Digi al-Analog Quan um Machine Lea ning
Lucas Lama a
Depa amen o de Física A ómica, Molecula y Nuclea , Facul ad de Física, Uni e sidad de Se illa, Se illa, Spain
Co espondence: Lucas Lama a ([email p o ec ed])
Recei ed: 17 No embe 2024 |Re ised: 30 Decembe 2024 |Accep ed: 20 Janua y 2025
Funding: MCIN/AEI, G an /Awa d Numbe : PID2022-136228NB-C21PID2022-136228NB-C22; ERDF A way o making Eu ope; Minis y o Digi al
T ans o ma ion and o Ci il Se ice o he Spanish Go e nmen ; Eu opean Union
Keywo ds: digi al-analog quan um p o ocols |machine lea ning |quan um echnologies
ABSTRACT
Machine lea ning algo i hms a e ex ensi ely used in an inc easing numbe o sys ems, applica ions, echnologies, and p oduc s,
bo h in indus y and in socie y as a whole. They enable compu ing de ices o lea n om p e ious expe ience and he e o e
imp o e hei pe o mance in a ce ain con ex o en i onmen . In his way, many use ul possibili ies ha e been made accessible.
Howe e , dealing wi h an inc easing amoun o da a poses di icul ies o classical de ices. Quan um sys ems may o e a way
o wa d, possibly enabling o scale up machine lea ning calcula ions in ce ain con ex s. On he con a y, quan um sys ems
hemsel es a e also ha d o scale up, due o decohe ence and he agili y o quan um supe posi ions. In he sho and mid e m,
i has been e idenced ha a quan um pa adigm ha combines e olu ion unde la ge analog blocks wi h disc e e quan um ga es,
may be ui ul o achie e new knowledge o classical and quan um sys ems wi h no need o ha ing a aul - ole an quan um
compu e . In his pe spec i e, we e iew some ecen wo ks ha employ his digi al-analog quan um pa adigm o ca y ou
e icien machine lea ning calcula ions wi h cu en quan um de ices.
1|Quan um Machine Lea ning
Machine lea ning is a knowledge ield, a ool, a compu ing pa -
adigm, a echnology, which is signi ican ly impac ing socie y a
la ge, by enabling a ple ho a o possibili ies, as well as challenges,
as wi h any new and highly powe ul echnology. Howe e , i s
ull deploymen is hampe ed by he ac ha an inc easing
amoun o da a is ha d o p ocess wi h cu en classical com-
pu e s, and consumes a signi ican amoun o ene gy. A possible
way o wa d o scale up machine lea ning calcula ions would be
ia he use o quan um de ices employing genuine quan um
p ope ies, such as supe posi ion and en anglemen . In his
way, i may be possible o ake ad an age o he speedup o quan-
um compu e s o ca y ou machine lea ning calcula ions in a
mo e e icien way, a leas in some ins ances. E en hough he
ield o quan um machine lea ning, which connec s machine
lea ning calcula ions o a quan um ha dwa e seeking o hese
ad an ages, is no absen o con o e sy and specula ion, as i
is eally ha d o p o e speedup wi h espec o he bes classical
machine lea ning algo i hms, his is a ield ha is ising much
expec a ion and hope ha wi h NISQ de ices one may bea
classical compu e s o use ul asks. Fo a ecen e iew o he
ield, see e . [1].
2|Digi al-Analog Quan um Pa adigm
Full- ledged, scalable, digi al quan um compu e s a e ha d o
achie e, as hey no mally equi e millions o physical qubi s
o encode aul - ole ance and e o -co ec ion p o ocols. On he
con a y, analog quan um simula o s a e ypically scalable bu
Dedica ed o he ounde s o Quan um Mechanics.
This is an open access a icle unde he e ms o he C ea i e Commons A ibu ion License, which pe mi s use, dis ibu ion and ep oduc ion in any medium, p o ided he o iginal wo k is
p ope ly ci ed.
© 2025 The Au ho (s). Ad anced In elligen Disco e y published by Wiley-VCH GmbH.
Ad anced In elligen Disco e y, 2025; 0:e2400023 1o 4
h ps://doi.o g/10.1002/aidi.202400023
Ad anced In elligen Disco e y
PERSPECTIVE
limi ed in he amoun o p oblems hey a e able o sol e, o ep o-
duce in a quan um simula ion. A possible combina ion o bo h
app oaches has eme ged in he pas ew yea s as a s a egy o he
sho and mid e m, which may enable NISQ quan um de ices o
achie e use ul asks, o ins ance, in lea ning new p ope ies o
quan um sys ems. The ocus on he ield o digi al-analog quan-
um p o ocols has been mos ly on quan um simula ions in he
pas ew yea s, wi h heo y p oposals and expe imen s in a a i-
e y o quan um pla o ms. Howe e , mos ecen ly se e al wo ks
ha aim a ca ying ou machine lea ning calcula ions ia digi al-
analog quan um p o ocols ha e appea ed in he li e a u e.
Fo an ea ly e iew on he ield o digi al-analog quan um sim-
ula ions see e . [2].
In Figu e 1, I plo a scheme o a gene ic ex eme digi al-analog
quan um p o ocol (EDAQP). This consis s o a combina ion o
he la ges analog blocks possible (a se o global na i e in e ac-
ions, namely, global uni a y ga es on all qubi s based on
he quan um pla o m na i e Hamil onian, U
1
(
1
), U
2
(
2
),…,
U
k
(
k
),…, and he smalles digi al ga es possible, ha is, single-
qubi ga es, R
i,j
, ac ing on qubi ja s ep i, which can be done wi h
he bes ideli ies in mos quan um pla o ms.
3|Digi al-Analog Quan um Machine Lea ning
In his pe spec i e, I gi e a non-exhaus i e o e iew o he ield
o quan um machine lea ning, when digi al-analog quan um
p o ocols a e employed o i s deploymen . I p opose o use
he name “Digi al-analog quan um machine lea ning”(DAQML)
o his esea ch a enue. I will ci e se e al pape s in his a ea as
well as b ie ly desc ibe each o hem.
In e . [3], he au ho s p opose o use a digi al-analog quan um
p o ocol o implemen a a ia ional quan um eigensol e o es i-
ma ing g ound s a e ene gies o molecules. A subsequen wo k
[4], analyzed he deploymen o a quan um gene ic algo i hm
ia he Qadence digi al-analog coding language o Rydbe g a om
a ays [5], o add ess also calcula ions o g ound s a e ene gies o
molecules, in a scalable way. This ins ance o digi al-analog
quan um p o ocol was o he EDAQP as shown in Figu e 1.
Re . [6] s udies he use o a digi al-analog quan um app oxima e
op imiza ion algo i hm o sol ing a se ies o asks wi h NISQ
de ices. In e . [7], he au ho s p opose o use a digi al-analog
quan um pa adigm o quan um ke nel implemen a ion in he
con ex o quan um machine lea ning. In e . [8], a quan um
Fou ie ans o m implemen a ion is p oposed ia he use o
digi al-analog quan um p o ocols, showing possible gains in
esou ces wi h espec o pu ely digi al ones, and enabling in his
way he use o his p imi i e in a a ie y o p o ocols, which may
also include quan um machine lea ning ones, in pa icula . This
is o ins ance analyzed in e . [9] wi h espec o he Ha ow-
Hassidim–Lloyd (HHL) p o ocol. In e . [10], he au ho s p opose
digi al-analog quan um lea ning algo i hms o Rydbe g a om
sys ems, which hey claim can combine he use ulness o quan-
um machine lea ning p o ocols in he nea e m, wi h he e i-
cien scalabili y ha is ecen ly being achie ed wi h Rydbe g
a oms.
Mos p e ious wo ks, which a e i s ins ances o he combina-
ion o quan um machine lea ning algo i hms wi h he digi al-
analog quan um pa adigm, show e idence ha his combina ion
can be ui ul, and pe haps, by enabling a be e scalabili y
o quan um machine lea ning algo i hms, may p o ide a quan-
um ad an age wi h espec o classical machine lea ning
calcula ions.
4|Implemen a ions o he Digi al-Analog
Quan um Pa adigm
Se e al expe imen s in quan um pla o ms ha e eached a signi -
ican ly la ge amoun o qubi s and/o Hilbe space dimension,
by combining s a e-o - he-a de elopmen s in hese pla o ms,
and he use o digi al-analog o simila quan um echniques.
Some o he quan um pla o ms employed a e apped ions,
supe conduc ing ci cui s, and Rydbe g a oms. This kind o
app oach has been conside ed in pa allel by se e al g oups,
which employ la ge analog blocks combined wi h disc e e ga es,
al hough some o hese echniques a e also e e ed o in he
li e a u e as “p og ammable analog quan um simula ions”.
Fo a ecen e iew ha desc ibes analog, digi al, and digi al-
analog quan um pa adigms, in he con ex o quan um simula-
ion expe imen s, see e . [11]. E en hough hese pionee ing
expe imen s ha e mainly been ocused on quan um simula ions,
simila se ups may add ess quan um machine lea ning asks as
he ones desc ibed abo e, wi h possible gains wi h espec o
classical compu e s a leas in some a o able si ua ions
[See, e.g., wo k done in s a ups PASQAL [12] and QUERA
[13] in his sense].
5|Ou look
The p e ious accoun o ecen li e a u e combining digi al-
analog quan um p o ocols wi h quan um machine lea ning
algo i hms, which I name “Digi al-Analog Quan um Machine
Lea ning”(DAQML), is jus an appe ize o wha may come
in he u u e. Wi h he inc easing e iciency o quan um pla -
o ms such as apped ions, supe conduc ing ci cui s, and
Rydbe g a oms, in achie ing de ices wi h ens, o e en, hund eds
o quan um bi s, he possibili y o combine la ge analog blocks
wi h digi al s eps, in he pa adigm ha we c ea ed mo e han
FIGURE 1 |scheme o a gene ic EDAQP. This consis s o a
combina ion o he la ges analog blocks possible, ha is, a se o
global na i e in e ac ions, namely, global uni a y ga es on all qubi s
based on he quan um pla o m na i e Hamil onian, U
1
(
1
), U
2
(
2
),…,
U
k
(
k
),…, and he smalles digi al ga es possible, ha is, single-qubi
ga es, which can be done wi h he bes ideli ies in mos quan um
pla o ms.
2o 4 Ad anced In elligen Disco e y, 2025
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10 yea s ago in he con ex o quan um simula ions, may ep e-
sen a signi ican s ep o wa d in he ield o quan um echnolo-
gies. Pe haps one o he i s app oaches ha could possibly
impac he indus y o ca ying ou use ul asks migh be he
one I desc ibe he e, o combining ou digi al-analog quan um
pa adigm wi h he ledgling ield o quan um machine lea ning.
This could, in u n, impac subsequen ly socie y a la ge, as well
as he scien i ic en e p ise, ia enhancing scien i ic disco e y, as
his e y new jou nal, Ad anced In elligen Disco e y, aims o
achie e.
In gene al e ms, conside ing analog blocks in a quan um p o o-
col allows one o making a much la ge numbe o quan um
sys ems o in e ac , and become en angled. Fo example, o long
chains o 20–60 apped ions, o 50 supe conduc ing qubi s, o
ens o hund eds o Rydbe g a oms, decomposing global en an-
gling ope a ions on o single and wo-qubi ga es is in gene al
highly ine icien , due o digi al e o s in oduced as well as
e o s p oduced in each wo-qubi ga e, such as u ning pulses
on and o , e c. O en, one may ca y ou quan um p o ocols
o which i su ices o combine la ge analog blocks, wi h a na i e
in e ac ion coupling all qubi s, wi h single-qubi ga es, pe haps
wi h a small numbe o addi ional wo-qubi ga es, and in his
si ua ion mos o en ideli ies will be be e han decomposing
e e y mul iqubi ga e on o elemen a y ga es. O cou se, his will
s ongly depend on each pa icula case.
Ne e heless, he digi al-analog quan um pa adigm includes he
digi al one and he analog one as pa icula cases, such ha ,
gi en ha he digi al one is p o en o be uni e sal o quan um
compu ing, digi al-analog quan um p o ocols a e i ially uni e -
sal as he digi al case is a pa icula case o he la e (e.g., o
analog blocks wi h iden i y ga es, ha ing he na i e in e ac ion
wi h negligible coupling o a bi a y ime). The main di icul y
in deploying hese kinds o p o ocols is elucida ing he op imal
mul iqubi , single, and (pe haps) wo-qubi ga e decomposi ion,
bu in gene al his signi ican ly enla ges he amoun o choices
a ailable in he expe imen al oolbox o op imizing he quan um
p o ocols. DAQML p o ocols will possibly pe o m be e when
one would need o ac on many qubi s collec i ely, p oducing
global en anglemen , and whene e single-qubi ga es would su -
ice o con e gence o he op imal solu ion, oge he wi h he
analog global blocks (possibly by adding a ew wo-qubi ga es,
when needed). A p omising scena io o his, as shown in e . [4],
is he quan um gene ic algo i hm ealm, whe e uni e sali y is no
so much needed a he le el o elemen a y ga es, bu con e gence
is achie ed ia ecombina ion and selec ion o he i es , while
some amoun o di e si y in e ms o single-qubi ga es and ime-
dependen global analog blocks is in oduced in each i e a ion in
a a he obus and andom way, whe e ga e impe ec ions do no
ma e much. In his conc e e scena io o quan um gene ic algo-
i hms, i may well be ha a known p oblem in quan um
machine lea ning p o ocols based on g adien descen con e -
gence, namely, ba en pla eaus, may be a oided as he gene ic
p o ocol does no ely on g adien s.
Rega ding po en ial u u e esea ch di ec ions in DAQML and,
gi en ha his kind o oolbox s ongly depends on he de elop-
men s o each quan um pla o m, i will be con enien o adap
he p o ocols o echnological ad ances in each quan um imple-
men a ion, namely, ions, supe conduc ing ci cui s, Rydbe g
a oms, quan um pho onics, e c. In gene al e ms, i can be con-
side ed ha la ge analog blocks and single-qubi ga es may be
obus in quan um machine lea ning gi en ha in many p o o-
cols one is i e a ing un il con e gence, and, as a as one is
app oaching he op imal, speci ic impe ec ions, o imings, in
he analog blocks, o he indi idual ga es, do no ma e so much.
A simila conside a ion may be done o he digi al-analog quan-
um pa adigm o quan um simula ions, gi en ha eal, la ge
quan um sys ems ypically ha e impe ec ions simila o hose
o he analog blocks o quan um pla o ms. The e o e, i is in gen-
e al no de imen al o ha e impe ec analog blocks, as a as one
can en ich hem wi h indi idual ga es, o p oduce, o example, a
ha d- o- each Heisenbe g model om an easily implemen able
Ising model.
In summa y, he digi al-analog quan um pa adigm is highly scal-
able and e o - obus , bo h o he DAQML and he quan um
simula ion applica ions. We hope ha he expe imen al quan um
communi y inds use ul ways o employ i o a new gene a ion o
highly complex, possibly use ul, quan um expe imen s.
Acknowledgmen s
The au ho acknowledges he suppo om g an s PID2022-136228NB-
C21 and PID2022-136228NB-C22 unded by MCIN/AEI/10.13039/
50110001103 and “ERDF A way o making Eu ope”. This wo k has also
been inancially suppo ed by he Minis y o Digi al T ans o ma ion
and o Ci il Se ice o he Spanish Go e nmen h ough he
QUANTUM ENIA p ojec call—Quan um Spain p ojec , and by he
Eu opean Union h ough he Reco e y, T ans o ma ion and Resilience
Plan—Nex Gene a ionEU wi hin he amewo k o he “Digi al Spain
2026 Agenda”.
Da a A ailabili y S a emen
The da a ha suppo he indings o his s udy a e a ailable om he
co esponding au ho upon easonable eques .
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4o 4 Ad anced In elligen Disco e y, 2025
29439981, 0, Downloaded om h ps://ad anced.onlinelib a y.wiley.com/doi/10.1002/aidi.202400023 by Readcube (Lab i a Inc.), Wiley Online Lib a y on [14/02/2025]. See he Te ms and Condi ions (h ps://onlinelib a y.wiley.com/ e ms-and-condi ions) on Wiley Online Lib a y o ules o use; OA a icles a e go e ned by he applicable C ea i e Commons License