icien sola cells,
9
and he de elopmen o highe -capaci y
ba e ies.10
6.2 Algo i hms o quan um chemis y
We desc ibe h ee o he mos impo an quan um simula ion algo i hms.
The i s is he T o e -Suzuki me hod, some imes called ‘T o e isa ion’,

84 In oduC Ion o Quan um Compu Ing o BusIness
which simula es ime e olu ion. In his case, we assume ha some co ec
ini ial s a e o he wo ld is encoded in he qubi s o some quan um compu e .
The T o e -Suzuki me hod is gua an eed o e u n a good app oxima ion
o he s a e a a la e ime, again encoded in he qubi egis e s.
The second algo i hm is quan um phase es ima ion (QPE), which epo s
he ene gy o a ce ain quan um s a e and can be used o p oduce a sys em’s
g ound s a e. As a sub ou ine, i equi es some ime e olu ion me hod, like
T o e -Suzuki. Un o una ely, QPE can only p o ide in o ma ion abou a
ce ain s a e i i ecei es an inpu ha is al eady a easonable app oxima ion
o his s a e. Especially in he con ex o desc ibing low-ene gy con igu a-
ions, his shi s he p oblem o p oducing good candida e g ound s a es.
The mos popula algo i hm o c ea ing s a es wi h ce ain p ope ies
(like e y low ene gies) is he a ia ional quan um eigensol e (VQE).
This is an example o a a ia ional quan um ci cui : a se ies o ga es ha
can be g adually changed un il he ou pu ma ches ce ain equi emen s.
Jus like o he a ia ional app oaches, i is a heu is ic algo i hm, lacking
igo ous gua an ees ha i will p oduce he desi ed ou pu in a easonable
ime. Howe e , i is a popula me hod oday hanks o i s ease o use and
he abili y o wo k wi h small, noisy compu e s.
C ea ing a good app oxima ion o a g ound s a e is, in gene al, NP-ha d.
This means ha i is ex emely unlikely ha a igo ous algo i hm exis s ha
can ind he g ound s a e o any quan um sys em. On he o he hand, he e
is good hope ha mo e heu is ic me hods (jus like VQE) will be ound ha
wo k well on ce ain subse s o sys ems. In ac , such heu is ic me hods
al eady o m he wo kho se o classical compu a ional chemis y, wi h ools
such as Densi y unc ional heo y (DFT), Con igu a ion In e ac ion (CI) and
Quan um Mon e Ca lo (QMC). These wo k o small sys ems bu a e o en
oo slow o s udy la ge sys ems such as p o eins o d ugs.
11
A wo ka ound is
o apply hese me hods o jus a small pa o he a ge sys em, employing
as e bu less accu a e me hods o o e see he la ge whole.
An example o a basic wo k low o ind a g ound s a e on a quan um
compu e could be as ollows. The i s s ep is o ain a VQE o ou pu
s a es wi h low ene gy.
12
These migh no be he exac g ound s a es, bu
hey will hope ully be simila (in ja gon, hey ha e a la ge o e lap wi h he
g ound s a e). As a second s ep, we append a QPE ci cui , which will no
only epo he ene gy o he VQE s a es, bu also has a ai p obabili y o
changing hese s a es in o pe ec g ound s a es (in ja gon: i p ojec s on o
he g ound s a e). Running he VQE + QPE combina ion a ew imes will
almos ce ainly gi e he lowes ene gy s a es, assuming he VQE p oduces
p ope app oxima ions o i .
applICa Ions In ChemIs y and ma e Ial sCIenCe 85
Fu he eading on simula ion algo i hms
Va ious mo e echnical and sophis ica ed me hods exis , o which we
e e o o he mo e echnical sou ces. These equi e expe knowledge o
quan um chemis y.
‘In oduc ion o Quan um Algo i hms o Physics and Chemis y’ (2012),13 a
pedagogical book chap e .
‘Quan um Algo i hms o Quan um Chemis y and Quan um Ma e ials
Science’ (2020),14 a scien i ic o e iew a icle.
6.3 A hype a ound quan um compu ing o clima e change
Some businesses make spec acula claims abou how quan um compu ing
could be a co ne s one in sol ing clima e change, hanks o he boos o R&D
on ba e ies, ca bon cap u e, and mo e e icien chemical ac o ies. Howe e ,
a ely do we see any e idence – mos seem o assume ha quan um compu -
e s simply spi ou bluep in s o e olu iona y sus ainable echnologies.
McKinsey akes he biscui wi h hei epo i led ‘Quan um compu ing
jus migh sa e he plane ’.15 The a icle igh ully selec s some o he mos
impac ul echnologies o educe CO2 emissions, like elec i ica ion o
anspo , imp o ed sola panels, and e en accines ha educe me hane
emissions by ca le (indeed, due o cow a s). The a icle concludes ha
he selec ed inno a ions could educe global wa ming om 1.7–1.8 °C by
2050 down o jus 1.5 °C. I is a mys e y o us why hey h ow in quan um
compu ing because he e is no men ion wha soe e abou why speci ically
quan um algo i hms would be he key enabling ac o . This exempli ies
wha we see mo e equen ly in popula a icles: quan um compu e s a e
depic ed simply as insanely as compu e s ha will magically sol e he
ba ie s o o he new echnologies on ou wishlis .
Wha a e he ue p ospec s o quan um compu ing in he con ex o
clima e change? Scep ics may poin ou ha echnological inno a ions alone
will no be su icien o a e a clima e disas e – we will emain agnos ic
86 In oduC Ion o Quan um Compu Ing o BusIness
in his deba e. A much mo e conc e e issue is he misma ch in imelines.
Clima e expe s ag ee ha , o limi global wa ming o no mo e han 1.5° C,
we need o ac ela i ely soon. Impe ial College London concludes on hei
websi e,16 e e encing he 2014 IPCC epo :
Limi ing wa ming o 1.5°C will only be possible i global emissions peak
wi hin he nex ew yea s, and hen s a o decline apidly, hal ing by 2030.
Ou chap e on imelines shows ha i is exceedingly unlikely ha signi i-
can quan um u ili y is possible anywhe e be o e he 2030s. Addi ionally,
i will ake se e al yea s be o e a compu a ional disco e y is su icien ly
ma u e o la ge-scale deploymen . Fo his eason, we don’ see quan um
compu e s as a good in es men agains clima e change, bu a he as a
long- e m de elopmen ha can help us ackle o he p oblems ha humani y
will ace in he u u e.
Do we eally ha e no conc e e applica ions in clima e science? Well, we do
ha e some conc e e leads. In he sea ch o a kille applica ion in chemis y,
pe haps he mos -s udied opic is he enzyme Ni ogenase. I s ac i e si e
is p ecisely a mul i-me al sys em ha classical me hods s uggle wi h,
and as we’ll soon see, i appea s in epu able plans o deca bonisa ion. To
unde s and he ele ance o his molecule, we need o di e in o he wo ld
o ood p oduc ion.
6.4 A case s udy o a po en ial kille applica ion: FeMoco
Today’s ag icul u e elies hea ily on he use o a i icial e ilise s. Wi hou
la ge-scale use o supplemen a y nu ien s, we would no be able o sus ain
in ensi e a ming p ac ices and eeding ou wo ld’s huge popula ion would
be p oblema ic. In ac , abou hal o he ni ogen a omsin ou body ha e
p e iously passed a e ilise ac o y!
Un o una ely, he p oduc ion o e ilise in ol es eno mous ene gy
consump ion and ca bon emissions. The main culp i is he ing edien
ammonia (NH3), o which we use as much as230 M on pe yea . Al hough
ou ai consis s mainly o molecula ni ogen (N2), plan s canno di ec ly
abso b his. Ins ead, hey ely on bac e ia (o , in he case o a i icial e ilise ,
humans) o pe o m so-called ni ogen ixa ion, b eaking he s ong iple
bond o molecula ni ogen and con e ing his in o ammonia. Mic oo gan-
isms can con e his in o u he ni ogen-con aining compounds ha he
oo sys em can abso b.
applICa Ions In ChemIs y and ma e Ial sCIenCe 87
P e y much all o he wo ld’s ammonia p oduc ion acili ies ollow he
so-called Habe -Bosch p ocess, whe e hyd ogen gas (H2) and ni ogen gas
(N2) eac oge he o o m ammonia. This me hod has he bene i ha i
can be implemen ed in la ge, high-yield p oduc ion lines bu comes wi h he
disad an age o i s s agge ing ene gy consump ion. The ine iciency s ems
om wo essen ial s eps: i s , p oducing su icien ly pu e hyd ogen and
ni ogen gasses, and la e , sepa a ing he H2 and N2 molecules in o indi idual
a oms. B eaking N2 is especially challenging due o i s s ong iple bond. As
an e ec , ac o ies ope a e a ex eme condi ions, wi h high empe a u es
(~400 deg ees Celsius) and high p essu e (o e 200 a mosphe es), d i en
mainly by na u al gas. As much as1.8% o he wo ld’s CO2 emissionis
caused by ac o ies pe o ming such eac ions, consuming a ound 3–5%
o he wo ld’s na u al gas p oduc ion!
Can’ his be done mo e e icien ly? We s ongly suspec so. Ce ain
bac e ia a e also capable o making ammonia, bu in a seemingly mo e
e icien way, wi hou high empe a u es o high p essu e. I would be
ex emely aluable o copy his ick.
To imi a e bac e ia, we need o be e unde s and a pa icula subs ance,
he FeMo co ac o (in sho : FeMoco), which ac s as a ca aly ic ac i e si e
du ing ammonia p oduc ion. A pe ec simula ion o FeMoco is no possible
on classical compu e s, as he s uc u e o oughly 120 s ongly eac ing
elec ons apidly becomes in ac able. In 2016, esea che s om ETH Zu ich
he chemical s uc u e o he emo co ac o o he
ni ogenase enzyme. igu e c edi s: smoke oo o www.
wikimedia.o g.
94 In oduC Ion o Quan um Compu Ing o BusIness
In asymme ic c yp og aphy, mo e o en calledpublic key c yp og aphy
(PKC), each pa icipan has wo keys: apublic keyand ap i a e key.
Thepublic keycan be sha ed wi h anyone, while hep i a e keymus be
kep sec e . Tha ’s why we use he sugges i e colou s g een (sa e o sha e)
and ed (keep p i a e!). I Alice wan s o send an enc yp ed message o
Bob, she usesBob’spublic key o enc yp he message. The message can
only be dec yp ed using Bob’s p i a e key, ensu ing ha only Bob can ead
he message.
The se ing wi h wo keys o e s mo e unc ionali y. Fo example, using
public key c yp og aphy, Alice could secu ely send a sec e key o Bob ha
hey can subsequen ly use o symme ic c yp og aphy, which is as e in
p ac ice. When public key c yp og aphy is buil o his pu pose, we call i
a key encapsula ion mechanism (KEM).
Fu he mo e, he p o ocol wo ks in ‘ e e se’. Alice can use he p i a e key o
enc yp a message, which hen anyone in he wo ld (including Bob) can
dec yp using he co espondingpublic key. Bob should hen be con iden ha
Alice is he only pe son who could ha e enc yp ed his message. Indeed, some-
hing enc yp ed wi h hep i a e keycanonly be dec yp ed wi h hepublic
key, and ice e sa. The enc yp ed message is much like a signa u e ha only
Alice can p oduce. This o ms he basis o digi al signa u es and ce i ica es.

he ImpaC on CyBe seCu I y 95
You can see public key c yp og aphy in ac ion whene e you isi a web
page. You b owse (like Ch ome o Fi e ox) will display ha he connec ion
is secu e, which means ha i e i ied ha he digi al signa u e is alid,
amongs o he hings. This gua an ees au hen ici y ( he page came om a
egis e ed se e ) and in eg i y ( he si e a i ed unchanged).
I should come somewha as a su p ise ha public key c yp og aphy is
e en possible a all! I ’s a small mi acle ha enc yp ion and dec yp ion wi h
wo o ally di e en keys can be made o wo k, hanks o some powe ul
ma hema ics. Howe e , i u ns ou ha he delica e ela ionship be ween
he wo keys is also a weak spo …
How good a e quan um compu e s a c acking c yp og aphy?
Symme ic-key c yp og aphyis qui e sa e agains quan um hacke s. The
bigges p oblems a e b u e- o ce a acks, whe e an a acke e ec i ely ies
e e y possible sec e key. Using a key size o 128 bi s, he o al numbe o
possible keys is 2128– ha ’s an incomp ehensibly la ge numbe , much mo e
han he numbe o a oms in a human body.
We know ha G o e ’s algo i hm speeds up b u e- o ce sea ch by
educing he numbe o a emp s om2128 o i s squa e oo , which
is264. This is some hing ha c yp og aphe s a e no happy abou , bu
conside ing he slowness and ex a o e head ha comes wi h quan um
compu e s, his doesn’ seem o be a p oblem in he o eseeable u u e.
S ill, o be on he sa e side, i is ecommended o double key leng hs,
hence, o use he same algo i hm wi h 256-bi keys. Changing his in
exis ing IT in as uc u e is ela i ely s aigh o wa d, al hough one
96 In oduC Ion o Quan um Compu Ing o BusIness
shouldn’ unde es ima e he ime and cos s o such changes wi hin
la ge o ganisa ions.
The si ua ion is en i ely di e en wi hpublic key c yp og aphy.The
mos -used algo i hms oday,RSAandECC, can be s aigh o wa dly
b oken by a la ge quan um compu e . We discussed he de ails o Sho ’s
algo i hmea lie and saw ha a ound 20 million qubi s and eigh hou s a e
needed o e ie e a sec e RSA key. Fo una ely, he e exis PKC sys ems
ha a e belie ed o be sa e agains quan um compu e s, and an ob ious
way o wa d is o s a using hese. We call such sys emspos -quan um
c yp og aphy, and despi e he con using name, hey’ e buil o wo k on
con en ional compu e s. We discuss he abbi hole o mig a ing o new
c yp og aphyin a di e en chap e .
Un o una ely, e en oday’s communica ion could be a isk due o a
p ac ice calledha es now, dec yp la e .Enc yp ed messages ha a e
sen o e a ne wo k can be in e cep ed and s o ed o many yea s, un il a
quan um compu e can e icien ly dec yp he messages. E en hough we
use public key enc yp ion mainly o es ablish empo a y keys o symme ic
c yp og aphy, a sma a acke could s ill e ace all he in e media e s eps
and e oac i ely spy on ou communica ion. I is unclea a wha scale
s o age o su icien ly de ailed in e ne da a is genuinely happening, bu i
seems plausible ha secu i y agencies o la ge na ions a e al eady doing his.
The ollowing able summa ises how ou c yp osys ems a e h ea ened:
Symme ic Public-key Quan um
ne wo ks
 oday (aes, … ) oday
( sa, eCC)
pQC Qkd
Sa e agains classical compu e s ✔ ✔ ✔ ✔
Sa e agains quan um compu e s ✔*
*wi h double
key leng hs
Unsa e ✔ ✔
Why don’ we swi ch o symme ic c yp og aphy?
Public key c yp og aphy sol es a e y undamen al p oblem: how can Alice
and Bob ag ee on a sec e key be o e hey ha e a means o enc yp ion in he
i s place? They canno jus send a new key o e he in e ne wi hou any
o m o enc yp ion, because anyone would be able o ead his. This is he
undamen al p oblem o key dis ibu ion. Le us look a he unc ionali y
o e ed by he wo ypes o c yp og aphy:
he ImpaC on CyBe seCu I y 97
Symme ic Public-key Quan um key dis ibu ion
Con iden iali y (p i acy) only wi h
p e-sha ed keys
✔ ✗
Au hen ica ion / In eg i y only wi h
p e-sha ed keys
✔ ✗
Es ablishing sec e keys ✗ ✔ ✔*
*only when ano he
mechanism akes ca e o
au hen ica ion.
I only we could somehow gi e Alice and Bob p e-sha ed keys in a secu e way,
we would esol e mos o hese p oblems. Wi hou public key c yp og aphy,
he e a e o he op ions:
– T us ed cou ie . Alice and Bob could mee e e y o he week o exchange
USB d i es wi h sec e codes.
– T us ed hi d pa y. Alice and Bob could bo h us a la ge ‘key se e ’.
I bo h sha e a sec e key wi h he key se e , hey can secu ely ask he
se e o gene a e a new sec e key ha hey can use oge he .
– Quan um key dis ibu ion. We discuss his solu ion u he below.
Un o una ely, us ed cou ie s o us ed hi d pa ies a e a ely an a -
ac i e al e na i e o public key c yp og aphy, especially when scaling
up o ne wo ks wi h housands o millions o connec ed use s. Cou ie s
a e simply oo slow o oday’s s anda ds, and single us ed pa ies would
pose a pa icula ly in e es ing a ge o a acke s.
7.3 Wha solu ions exis ?
The e is a clea need o pos -quan um c yp og aphy o eplace com-
monly used c yp osys ems like RSA and ECC. Fo una ely, back in 2016,
he Ame ican Na ional Ins i u e o S anda ds and Technology (NIST)
s a ed a compe i ion o selec a new c yp osys em, which should balance
sa e y and p ac ical usabili y ( o example, i should no be oo slow o
memo y-ine icien ). They in i ed expe s om a ound he globe o p opose
c yp og aphic algo i hms, which pee s assessed. Fou ounds and se e al
b oken algo i hms la e , NIST selec ed a i s se o winne s ha a e sui able
o la ge-scale use. As o Augus 2024, he i s h ee PQC algo i hms a e
now o icial NIST s anda ds.
98 In oduC Ion o Quan um Compu Ing o BusIness
E en hough his e o was coo dina ed by an Ame ican ins i u e,
he p ocess was backed and ca ied ou by c yp og aphe s om a ound
he wo ld. A b oad majo i y o cybe secu i y expe s ha e con idence in
NIST’s compe i ion and ecommend he inal s anda ds. Na ional secu i y
o ganisa ions om o he coun ies like BSI (Ge many) and ANSSI (F ance)
may p e e di e en algo i hms bu ha e also explici ly s a ed ha his does
no mean ha hey conside NIST’s s anda ds unsa e.
The esul s o he compe i ion a e as ollows. Fi s ly, NIST selec ed one
Key Encapsula ion Mechanism ha can be used o es ablish sec e keys o e
an unenc yp ed connec ion – emembe he p oblem o communica ing
wi h a web shop ha you had ne e encoun e ed be o e.
Func ionali y NIST Name P oblem amily
Documen a ion
O iginal name
key encapsula-
ion mechanism
ml-kem module-la ice
based
Ips 203 C ys als-kybe
Secondly, NIST selec ed h ee di e en Digi al Signa u e Algo i hms. These
a e used o au hen ica ion and in eg i y – emembe how we don’ wan
ou messages o be al e ed in ansi o how we wan o p e en malwa e
injec ed in so wa e upda es.
Func ionali y NIST Name Algo i hm amily
Documen a ion
O iginal
name
digi al signa u es
algo i hm
ml-dsa module-la ice
based
Ips 204 C ys als-
dili hium
digi al signa u es
algo i hm
slh-dsa s a eless hash-Based Ips 205 sphInCs+
digi al signa u es
algo i hm
n-dsa as - ou ie
ans o m o e
n u-la ice based
Ips 206 alCon
You migh wonde why h ee algo i hms we e selec ed. Un o una ely, all
h ee s anda ds come wi h downsides, o example, because he keys can
ake up mo e memo y o because he pe o mance ( ime o sign o e i y)
is wo se. The eal-wo ld impac will di e pe use case. ML-DSA is he main
c yp osys em ecommended o gene al use, whe eas SLH-DSA and FN-DSA
may be bene icial in speci ic ci cums ances.
he ImpaC on CyBe seCu I y 99
A e he new s anda ds conside ed sa e?
The sho answe is yes: he new PQC s anda ds a e conside ed eady o use,
and choosing algo i hms such as ML-KEM o ML-DSA is widely ega ded
as a sound decision. The e may be excep ions in speci ic high-secu i y
scena ios, bu i you a e ope a ing in such a con ex , you a e likely al eady
awa e o hese nuances.
Howe e , he e seems o be some unce ain y wi hin he c yp og aphic
communi y ega ding whe he he new PQC s anda ds will be as eliable as
ou us ed RSA o ECC. The new s anda ds ha e no ye s ood he es o ime,
and i is possible ha unexpec ed weaknesses – whe he mino implemen a-
ion laws o undamen al ulne abili ies – may s ill be p esen . To illus a e,
a PQC me hod called SIKE1 was in he ace o become a new NIST s anda d
and made i all he way o he ou h ound un il i was p o en unsa e.
To mi iga e any unexpec ed ulne abili ies in he new s anda ds, mos au-
ho i ies ecommend a hyb id implemen a ion ha combines he s eng hs
o bo h con en ional and pos -quan um PKC. Mo eo e , o ganisa ions a e
gene ally ad ised o in es in c yp og aphic agili y, a b oad e m used o
desc ibe he abili y o easily upda e cybe secu i y de ences.
The abo e may sound somewha nega i e, bu we don’ expec he sligh ly
lowe us o s and in he way o adop ion. C yp og aphic algo i hms
hemsel es a e a ely he weakes poin , so i seems wise o ocus on o he
po en ial ulne abili ies ins ead.
Wha abou Quan um Key Dis ibu ion (QKD)?
Quan um key dis ibu ion is also p esen ed as a solu ion o key exchange,
making i a po en ial al e na i e o RSA, ECC and ML-KEM.
S ill, many secu i y au ho i ieswa nagains adop ingQKD oday. Al-
hough he idea is p omising, oday’s ha dwa e is s ill imma u e. Mo eo e ,
QKD doesn’ p o ide any unc ionali y o digi al signa u es, hus we will
need he mig a ion o PQC anyway.
I is somewha o a pi y ha QKD is no so ma u e ye , because i would
be a iable weapon agains Ha es Now, Dec yp La e . Ne e heless, since
a quan um h ea could be he e as soon as he ea ly 2030s, expe s wa n
ha companies and go e nmen s should ix hei PQC i s . A a la e s age,
QKD can be conside ed as an add-on o u he secu i y.
Wha abou Quan um Random Numbe Gene a o s (QRNG)?
Good andom numbe gene a o s a e excep ionally impo an in c yp og a
-
phy, and QRNGs could p o ide a good al e na i e o heha dwa e andom
numbe gene a o s ha a e widely used oday.

100 In oduC Ion o Quan um Compu Ing o BusIness
Howe e , all hey do is gene a e andom numbe s – ha doesn’ make any
p o ocol in i sel quan um-sa e. As a gene al wa ning:p oduc s wi h ‘quan-
um’ in he name do no au oma ically p o ec agains Sho ’s algo i hm!
7.4 Conclusion
C yp og aphy is s ongly in e wined wi h quan um compu ing h ough
G o e ’s algo i hm, Sho ’s algo i hm, and Quan um Key Dis ibu ion.
Secu i y expe s ecommend ha he e is an ob ious way o wa d:
– Replace cu en public key c yp og aphy wi h new, quan um-sa e
p o ocols (PQC);
– Double key leng hs in symme ic c yp og aphy.
Especially he i s bulle is a majo challenge. The e a e many legacy sys ems
on he in e ne ha can no be upda ed so easily. Billions o de ices a e
all in e connec ed, so upda ing one de ice may cause incompa ibili ies
somewhe e else. Mo eo e , PQC p o ocols will likely equi e mo e CPU
powe , memo y, and bandwid h han oday’s us ed me hods. Companies
may need o upda e he co e code o hund eds o e en housands o applica-
ions. Las ly, he new p o ocols ha en’ been es ed as ex ensi ely as ou
con en ional me hods, so i is no unlikely ha new secu i y issues will be
ound. Be o e hey a e e en buil , quan um compu e s a e al eady causing
headaches o c yp og aphe s and cybe secu i y manage s.
7.5 Fu he eading
Cloud la e’s esou ce page ‘The S a e o he Pos -Quan um In e ne ‘ explains
many aspec s o he mig a ion o pos -quan um c yp og aphy.
he nsa publishes ecommenda ions on which c yp og aphic algo i hms
should be used and ske ches a conc e e imeline abou when go e nmen-
al secu i y sys ems should be upda ed.
he ImpaC on CyBe seCu I y 101
The PQC Mig a ion Handbook is a ee guide o co po a e manage s on
how o ackle he upcoming c yp og aphy mig a ion, w i en by du ch
esea ch o ganisa ions no, CWI, and he sec e se ice aI d.
In he con ex o ha es now, dec yp la e , he u gency o mig a e
depends on how long you da a should emain con iden ial, acco ding o
mosca’s heo em.
7.6 No e
1. Goodin, Dan. ‘Pos -Quan um Enc yp ion Con ende Is Taken ou by Single-Co e
PC and 1 Hou ’. A s Technica, 2Augus 2022. h ps://a s echnica.com/in o ma ion-
echnology/2022/08/sike-once-a-pos -quan um-enc yp ion-con ende -is-koed-in-
nis -smackdown/.
8 Applica ions o quan um ne wo ks
I we’ e building compu e s ha deal wi h qubi s, supe posi ion, and en an-
glemen , wouldn’ hese compu e s also need some way o send qubi s o
each o he ? This is he d eam o he quan um in e ne : a ne wo k pa allel
o ou well-known classical in e ne ha allows he ansmission o qubi s.
The e is a bi o a pa adox he e. On he one hand, a ull-blown quan um
in e ne ha s e ches ac oss he globe is e y, e y a away – i will equi e
quan um epea e s o b idge longe dis ances, pu i ica ion mechanisms
o epai impe ec ions, and many mo e echnologies ha we’ e only jus
igu ing ou . On he o he hand, i is o en said ha quan um ne wo ks
ha e a highe Technology Readiness Le el han compu ing. Tha sounds
like a con adic ion, igh ?
The main explana ion is ha he e a e some applica ions o small-scale
‘impe ec ’ quan um ne wo ks, pa icula ly in he con ex o c yp og aphy.
In a sense, quan um ne wo king applica ions ha e always been ahead
o quan um compu ing. Al eady in 1984, long be o e quan um compu e s
we e se iously conside ed, quan um pionee s Cha les Benne and Gilles
B assa d disco e ed a me hod o secu ely nego ia e a sec e key ( hink o a
passwo d) be ween wo dis an pa ies based on sending indi idual pho ons.
Thei esul is now amously known as heBB’84 p o ocol. Simila ly, he
comme cialisa ion o ne wo k echnologies has long been ahead o compu -
ing. Ea ly quan um s a ups like MagiQ Technologies and ID Quan ique
we e ounded a ound he s a o his cen u y, and hei i s comme cial
ne wo king p oduc s we e b ough o he ma ke in 2003 and 2004. This
echnology, whe e a quan um ne wo k is used o gene a e a sec e key a
wo endpoin s, is called Quan um Key Dis ibu ion (QKD) – an applica ion
ha we will add ess in much mo e de ail below.
8.1 The p omises o he quan um in e ne
The e is a long lis o a gumen s why we should be exci ed abou he quan um
in e ne . He e a e some o he applica ions ha we hea mos equen ly:
– Clus e ing quan um compu e s: By connec ing mul iple smalle com-
pu e s, one migh build a much la ge compu e wi h mo e combined
memo y, allowing i o ackle mo e complex p oblems.
– Secu ing classical communica ion.The main con ende he e is Quan um
Key Dis ibu ion (QKD), some imes dubbed he ‘unhackable’ ne wo k.This
110 In oduC Ion o Quan um Compu Ing o BusIness

op ImIsa Ion and aI: Wha a e CompanIes doIng oday? 111
Big O no a ion (see he Box ‘Wha does asymp o ic un ime mean?’ in
he chap e on applica ions), making i s aigh o wa d o ecognise and
compa e he e iciency o algo i hms.
F om he pe spec i e o asymp o ic scaling, a b oad spec um o quan um
algo i hms exis s ha could speed up op imisa ion asks. Scien i ically, i is
down igh ascina ing ha hese algo i hms can p o ide such ad an ages,
using he laws o exo ic physics o sa e illions o compu a ional s eps.
Howe e , his book is abou quan um compu ing o business, so while we
app ecia e he ma els o na u e, a he end o he day, we wan o know
wha he mos p ac ical way o sol e ou p oblems is. No ma e wha abs ac
ma hema ics says, all we ca e abou is he ac ual wall clock ime o ou
speci ic niche o p oblems.
A his poin , he compe i ion om classical compu e s becomes ie ce.
Today’s p ocesso s om companies like AMD o N idia a e so incomp e-
hensibly as ha a quan um algo i hm mus be qui e special be o e i can
o e come he ela i e slowness o a quan um compu e . Mo eo e , quan um
compu e s will ha e a ai amoun o o e head om e o co ec ion ha
con en ional compu e s don’ ha e o wo y abou . I we’ e looking a wall
clock ime, he ace be ween quan um and classical is much igh e !
E en when we compa e classical algo i hms, asymp o ic complexi y isn’
always he bes indica o . Fo example, he Coppe smi h-Winog ad algo i hm
can mul iply huge ma ices ela i ely e icien ly – asymp o ically, i ’s much
as e han he naï e b u e- o ce me hods used oday. La ge ma ices a e
abundan in compu a ionally hung y ields like enginee ing and AI, so one
migh expec Coppe smi h-Winog ad o be widely adop ed. Ne e heless,
i appea s ha ha dly any p o essional so wa e implemen a ions ac ually
use his algo i hm, no any o i s ela i es.1 I u ns ou o be di icul o
wo k wi hand enabling i s speedup equi es e en la ge ma ices han we
handle oday. Asymp o ic complexi y is a use ul ool, bu no sil e bulle .
Mo eo e , he heo y o asymp o ic complexi y is unsui able when
compa ing heu is ic algo i hms. Fo example, a class o p oblems ha we
call ‘NP-comple e’ is ha d o sol e in heo y, while we ha e so wa e ools
like Gu obi and CPLEX ha sol e such p oblems qui e well on a daily basis.
The only uly ai compa ison is benchma king. I in ol es s anda dised
es s o indica e he pe o mance o an algo i hm o a machine. The es s
could be as simple as a se o e e ence p oblems ha should be sol ed as
quickly as possible. Fo example, supe compu e s a e commonly compa ed
h ough he LINPACK benchma k, whe eas algo i hms o he T a eling
Salesman P oblem can be es ed in TSPlib. The ield o AI has been playing
his game o a long ime, ocusing on uzzy p oblems like p oducing na u al
112 In oduC Ion o Quan um Compu Ing o BusIness
English ex s o ecognising wha ’s on an image – s u ha ’s ha d o o mally
de ine in ma hema ics. Fo example, neu al ne wo k a chi ec u es o
image ecogni ion canno be aken se iously un il hey ha e been es ed
on s anda dised da ase s like MNIST and ImageNe .
To assess he ad an age o quan um compu e s, we’ll need o compa e
hem o classical machines in simila benchma ks. Un o una ely, oday’s
ha dwa e is a om adequa e, and, so a , he bes compa isons a e based
on esou ce es ima es and heu is ic a gumen s. Today, i seems nea ly
impossible o p o e he u ili y o a quan um op imisa ion algo i hm.
Ne e heless, i is no ha d o ind a icles ha boldly claim a business-
eady speedup wi h jus a ew housand noisy qubi s, and we s ongly
ecommend being scep ical abou such sou ces. The e a e many ways in
which such esul s can be misleading. Fo example, many a icles me ely
epo ha a quan um compu e can sol e a p oblembu ail o quan i y
how as o accu a e i is in compa ison o he bes -known classical me hod.
These a icles can s ill ha e e y sugges i e i les ha make one belie e ha a
quan um compu e is as e . Some imes, esea che s compa e hei quan um
algo i hm only o ‘weak’ con ende s, like classical b u e o ce sea ch o a
simpli ied algo i hm ha ’s a ely used in p ac ice. Such si ua ions a e likely
o occu when analysing some obscu e da ase o sol ing a p oblem ha
nobody has se iously looked a be o e. Occasionally, a quan um algo i hm
is benchma ked agains a classical machine lea ning model ained by he
same esea che s. Op imising AI me hods is inicky, and such epo s make
us scep ical abou whe he he classical me hod was ea ed jus as ca e ully
as he quan um app oach. All o hese examples indica e he impo ance
o es ing quan um algo i hms agains well-s udied classical app oaches.
This all sounds qui e nega i e, bu we s ill see i as a posi i e de elopmen
when companies pe o m ea ly explo a ions o quan um algo i hms, o en
es ing accessible algo i hms like a ia ional ci cui s on sec o -speci ic
oy p oblems. Quan um compu ing can be inc edibly complex, and i will
ake ime o gain expe ience, ain a quali ied wo k o ce, and ackle all he
ba ie s ha s and in he way o aking a quan um algo i hm o p oduc ion.
I would be bes o he ield i e e yone is hones when he ou come o a
p oo -o -concep is p ima ily a se o lea ned lessons, wi hou in la ing he
esul as a e olu iona y speedup.
To conclude, quan um algo i hms will need o p o e hei wo h in s anda d-
ised benchma ks, simila o how leading AI me hods a e assessed oday. While
we a e wai ing o he ha dwa e o ma u e, he mos ele an in o ma ion
comes om igo ous esou ce es ima es. One should be ca e ul wi h claims
pu ely based on an algo i hm’s pe o mance on ela i ely small-scale p oblems.
op ImIsa Ion and aI: Wha a e CompanIes doIng oday? 113
Fu he eading
he scien i ic pape ‘Be e han classical? he sub le a o benchma king
quan um machine lea ning models’ pe o ms a sys ema ic es on se e al
quan um machine lea ning models.
oli ie ez a y p oposes a amewo k o assess quan um compu e case
s udies.
me iq is a pla o m ha collec s se e al ea ly quan um benchma ks.
( echnical) he Quan um economic de elopmen
Conso ium (Qed-C) p oposes benchma ks based on
se e al op imisa ion asks.
mic oso azu e ea u es a esou ce es ima o ha helps gauge he
numbe o qubi s and heamoun o ime needed o un ce ain quan um
algo i hms.
9.2 Whe e should we look o a new kille applica ion?
Well, we simply don’ know! Howe e , some use ul echnical hin s may be:
– We’d mos likely equi e anexponen ial, a la ge polynomial,o someheu-
is icspeedup. This is much mo e likely achie ed on p oblems whe e we
don’ al eady know e y e icien classical algo i hms.
– When eading da a is a limi ing ac o ( o example, in big da a applica-
ions), quan um compu e s appea o be ela i ely slow. Ge ing he da a
in o a quan um compu e seems o ake a leas as long as p ocessing
he da a on a much cheape supe compu e . This holds, o example,
when sea ching h ough a la ge da abase, bu also o da a-in ensi e
simula ions like wea he o ecas ing.
114 In oduC Ion o Quan um Compu Ing o BusIness
– Simila ly, i he desi ed ou pu is a la ge amoun o da a (such as a e y
la ge lis o able), hen a quan um compu e is likely no e icien .
Mos quan um algo i hms look a a global p ope y o a unc ion o
da ase ha can be encoded in a e y small ou pu (like Deu sch-
Jozsa o Sho ’s algo i hm when in e p e ed as inding he pe iod o a
unc ion).
– Some people would say ha i quan um compu e s a e no ‘ as e ’, pe haps
hey migh sol e a p oblem ‘mo e accu a ely’ ( o example, hey migh
p oduce a mo e eliable o ecas ). Howe e , when we look a speedups,
hen accu acy is al eady aken in o accoun : we compa e he numbe o
needed o achie e a gi en accu acy.
– Classical compu e s a e al eady inc edibly as , and he bo leneck o
many eal-wo ld compu a ional p oblems is no in a compu e ’s clock
speed. I an applica ion does equi e a supe compu e oday, hen i ’s
unlikely ha anyone will in es in a quan um compu e soon.
9.3 Examples o esul s in di e en sec o s
To gain u he unde s anding o he comme cial applica ions o quan um
compu e s, we each a poin whe e we can no longe p o ide any gene ic
wisdom. The bes way o unde s and his ield is by looking a a ious
examples. In his sec ion, we p esen h ee indus ies ha a e commonly
men ioned in he con ex o quan um applica ions: pha maceu icals,
inance, and ene gy. Fo each o hese, we b ie ly highligh ypical use
cases and discuss one o wo echnical epo s.
The epo s a e picked o no pa icula easonexcep ha hey should
p o ide a decen amoun o echnical in o ma ion – much mo e han a
ypical p ess elease o blog pos would. Mo eo e , hese epo s co e a
b oad spec um o esul s, ackling di e en p oblems, ea u ing di e en
ypes o companies, and aking di e en pe spec i es on he deg ee o u ili y
ha quan um compu e s would o e . We limi ou sel es o use cases in
op imisa ion and AI, because quan um simula ion and cybe secu i y a e
al eady co e ed in mo e dep h in di e en chap e s.
No e
he applica ion a eas and use cases highligh ed he e a e specula i e: he e is
no ha d gua an ee ha quan um compu e s will o e signi ican ad an ages
op ImIsa Ion and aI: Wha a e CompanIes doIng oday? 115
o hese applica ions. We selec ed he examples below because hey ha e
no able po en ial, meaning ha u he in es iga ion is jus i ied (and will likely
happen in he ollowing yea s).
mo eo e , his sec ion is mean o gi e examples, and i ’s a om exhaus i e.
Pha maceu ical indus y & heal h
The pha maceu ical sec o seems willing o make long- e m in es men s,
mainly because IP and pa en s can be e y p o i able. Indeed, he la ge
co po a ions ile some 50–100 ‘quan um’ pa en s each yea .
2
Pa o he
en husiasm is jus i ied because compu a ional chemis y R&D is pa o
hei co e business. The b oade heal h indus y, including pa ies like
hospi als and manu ac u e s o medical equipmen , may ha e less ocus
on quan um simula ions bu a e s ill equen ly men ioned.
Some o he mos s udied hemes include:
– Compu e -aided d ug disco e y, whe e a (quan um) compu e simula es
how a p oposed d ug eac s wi h compounds in he human body.
In pa icula , quan um-mechanical in e ac ions may be ele an
when es ima ing he binding s eng h be ween a d ug and biological
compounds;
– Op imising s a egies o d ug syn hesis;
– Simula ion o he molecula spec a expec ed in NMR o spec oscopy
expe imen s.
E en hough he chemical na u e o d ug design lends i sel well o exponen-
ial speedups, some es ain is wa an ed. The mos impo an quan um
speedups a e expec ed o s ongly co ela ed sys ems ha exhibi la ge
amoun s o supe posi ion and en anglemen . A ecen o e iew a icle
s a es he ollowing abou d ug design:3
[Classical me hods] o e good-enough accu acy o mos sys ems. This
is because mos o al d ugs a e small closed-shell o ganic molecules ( hey
need o pass h ough he gu wall o be abso bed) which gene ally lack
s ong co ela ion.
This leads hem o conclude:
[I] he ad an age o quan um compu e s is limi ed o s ongly co ela ed
sys ems, hey migh ha e limi ed p ac ical signi icance in d ug design.

116 In oduC Ion o Quan um Compu Ing o BusIness
Ne e heless, he e a e s ill plen i ul compu a ional challenges ha classical
compu e s ha en’ sol ed, bo h in he a eas o quan um simula ion and
op imisa ion. Whe he quan um compu e s will add ess jus a small niche
o s ongly co ela ed sys ems o p o e o ha e b oade applicabili y is s ill
an open ques ion.
Example esul s
Explo ing he Ad an ages o Quan um Gene a i e Ad e sa ial Ne wo ks
in Gene a i e Chemis y
The pape is based on Gene a i e Ad e sa ial Ne wo ks (GAN), whe e wo neu al ne wo ks
a e ained simul aneously. One ne wo k is a ‘disc imina o ’, which has o de ec whe he a
s uc u e (g aph) o a molecule de i es ei he om a ixed da ase o whe he i is c ea ed
by he o he ne wo k, he ‘gene a o ’. By aining bo h ne wo ks in pa allel, hey become
inc easingly adep a hei ask, such ha e en ually, he gene a o mimics na u al molecule
s uc u es e y accu a ely.
The pape cons uc s he GANs pa ially om a ia ional quan um ci cui s (VQC)
and sees imp o emen s in some benchma ks. No e ha his has only been es ed o ela-
i ely small molecules.
My subjec i e iew is ha his looks like an o e all in e es ing app oach. The abs ac does
ge us scep ical due o a claim ha he au ho s ‘demons a e he quan um ad an age o a
VQC in he disc imina o o GAN’ because he VQC pe o ms ce ain asks be e han a clas-
sical neu al ne wo k while using ewe in e nal pa ame e s. A compa ison o jus one sel -
w i en classical con ende is ne e ai . Mo eo e , a quan um model wi h ewe pa ame e s
can s ill ake mo e ime and esou ces o ain o op imise.
P ess elease: h ps://zapa a.ai/news/zapa a- oxconn-insilico-medicine-
uni e si y- o on o-quan um-gene a i e-ai- o -d ug-disco e y/.
Pape e e ence: Kao, Po-Yu, Ya-Chu Yang, Wei-Yin Chiang, Jen-Yueh
Hsiao, Yudong Cao, Alex Alipe , Feng Ren, e al. ‘Explo ing he Ad an ages
o Quan um Gene a i e Ad e sa ial Ne wo ks in Gene a i e Chemis y’.
Jou nal o Chemical In o ma ion and Modeling 63, no.11 (12June2023):
3307–3318. h ps://doi.o g/10.1021/acs.jcim.3c00562.
O ganisa ions in ol ed: Insilico Medicine, Foxconn, Zapa a.
op ImIsa Ion and aI: Wha a e CompanIes doIng oday? 117
Hyb id Quan um Image Classi ica ion and Fede a ed Lea ning o Hepa ic
S ea osis Diagnosis
In his wo k, he au ho s ain a neu al ne wo k o assess pho os o li e s wi h he aim o
diagnosing non-alcoholic a y li e disease (NAFLD). They compa e a s anda d (classical)
con olu ional neu al ne wo k wi h a ‘hyb id’ model ha con ains a a ia ional quan um
laye . The pape claims ha he quan um e sion is mo e accu a e by 1.8pe cen age poin s.
My pe sonal e alua ion would be qui e posi i e i i we en’ o an impo an de ail ha
he quan um laye uses jus i e qubi s. I seems unlikely ha such an a chi ec u e would
ou pe o m classical me hods in a ai compa ison, especially because simula ing i e qubi s
is i ial o a classical compu e . A possible explana ion is ha he classical ne wo k wasn’
p ope ly op imised (and he pape doesn’ sha e he necessa y de ails o check his). This
hypo hesis seems suppo ed by one o he pape ’s own plo s, whe e he classical model’s
accu acies d op when i gains access o mo e aining da a. This shows why i ’s impo an o
compa e algo i hms on well-s udied benchma ks.
P ess elease: h ps://www.einp esswi e.com/a icle/735111499/quan um-
algo i hm-ou pe o ms-cu en -me hod-o -iden i ying-heal hy-li e s- o -
ansplan .
Pape e e ence: Lusnig, Luca, Asel Sagingalie a, Mikhail Su mach, Ta jana
P o ase ich, O idiu Michiu, Joseph McLoughlin, Ch is ophe Mansell, e al.
‘Hyb id Quan um Image Classi ica ion and Fede a ed Lea ning o Hepa ic
S ea osis Diagnosis’. Diagnos ics 14, no.5 (6Ma ch2024): 558. h ps://doi.
o g/10.3390/diagnos ics14050558.
O ganisa ions in ol ed: Te a Quan um, Uni e si y o T ies e
See also:
(scien i ic o e iew a icle) ‘d ug design on quan um
compu e s’, h ps://www.na u e.com/a icles/
s41567-024-02411-5 (open access: h ps://a xi .o g/
abs/2301.04114).
(scien i ic o e iew a icle) ‘Quan um Compu ing o molecula Biology’,
h ps://doi.o g/10.1002/cbic.202300120.
118 In oduC Ion o Quan um Compu Ing o BusIness
Finance
The e is an ex ensi e body o li e a u e on applica ions in he inancial
se ices sec o . Ou in ui ion ells us ha his is mainly hanks o wo
op-down easons: small algo i hmic imp o emen s can quickly lead o la ge
mone a y gains, and ins i u ions like banks ha e ela i ely long in es men
ho izons, making hem mo e willing o in es in echnologies ha could
be se e al yea s away. Un o una ely, a his poin , he e is li le e idence
o igo ous exponen ial speedups in his sec o , so he ocus is p ima ily
on polynomial and heu is ic imp o emen s.
Some o he mos commonly s udied hemes include:
– Op imising in es men po olios ( o high p o i and low isk);
– Analysing isk and s udying u u e ma ke scena ios;
– Es ima ing he p ice o complex asse s, such as op ions;
– F aud de ec ion.
Example esul s
Quan um Deep Hedging
A hedge is an in es men chosen speci ically o o se he po en ial o loss in o he in es -
men s. Fo example, a bank wi h many asse s in a ola ile ma ke migh also in es in a sec-
o ha ypically mo es in heopposi e di ec ion. The p oblem can be cas in a con en ional
ein o cemen lea ning amewo k, whe e a compu e p og am makes i ual in es men
decisions and ecei es ewa ds depending on i s pe o mance, allowing i o lea n be e
s a egies. Deep hedging is an exis ing classical me hod o ain a good so wa e agen us-
ing deep (mul i-laye ) neu al ne wo ks.
This pape in es iga es he po en ial o quan um compu e s in his a ea. Amongs o he
hings, he au ho s eplace ce ain ne wo k laye s wi h quan um a ian s. Compa ed o he
classical app oach, hey achie e compa able sco es while using ewe ainable pa ame e s.
They also p oduce quali a i ely di e en in es men s a egies, hence o e ing some hing
unique compa ed o he con en ional app oach. The new me hods a e es ed on Quan-
inuum’s H1–1 and H1–2 apped ion compu e s using up o 16 qubi s.
Ou subjec i e in e p e a ion is ha his is an in e es ing and sound pape ha ocuses
on igo ous analysis a he han ex a agan claims. As a downside, we a e no awa e
o any s anda dised benchma k in his ield, no is he e e idence ha he quan um
app oach could lead o as e compu a ion imes (as he educ ion in pa ame e s sug-
ges s).
op ImIsa Ion and aI: Wha a e CompanIes doIng oday? 119
P ess elease: h ps://www.jpmo gan.com/ echnology/news/jpmo gan-
chase -qcwa e-e ol e-hedging- o -a-quan um- u u e.
Pape e e ence: Che a , El Amine, Snehal Raj, Io danis Ke enidis, Abhishek
Shekha , Ben Wood, Jon Dee, Shou anik Chak aba i e al. ‘Quan um
Deep Hedging’. Quan um 7 (29No embe 2023): 1191. h ps://doi.
o g/10.22331/q-2023-11-29-1191.
o ganisa ions in ol ed: Jpmo gan Chase, QCWa e, uni e si é de pa is
Quan um po olio op imisa ion by Ci i Inno a ion Labs and Classiq
The po olio op imisa ion p oblem is as ollows. You ecei e a lis o possible s ocks you
may in es in and a p obabilis ic ou look o hei expec ed gains and ola ili y (i.e. he
iskiness o he s ock). The gains can be co ela ed. Gi en ha you’ e allowed only o ake a
ce ain amoun o isk, wha would be he op imal se o s ocks o in es in?
In his wo k, he au ho s op imise asse s using he Quan um App oxima e Op imisa ion
Algo i hm, an example o a a ia ional quan um ci cui . The e a e no me hodological in-
no a ions, bu he au ho s do a good job o combining exis ing building blocks in o a ull
end- o-end implemen a ion: he algo i hm is w i en in a high-le el so wa e package (by
Classiq), using eal-wo ld da a (by Yahoo inance) in a s anda d Py hon da a p ocessing
pipeline (using Pandas), and unning he esul ing quan um p og am h ough he cloud
( h ough AWS, albei on a classical simula ion in his case). The e is no compa ison wi h any
classical me hods.
In ou subjec i e in e p e a ion, his is mo e a ma ke ing ou ing (showcasing he echni-
cal wi o he pa ies in ol ed) han a newswo hy esul . None heless, se e al news ou le s
picked his up, mos likely hanks o he la ge companies in ol ed.
P ess elease: h ps://www.classiq.io/insigh s/ci i-and-classiq-ad ance-
quan um -solu ions - o -po olio-op imiza ion-using-amazon-b ake .
Blog e e ence: ‘Ci i and Classiq Ad ance Quan um Solu ions o Po olio
Op imiza ion Using Amazon B ake | AWS Quan um Technologies Blog’,
7Feb ua y2024. h ps://aws.amazon.com/blogs/quan um-compu ing/
ci i-and-classiq-ad ance-quan um-solu ions- o -po olio-op imiza ion/.

10 Quan um ha dwa e
Con en ional compu e ha dwa e is ex emely eliable. P o essional se e s
a e supposed o un non-s op o yea s wi hou any ha dwa e ailu es. I
you ake a new p oduc ou o a box, you can be easonably su e ha i will
wo k p ecisely as ad e ised – and i does no , i should be s aigh o wa d
o eplace. Mo eo e , classical IT is ex emely well-s anda dised. No ma e
wha supplie you buy a compu e om, you can be easonably su e you can
un you a ou i e applica ions on hem. Thanks o such high eliabili y
and clea compa ibili y, i is a he easy o compa e di e en machines,
o example, by looking a speed (e.g. loa ing-poin ope a ions pe second,
FLOPS) and memo y size.
We will see ha his is adically di e en o quan um compu e s. De ices
make mis akes, ha e limi ed unc ionali ies, and memo y is sca ce compa ed
o classical compu ing s anda ds. Se e al manu ac u e s ocus on niche
applica ions, making ade-o s in ce ain ea u es o enhance pe o mance
in o he s. In his chap e , we ake a high-le el pe spec i e a quan um
compu ing ha dwa e. We add ess he wo mos impo an aspec s:
– Wha unc ionali y does a de ice ha e?
– Wha ype o qubi s a e used?
10.1 Di e en unc ionali ies
The igu e below shows h ee di e en unc ionali ies ha quan um
compu e s can ha e ( op, ed), along wi h some examples o p oduc s on
he ma ke (yellow), buil om di e en building blocks. This lis is by no
means comple e! I should, a bes , gi e an indica ion o he cu en s a e
o he a . Le us s a by aking a close look a he unc ionali ies.
Ou bigges d eam is o ha e a‘uni e sal quan um compu e ’. The wo d
‘uni e sal’ indica es ha i can execu e any quan um algo i hm (o , echni-
cally, i can app oxima e any algo i hm’s ou pu o a bi a y p ecision). Fo
compa ison, you lap op, phone, and e en a mode n co ee machine a e
uni e sal classical compu e s, making hem capable o unning any classical
applica ion you can hink o : sp eadshee s, 3D games, da a enc yp ion, and
so on. Simila ly, a p ope uni e sal quan um compu e is sui able o any
quan um applica ion, ega dless o whe he i is al eady known oday o
in en ed in he u u e.
128 In oduC Ion o Quan um Compu Ing o BusIness
The de ini ion o ‘uni e sal’ is blind o some de ails, such as memo y limi a-
ions (i assumes you will ne e un ou o RAM), and omi s edious de ails
abou so wa e compa ibili y (a PlayS a ion game won’ un on an Xbox).
In ou high-le el o e iew, such de ails a e unimpo an : he main poin is
ha he e also exis de ices ha canno  un jus any algo i hm.
Does a uni e sal compu e need o be ‘ga e-based’?
no, he e a e a ious compu a ional models ha a e uni e sal.
he e a e di e en ways o make a ‘uni e sal quan um compu e ’. he
mos popula way is o use aga e-basedapp oach, whe e elemen a y op-
e a ions (‘ga es’) change he da a s o ed one o wo qubi s a a ime. his
pe spec i e is mos in ui i e o hose used o con en ional logical ci cui s
(wi h and, o and no ga es), and mos quan um algo i hms a e p e-
sen ed in his language. o he al e na i es includeadiaba iccompu a ion
andmeasu emen -basedcompu a ion, which can heo e ically un any
algo i hm w i en o a ga e-basedcompu e wi hou issues and ice e sa.
Cu en ly, ga e-based compu e s a e by a he mos widesp ead and ap-
pea o be he mos popula app oach in he ace owa ds a million-qubi
quan um compu e : nea ly all la ge ech companies ely on his a chi ec-
u e. he e is one impo an excep ion. some pho onics s a ups a e
wo king owa ds measu emen -based compu ing, as his o e comes he
challenges in pe o ming ‘en angling’ quan um ga es wi h pho ons. In he
ollowing, we will ocus mos ly on ga e-based compu e s.
Quan um ha dWa e 129
No ma e wha a chi ec u e o qubi ype you pick, oday’s echnology
will only allow you o un ela i ely sho compu a ions. This is due o he
inhe en impe ec ions in qubi cons uc ion and con ol me hods. The
impe ec ions cause e o s o accumula e, so a e some numbe o s eps,
he esul is almos su ely co up ed and unusable. Fo longe compu a ions,
ixing e o s on he ly is essen ial, using so-callede o co ec ion.
A he ime o w i ing, we li e in he so-called NISQ e a, wi hNoisy
In e media e-Scale Quan um de ices. Many a e heo e ically ully uni-
e sal, excep ha hey a e limi ed bo h in he numbe o qubi s and, mos
o all, in he numbe o s eps hey can execu e. Companies like IBM, IonQ,
Quan inuum, and Pasqal all ha e NISQ compu e s a ailable o es o e
he cloud.
A uni e sal compu e is a jack-o -all- ades, bu i excels a no hing.
Enginee s can makespecial-pu pose de ices ha imp o e in ce ain a eas
(like he numbe o qubi s o clock speed) by omi ing ce ain unc ionali ies.
Aquan um simula o specialises in mimicking he beha iou o a pa icula
class o ma e ials o molecules. The p ecise capabili ies can be desc ibed in
he ma hema ical language o a ‘Hamil onian’ ha speci ies which ma e ials
quali y. Fo example, Ha a d-spino QuE a o e s a quan um simula o
o e he cloud ha mimics a quan um Ising model.
1
Today’s simula o s
(like QuE a’s) a e ai ly simila o a uni e sal NISQ compu e , missing only
a ew essen ial ing edien s, and simila ly ha ing es ic ions due o noise.
Al hough hey look simila , hey a e no designed o un con en ional
(ga e-based) algo i hms.
The ja gon a ound simula o s can be a bi con using. Fi s ly, he e m
‘quan um simula ion’ is also used when a classical compu e ies o calcula e
he ou pu o a quan um algo i hm. To di e en ia e, some p e e he e m
‘emula ion’ o such classical app oaches.Secondly, we o en hea a dis inc-
ion be ween ‘analogue’ and ‘digi al’ simula ion. I onically, bo h app oaches
end o disc e ise in o ma ion o e disc e e qubi s (which we call digi al). In
p ac ice, he e ms a e a he used o dis inguish be ween con inuous and
disc e e ime s eps. An analogue simula ion would use longe , con inuous
ope a ions on he qubi s, whe eas a digi al simula ion uses quan um ga es
ha ac in sho , disc e e bu s s on he qubi s.
Ano he special-pu pose de ice is hequan um anneale ,popula ised
mainly by he Canadian scale-up D-Wa e. These special-pu pose de ices
can sol e a speci ic class o op imisa ion p oblems ha goes by he name
o QUBO: quad a ic uncons ained bina y op imisa ion. The e is a well-
de eloped heo y o mapping a ious indus ial p oblems in o he QUBO
Quan um ha dWa e 131
o malism, making anneale s ai ly e sa ile machines. Howe e , quan um
anneale s will ne e be able o ake ad an age o he a ious o he quan um
algo i hms ou he e: e en wi h enough qubi s, we won’ see hem c acking
codes using Sho ’s algo i hm.
Fu he eading
d-Wa e’s in oduc ion o i s quan um annealing pla o m
scale-up pasqal epo s on a ma e ial science
simula ion wi h 196 qubi s. In ano he a icle, hey
explain why an ‘analogue’ quan um simula ion has i s
ad an ages.
Que a makes a 256 qubi simula o a ailable o e he Cloud.
10.2 Di e en building blocks
Ano he impo an ques ion conce ns he ma e ials used o c ea e qubi s. Sci-
en is s ha e cooked up se e al compe ing app oaches, such as supe conduc ing
ma e ials, pho ons, indi idual a oms, o ions, each wi h hei own s eng hs
and weaknesses. When compa ing di e en qubi s, we use he e minology
o qubi implemen a ion, he qubi ype, o (wha we p e e ) qubi pla o m.
The con en ional compu e elec onics indus y has se led on a single
choice o ma e ial and manu ac u ing p ocess: essen ially, all compu e
chips a e made using li hog aphy on silicon wa e s. On he con a y, he e
is an ongoing ace be ween wildly di e en qubi pla o ms, and i is s ill
unclea which will e en ually be he winne — o whe he we will con e ge
o a single winne a all.
The e is ascina ing physics behind he di e en ha dwa e ypes, bu we
won’ del e in o ha in his non- echnical book (would you ca e o he wise
wha ma e ial you classical CPU is made o ?). Howe e , as soon as you wan

132 In oduC Ion o Quan um Compu Ing o BusIness
o es a p o o ype quan um p og am on eal-wo ld NISQ ha dwa e, you
p obably wan o lea n mo e de ails. In e es ed eade s a e in i ed o ake
a look a he e e ences below.
I is in e es ing o no e ha all hese di e en unc ionali ies (uni e sal
compu e s, anneale s, and simula o s) can, in p inciple, be buil using any
ype o qubi . Re u ning o he igu e a he op, you can see ha speci ic
qubi pla o ms ha e been used o mul iple pu poses, and i ’s likely ha
he emp y ields will also be popula ed in he u u e.
10.3 Fu he eading
di e en ypes o qubi s explained by si ed.eu
di e en ypes o qubi s a IQC Wa e loo
di e en ypes o qubi s on Wikipedia
a mooC abou di e en ha dwa e ypes by u del
10.4 No e
1. Gemelke, N. and Lukin, A. (2022) Hamil onian simula ion on QuE a’s 256-qubi Aquila
machine, QuE a. h ps://www.que a.com/e en s/hamil onian-simula ion-on-que as-
256-qubi -aquila-machine.
11 E o co ec ion
A a glance
o un long compu a ions, we need o d ama ically educe he likelihood
o e o in each compu a ional s ep – no jus a li le bi , bu by a ac o o
millions.
e o co ec ion is he mos e ec i e me hod o achie e ex emely low
e o p obabili ies. I combines a small numbe o ‘physical’ qubi s ( hink
o se e al hund ed) in o a single ‘logical’ qubi ha supp esses e o sexpo-
nen ially.
logical qubi s a e s ill no pe ec : he ‘numbe o s eps’ ha hey can
su i e is an impo an speci ica ion ha de e mines whe he hey can a
pa icula applica ion.
I ’s 2024 and we’ e seeing a majo shi in he oad maps o quan um com-
pu e manu ac u e s. Se e al companies no longe pu hei ba e qubi s in
he spo ligh , bu ins ead ocus on logical qubi s. E o co ec ion seems o
be an essen ial componen o la ge-scale quan um compu ing, adding ye
ano he ace in which hese de ices di e om hei classical coun e pa s.
Al hough his is a ela i ely ad anced opic, we ind i so impo an ha i
dese es a dedica ed chap e in his book.
As wi h many aspec s o quan um compu ing, e o co ec ion can be
a he con using. A s a emen ( ha is inco ec !), which we o en hea is:
Logical qubi s (o e o -co ec ed qubi s) a e esilien o e o s ha occu
du ing a compu a ion. Once we ha e logical qubi s, we can inc ease he
leng h o ou compu a ions inde ini ely.
Wha ’s he p oblem he e? Well, no e e y logical qubi is c ea ed equally.
In he nea u u e, we expec o see logical qubi s ha a e pe haps 2x mo e
accu a e han oday’s ba e ha dwa e qubi s, and la e 10x, and in he u u e
pe haps 1000x. E o co ec ion is a ick o educe he p obabili y o e o s,
bu i will no elimina e e o s comple ely. In he ollowing decade, we
expec g adual imp o emen s, hope ully down o e o a es o 10-10and
below.
134 In oduC Ion o Quan um Compu Ing o BusIness
11.1 Wha is e o co ec ion?
In quan um e o co ec ion, we combine some numbe ( hink o hund eds
o housands) o ‘physical’ha dwa e qubi s in o a i ual‘logical’qubi .
The logical qubi s a e he in o ma ion ca ie s used in an algo i hm o
applica ion. E o co ec ion me hods can de ec whene e iny e o s
occu in he logical qubi , which can hen be ‘ epai ed’ wi h s aigh o wa d
ope a ions. Unde he assump ion ha he p obabili y o ha dwa e e o s
is su icien ly low (below a ce ain e o h eshold), he o e all accu acy
imp o es exponen ially as we employ mo e physical qubi s o make a logical
qubi . Hence, we ob ain a e y a ou able ade-o be ween he numbe o
usable qubi s and he accu acy o he qubi s.
Doesn’ measu ing a quan um s a e des oy he in o ma ion in he qubi s?
Indeed, i we nai ely measu e all he physical qubi s, we des oy po en ially
aluable in o ma ion encoded in he qubi s. Howe e , quan um e o co -
ec ion uses an ingenious way o measu e only whe he o no an e o
occu ed. I lea ns no hing abou he ac ual in o ma ion con en o he qubi .
I u ns ou ha his way, he da a s o ed in he logical qubi is no a ec ed.
Why a e e o s so much o a p oblem?How do e o s sc ew up ou
compu a ions?
In sho , e en iny e o s a e a p oblem because we wan o pe o m an
as onishing numbe o quan um ope a ions successi ely — hink o billions
o illions o hem.
Le ’s make his mo e conc e e. A compu e p og am is essen ially a
sequence o ‘s eps’, each o which a compu e knows how o pe o m. We
say ha a p og am o algo i hm has awid h,which is he numbe o qubi s
i equi es. I also has adep h,which is he numbe o consecu i e s eps
ha need o be pe o med. You may in e p e one s ep in ea ly ha dwa e
as a single quan um ga e (al hough, in p ac ice, ga es may be pe o med in
pa allel, making heimpac o e o s sligh ly mo e complica ed).
e o Co eC Ion 135
Wid h (numbe o bi s)
Dep h (numbe o s eps)
Se a = 450
Compu e b = a * 2
Compu e c = a * b
Compu e d = c + a
…
…
The concep o ‘wid h’ is p e y s aigh o wa d: i he compu e doesn’
ha e enough memo y, i canno un he p og am. Dealing wi h ‘dep h’ is
ha de . To un a p og am o 10
9
s eps, we need o limi e o s o oughly he
in e se, say, a p obabili y o 10-9pe s ep. I he e o is la ge , i becomes
ex emely unlikely ha he quan um compu e will p oduce he co ec
ou come. These a e no ha d numbe s: a compu e wi h 10-10e o would
be a signi ican imp o emen ( esul ing in much ewe mis akes), and a
compu e wi h 10
-8
e o migh be pushed o also ind he co ec answe
a e many ies. Howe e , as he imbalance be ween dep h and e o g ows,
he p obabili y o inding a co ec ou come is educedexponen ially. We
illus a e his in mo e de ail in he box below.
To illus a e, why do we need such small e o a es?
le ’s look a a simple model o a compu e , which is no unlike wha hap-
pens inside a quan um compu e o a mode n (classical) Cpu. as abo e,
he compu e is supposed o wo k h ough a lis o ins uc ions. We can
conside a ious speci ica ions o a compu e :
– he a ailable memo y, measu ed in bi s (o pe haps megaby es o giga-
by es, i you like).

e o Co eC Ion 143
LDPC codes a e now apidly gaining a en ion. They build on a la ge
body o classical knowledge and could ha e ( heo e ically) mo e a ou able
scaling p ope ies o e he su ace code.
Which code will e en ually become he s anda d (i any) is s ill comple ely
open.
Wha a e he main challenges?
Fi s ly, we would need jus sligh lymo e accu a e ha dwa e. We men ioned
a ce ain accu acy h esholdea lie : s a e-o - he-a ha dwa e seems o be
close o his h eshold bu no com o ably o e i . Secondly, e o co ec ion
also equi es signi ican classical compu ing powe , which needs o sol e
a ai ly complex ‘decoding’ p oblem wi hin ex emely small ime bounds
(wi hin jus a ew clock cycles o a mode n CPU). Classical decoding needs
o become mo e ma u e, bo h a he ha dwa e and he so wa e le el. I
is likely ha pu pose-buil ha dwa e will need o be de eloped, which o
some pla o ms migh be placed inside a c yogenic en i onmen (placing
s ingen bounds on hea dissipa ion). Theo e ical b eak h oughs can s ill
educe he equi emen s o classical p ocessing.
Las ly, i u ns ou ha ‘mid-ci cui measu emen s’ a e echnically chal-
lenging. Wi hou in e media e measu emen s, one migh e oac i ely de ec
e o s, bu one canno epai hem. We should also wa n ha many ela ed
e ms exis , such as ‘e o mi iga ion’ and ‘e o supp ession’. They migh
be use ul o inc emen al ideli y imp o emen s, bu hey don’ b ing an
exponen ial inc ease in dep h like p ope e o co ec ion does.
11.4 Conclusion
The bo om line is ha one shouldn’ nai ely ake ‘logical qubi s’ as pe ec
building blocks ha will un inde ini ely. A logical qubi is no gua an ee
ha a compu e has any capabili ies; i me ely indica es ha some kind
o e o co ec ion is applied (and i doesn’ say any hing abou how well
he co ec ion wo ks). A much mo e in e es ing me ic is he p obabili y
o e o in a single s ep (in ja gon: he ideli y o an ope a ion), which gi es
a easonable indica ion o he numbe o s eps ha a de ice can handle!
144 In oduC Ion o Quan um Compu Ing o BusIness
11.5 Fu he eading
‘The Quan um Th ea Timeline Repo ’asked se e al expe s wha hey ind
he mos likely app oach o aul - ole ance (sec ion 4.5).
B i ish s a up i e lane builds a ha dwa e chip ha
decodes which e o occu ed on logical qubi s.
hey p o ide an accessible p ess elease and a mo e
echnical scien i ic a icle.
C aig gidney (google) has amo e echnical blog pos on why adding
physical qubi s will emain ele an in he ollowing decades.
( echnical) somescien i icwo k speaks o ‘ea ly aul - ole an ’ quan um compu ing, such as:
‘ea ly aul - ole an Quan um Compu ing’, discussing how we can squeeze
as much as possible ou o limi ed de ices.
‘Assessing he Bene i s and Risks o Quan um Compu e s’ akes a simila wid h
x dep h app oach as we do he e, bu uses i o assess wha applica ions will
be wi hin each i s .
12 Wha s eps should you o ganisa ion
ake?
In he p e ious chap e s, we discussed heuse cases, he h ea s,and
he imelines o quan um echnologies. We will now look a he s a egic
pe spec i e o a ypical non-quan um en e p ise. We will assume a ypical
la ge-scale o ganisa ion ha does no sell IT p oduc s pe se, bu elies
hea ily on compu ing in as uc u e o op imise i s ope a ions, supe ise
p ocesses, communica e wi h supplie s and clien s, and po en ially in es in
compu e -aided R&D. While hese o ganisa ions may be exci ed abou he
po en ial o quan um compu ing, hey may also eel ulne able – whe he
due o compe i o s ad ancing ahead o due o hacke s a acking legacy
c yp og aphy.
We ou line he ypical p ocess an o ganiza ion unde akes in h ee s eps.
The i s s eps, like g owing expe ise, inding adequa e s a , and doing
i s p oo -o -concep s udies, will be la gely sec o -independen . Fu he
s eps can become mo e o ganisa ion-speci ic, and we will highligh se e al
ools o ailo ed assessmen
C yp og aphy
Quan um applica ions
1. No- eg e mo es
Appoin a wo king g oup
Assess he u gency o PQC
Read up and lea n
C ea e awa eness
2a. P epa a ion s eps
Find impac ul use-cases
Ske ch a oad map
3a. Implemen a ion
2b. P epa a ion s eps
C ea e an in en o y
Fo m a mig a ion plan
3b. Implemen a ion
Mig a e o pos -quan um
c yp og aphy
12.1 Common i s s eps
S ep 1: S a wi h no- eg e mo es
Mos companies s a wi h ea ly s eps aimed a be e unde s anding he
si ua ion. These can be done wi h e y li le inancial isk.
146 In oduC Ion o Quan um Compu Ing o BusIness
Some mus -do ac ions:
– Appoin a quan um lead o a quan um wo king g oup asked wi h ol-
lowing he de elopmen s.
– Read up and lea n. I you’ e come his a in his book, you’ e al eady doing
a an as ic job.We ha e a sepa a e chap e on u he lea ning esou ces.
– C ea e in e nal awa eness. Many employees will enjoy inspi a ional alks,
ou s o demons a ions ha academics o quan um manu ac u e s can
p o ide.
Op ionally:
– Pu quan um on he agenda wi h senio managemen .
– In ol e collabo a o s, supplie s and endo s, and make you in e es in
quan um known. I is o you bene i i supplie s a e well-p epa ed.
– Pa icipa e in a wo kshop, hacka hon, o simila e en .
In e ms o mo e conc e e ollow-up ac ions, i makes sense o spli you
quan um jou ney in o wo di e en ca ego ies:
a. P epa ing o quan um applica ions,whe e he goal is o le e age quan um
echnologies o gain some compe i i e ad an age ( o example, by s eng h-
ening you R&D, u he op imising you logis ics, imp o ing a p oduc , e c).
b. Mig a ing oquan um-sa e c yp og aphy, whe e he goal is o keep you
IT secu e agains a acke s wi h a quan um compu e .
These endea ou s se e e y di e en pu poses and a e likely spea headed
by di e en depa men s. Hence, i seems logical o b eak hese down in o
sepa a e p ojec s. We discuss u he s eps in bo h di ec ions sepa a ely.
12.2 P epa e o use quan um applica ions
S ep 2a: Explo e use cases
A his s age, mos o ganisa ions will wan o make low- eg e mo es ha ge
hem p epa ed o le e age quan um echnologies ai ly soon a e p ac ical
u ili y becomes a ailable. Some o he bo lenecks could be he lack o
in-house knowledge, a limi ed a ailable wo k o ce, o a long imeline o
in eg a e quan um applica ions in p oduc ion en i onmen s.
Mus do:
– Iden i y he mos impac ul use cases in you sec o .
– Ske ch a oad map o he coming yea s.
Wha s eps should you o ganIsa Ion ake? 147
Op ionally:
– S a conc e e p oo -o -concep p ojec s. Righ now, hese a e unlikely
o o e p ac ical u ili y and will likely ackle jus a oy p oblem. How-
e e , hese help build expe ience in se ing up quan um p ojec s and
can unco e ‘unknown unknowns’. Fo s a wi h a s ong physics o
ma hema ics backg ound, i is ela i ely accessible (and un!) o ge
acquain ed wi h quan um p og amming packages andimplemen a i s
es algo i hm.
– Find s a egic pa ne s. O ganisa ions can sa e cos s by collabo a ing
on ea ly, p e-compe i i e explo a ion.
– C ea e PR! We no ice ha many companies a e ac i ely p omo ing
hei ea ly esul s on quan um applica ions, e en i hese do no o e
signi ican ad an ages ye .
– Hi e s a wi h a s ong backg ound in quan um echnologies who
unde s and he ma ke , ha e he igh skills o lead p oo -o -concep
s udies, and can o e ad ice o s a egic decisions.
S ep 3a: Implemen ing ac ual applica ions, whene e eady
F om he e onwa ds, i ge s inc easingly di icul o gi e conc e e ad ice, as
p io i ies may depend on you business and on he way he ield o quan um
compu ing will p og ess. Se e al sou ces will simply ell you do ‘de elop a
long- e m s a egy’ o simila . O he s highligh he need o ‘ emain agile’
o quickly adap o his apidly e ol ing ield.
Fo inspi a ion o a do on he ho izon, you may hink owa ds a com-
pe ence cen e o quan um compu ing, simila o how many companies
ha e special depa men s o da a science and/o AI. A conc e e ask could
be o elabo a e on he lis o impac ul use cases om he p e ious s ep,
benchma king he pe o mance o a ious quan um and classical so wa e
ools. Ano he ask could be o p o essionalise an ea lie p oo -o -concep
p ojec , b inging i close o implemen a ion in a p oduc ion en i onmen .
Iden i ying ui ul use cases
F om a op-down pe spec i e, i is a good exe cise o iden i y you cu en
needs in high-pe o mance compu ing.Wha do you cu en ly spend you
compu ing budge on? A e he e any a eas whe e new ools in compu a ion
o modelling could p o ide se ious business alue ( o example, by being
as e , ackling bigge p oblems, o deli e ing highe accu acy)? Which
quan i ies would you ideally ha e calcula ed bu a e beyond he each
o cu en compu e s?This esul s in a longlis o use cases whe e new
compu a ional ools a e wo h u he in es iga ion. The nex s ep would

148 In oduC Ion o Quan um Compu Ing o BusIness
be o esea ch o wha ex en a quan um compu e (o whiche e o he new
compu a ional ool) o e s any ad an age.
We ecommend his op-down app oach because i can lead o conclusions
soone , especially because i a oids s udying use cases ha a e no wo h
you ime ( o example, because addi ional compu a ional powe p o ides
li le alue).
I is also possible o ake a bo om-up app oach. Looking a he a ailable
quan um algo i hms, which would speed up p ocesses in you exis ing IT?
Would any o hem p o ide alue o you business? This mo e echnical
pe spec i e equi es some in-dep h quan um expe ise bu can de ini ely be
wo h he e o , especially i you ha e people wi h he igh skills a ailable.
he Quan um applica ion lab is a collabo a ion be ween a ious du ch e-
sea ch o ganisa ions. hey in i e end-use s o explo e he bene i s o quan um
compu e s in p ojec s ha las anywhe e be ween h ee and wel e mon hs,
anging be ween a i s explo a ion o use cases o ad anced de elopmen o
quan um p o o ype so wa e. se e al example p ojec s can be ound on hei
websi e: www.quan umapplica ionlab.com.
Fu he eading
scien is s p opose a amewo k o disco e which eal-wo ld p oblems a e
po en ially accele a ed by quan um compu e s.
Consul an oli ie ez a y p oposes a amewo k o assess he ma u i y o
quan um compu ing case s udies.
(you ube) a eco ding o Quan um.ams e dam’s online semina ‘Wha do
companies ge ou o quan um p ojec s oday?’
Wha s eps should you o ganIsa Ion ake? 149
Wha does an R&D collabo a ion wi h academia look like?
se e al end-use s ha e s a ed collabo a ions wi h uni e si ies o be e
unde s and he use cases o quan um compu ing. his is o en a win-win
si ua ion, as companies can lea n om enowned expe s a ela i ely low
cos s, whe eas academics bene i om addi ional unding and showcasing
ha hei esea ch has p ac ical in e es s. mo eo e , many coun ies p o-
ide subsidies o so-called ‘public-p i a e pa ne ships’. Below, we ske ch a
pe sonal expe ience wi h he p ocess o s a ing such a pa ne ship.
you will mos likely be dealing wi h a uni e si y’s ech ans e o ice ( o),
which specialises in making in-house knowledge a ailable ex e nally. as a
i s s ep, i is impo an o ag ee on he scope o he p ojec : wha a e he
esea ch ques ions, wha a e he expec ed ou comes, how long will he p o-
jec un, and so o h. Ideally, his would be a discussion be ween an expe
om you o ganisa ion and a uni e si y’s (assis an ) p o esso . he p o esso
will mos likely ake a supe ising ole, as he ac ual wo k is o en execu ed
by a junio esea che employed as a phd candida e o a pos doc o al (pd)
esea che . phd p og ammes ake ela i ely long, 3–5 yea s depending on
you locale, and i may ake some ime be o e he i s esul s come in. pos -
doc p ojec s o en ake 1–3 yea s and can lead o esul s soone , bu as o
2024, i can be much ha de o hi e a pos doc wi h he igh compe encies.
When he opic and du a ion o he p ojec a e clea , i is impo an o dis-
cuss de ails a ound in ellec ual p ope y (Ip), o en done by legal expe s.
o uni e si ies, i is impo an ha esea che s can keep building upon he
p ojec ’s Ip in an academic se ing. mo eo e , hey will demand ha he
esul s can be published in scien i ic jou nals. a he same ime, a paying
company will wan su icien op ions o pa en new disco e ies and will
equi e exclusi e use o he Ip wi hin hei sec o . hese demands do no
necessa ily con lic wi h each o he , and in p inciple, i should be possible
o ind an a angemen ha sa is ies bo h pa ies.
a s aigh o wa d way o ensu e ha he company lea ns om he academ-
ic de elopmen s is by o ganising mee ings o wo kshops h oughou he col-
labo a ion p ojec , in which he ongoing &d is discussed wi h company s a .
he occasional dialogue wi h company s a is a guably mo e impo an han
a shiny inal epo o pape , which isks disappea ing in someone’s d awe .
12.3 Mig a ing o pos -quan um c yp og aphy
This sec ion elies on echnical knowledge om he p e ious chap e on
cybe secu i y.
Wha s eps should you o ganIsa Ion ake? 151
S ep 2b: P epa e you mig a ion
C yp og aphy is a comple ely di e en beas , wi h a mo e conc e e goal, and
mo e u gen imelines o mos o ganisa ions. Con a y o he applica ions in
he p e ious sec ion, he c yp og aphy mig a ion is no op ional. Fo una ely,
mos o ganisa ions ace he same p oblem, and he e is ample esea ch
on e ec i e s eps. The co e challenge is o upg ade all exis ing public key
c yp og aphy o Pos -Quan um C yp og aphy (PQC) in he nex decade,
which could be sp ead o e hund eds o housands o di e en applica ions.
Many businesses, especially hose dealing wi h c i ical in as uc u e, may
addi ionally deal wi h egula o s who may o may no ha e guidelines eady.
Mo eo e , IT ansi ions can be inc edibly slow – i is no uncommon o see
plans ha co e i e o e en en yea s.1
Au ho i ies seem o ag ee ha he ollowing ini ial s eps should be aken
u gen ly by all la ge o ganisa ions.
– C ea e awa eness: make su e ha he quan um h ea is well-unde s ood
in you secu i y depa men s and among IT manage s and p oduc owne s
h oughou he o ganisa ion.
– C ea e an in en o y o c yp og aphic asse s used wi hin he o ganisa ion.
This should include bo h so wa e and ha dwa e and should clea ly speci y
he used algo i hms, whe he de eloped in-house o pu chased om a
endo . Some pa ies e e o a ‘c yp og aphic bill o ma e ials’ (CBOM).
– De e mine he isk and u gency o PQC mig a ion. Mos o ganisa ions
al eady pe o m egula isk assessmen s o hei IT in as uc u e. Ad-
di ionally, o ganisa ions should assess whe he hey classi y as an u gen
adop e o PQC (see below).
– C ea e a mig a ion plan. This is a mo e
complex s ep, which should a leas p io i-
ise which asse s mus be mig a ed i s and
indica e whe he he mig a ion o all u gen
sys ems can be ealis ically achie ed in
ime, be o e he a i al o c yp og aphically
ele an quan um compu e s.
Fo mo e de ails, we ecommend ollowing
hePQC Mig a ion Handbook, a ee guide
w i en by he Du ch sec e se ice AIVD and
esea ch o ganisa ions CWI and TNO.Secu i y
au ho i ies in o he coun ies ha e made simila
guidance a ailable.