Enhanced Connectivity of Quantum Hardware with Digital-Analog Control

PHYSICAL REVIEW RESEARCH 2, 033103 (2020)
Enhanced connec i i y o quan um ha dwa e wi h digi al-analog con ol
Asie Galicia ,1Bo ja Ramon ,1En ique Solano,1,2,3,4and Mikel Sanz 1,2,3,*
1Depa men o Physical Chemis y, Uni e si y o he Basque Coun y, Apa ado 644, 48080 Bilbao, Spain
2IQM, Nymphenbu ge s asse 86, 80636 Munich, Ge many
3IKERBASQUE, Basque Founda ion o Science, Ma ia Diaz de Ha o 3, 48013 Bilbao, Spain
4In e na ional Cen e o Quan um A i icial In elligence o Science and Technology (QuA is )
and Depa men o Physics, Shanghai Uni e si y, 200444 Shanghai, China
(Recei ed 23 Decembe 2019; accep ed 23 June 2020; published 20 July 2020)
Quan um compu e s based on supe conduc ing ci cui s a e expe iencing apid de elopmen , wi h he aim
o ou pe o m classical compu e s in ce ain use ul asks in he nea u u e. Howe e , he cu en ly a ailable
chip ab ica ion echnologies limi he capabili y o ga he ing a la ge numbe o high-quali y qubi s in a single
supe conduc ing chip, a equi emen o implemen ing quan um e o co ec ion. Fu he mo e, achie ing high
connec i i y in a chip poses a o midable echnological challenge. He e, we p opose a hyb id digi al-analog
quan um algo i hm ha enhances he physical connec i i y among qubi s coupled by an a bi a y inhomogeneous
nea es -neighbo Ising Hamil onian and gene a es an a bi a y all- o-all Ising Hamil onian only by employing
single-qubi o a ions. Addi ionally, we op imize he p oposed algo i hm in he numbe o analog blocks and
in he ime equi ed o he simula ion. These esul s ake ad an age o he na u al e olu ion o he sys em
by combining he lexibili y o digi al s eps wi h he obus ness o analog quan um compu ing, allowing us o
imp o e he connec i i y o he ha dwa e and he e iciency o quan um algo i hms.
DOI: 10.1103/PhysRe Resea ch.2.033103
I. INTRODUCTION
Quan um compu a ion has eme ged in ecen yea s as a
p omising echnology which aims a sol ing p oblems such as
he ac o iza ion o a composi e numbe [1], s udying quan um
ield heo ies [2,3], simula ing quan um chemis y [4,5] and
luid dynamics [6], and simula ing complex sys ems [7–10]
mo e e icien ly han classical compu a ion. Ano he eason
o he inc easing in e es in he ield o quan um compu a ion
is due o G o e ’s algo i hm [11], which shows quad a ic
speedups when compa ed agains classical sea ch algo i hms.
This echnology can be implemen ed in di e en quan um
pla o ms, such as apped ions, pho onics, o supe conduc ing
qubi s, among o he s. In his las pla o m, s ong e o s ha e
been pe o med o show an example in which a quan um
p ocesso ou pe o ms any classical compu e , a miles one
ecen ly achie ed by Google [12]. This lou ishing echnol-
ogy s ill p esen s a conside able numbe o challenges o be
sol ed, such as inc easing he numbe o high-quali y qubi s
in a single quan um p ocesso o achie ing in e ac ions among
all qubi s. These p oblems cha ac e ize he so-called noise
in e media e-scale quan um (NISQ) de ices [13], quan um
chips comp ising 50–100 qubi s, which a e s ill a ec ed by
signi ican noise.
*[email p o ec ed]
Published by he Ame ican Physical Socie y unde he e ms o he
C ea i e Commons A ibu ion 4.0 In e na ional license. Fu he
dis ibu ion o his wo k mus main ain a ibu ion o he au ho (s)
and he published a icle’s i le, jou nal ci a ion, and DOI.
Wi h he objec i e o op imizing he cu en ly a ailable
esou ces o quan um compu a ion and, ul ima ely, imple-
men ing a use ul quan um compu e in he NISQ e a, an
al e na i e pa adigm o quan um compu ing, called digi al-
analog quan um compu a ion (DAQC) [14–16], has ecen ly
been p oposed. The DAQC pa adigm aims a imp o ing he
ideli y o he algo i hm by exploi ing he obus ness o he
na u al dynamics p o ided by he quan um pla o m oge he
wi h he lexibili y gi en by he as implemen a ion o single
qubi o a ion (SQR).
Recen ly, a cons uc i e me hod o achie e uni e sal quan-
um compu a ion employing an all- o-all (ATA) Ising Hamil-
onian as he analog esou ce has been p oposed in Re . [15].
A e wa d, he implemen a ion o ele an quan um algo-
i hms wi hin he DAQC pa adigm has also been add essed
wi h he digi al-analog quan um Fou ie ans o m (QFT)
[17] and he quan um app oxima e op imiza ion algo i hm
(QAOA) [18]. In hese e e ences, he au ho s nume ically
compa ed he pe o mance o he DAQC pa adigm agains he
pu ely digi al one unde sensible noise sou ces, showing he
o me gi es ema kably be e esul s when he sys em scales
up wi h he numbe o qubi s.
The in insic connec i i y among qubi s in a quan um
pla o m is no necessa ily well desc ibed by an ATA ho-
mogeneous Hamil onian. In ac , a ealis ic quan um chip is
expec ed o p esen lowe connec i i y ocused on app oxi-
ma ely nea es -neighbo (NN) in e ac ions, since ATA con-
nec ions equi e a p ohibi i ely inc easing amoun o wi ing
among qubi s.
In his pape , we design an algo i hm ha op imally sim-
ula es an a bi a y ATA Ising Hamil onian employing as a
esou ce a gi en inhomogeneous NN Ising model and SQR.
2643-1564/2020/2(3)/033103(11) 033103-1 Published by he Ame ican Physical Socie y
GALICIA, RAMON, SOLANO, AND SANZ PHYSICAL REVIEW RESEARCH 2, 033103 (2020)
1
2
3
4
5
(a)
1
2
3
4
5
(b)
1
2
3
4
5
(c)
FIG. 1. Examples o Ising Hamil onians as hei g aph ep e-
sen a ion. Panel (a) shows a comple e K5g aph, which ep esen s
an all- o-all Hamil onian HATA (g) o i e qubi s. Panel (b) shows a
Hamil onian pa h which ep esen s a nea es -neighbo Hamil onian
HNN(g) o i e qubi s. Panel (c) shows ano he Hamil onian pa h
wi h he e ex pe mu a ion P=[1,3,4,2,5].
This is achie ed employing O(5L2) analog blocks, i.e., NN
e olu ions, wi h Lbeing he numbe o qubi s o he chip.
E en hough he pa icula dynamics conside ed as a esou ce
is he ZZ Ising Hamil onian, he p oposed algo i hm could
be ex ended o o he dynamics, such as he XX +YY Ising
Hamil onian. The simula ion o he Hamil onian is op imal
in he dependence o he numbe o analog blocks and he
simula ion ime equi ed o hese analog blocks.
II. GRAPH REPRESENTATION
OF AN ISING HAMILTONIAN
The Ising Hamil onian o Lqubi s can be in e p e ed as
a weigh ed g aph o L e ices, whe e he weigh o he edge
connec ing he e ex i o he e ex jis gij. I wo e ices i
and ja e no connec ed, gij =0.
In his ep esen a ion, an ATA Ising Hamil onian o L
qubi s becomes a comple e g aph KL, i.e., a g aph wi h edges
among e e y possible e ex wi hou epe i ion. On he o he
hand, he NN Ising Hamil onian is ep esen ed as a Hamil o-
nian pa h, ha is, a pa h isi ing all he possible e ices only
once.
I is no ewo hy o men ion ha an a bi a y Hamil onian
pa h is ep esen ed by a pe mu a ion o all he e ices in he
g aph. To eco e a Hamil onian pa h om a gi en e ex
pe mu a ion, i su ices o connec wi h an edge he e ices
ha a e adjacen in he pe mu a ion. An example o a comple e
g aph, oge he wi h wo Hamil onian pa hs, is ep esen ed
in Fig. 1, whe e we also show he e ex pe mu a ion o he
Hamil onian pa hs.
No ice ha we a e cu en ly dealing wi h ZZ in e ac ions,
and hus di e en Hamil onian pa h e olu ions commu e.
This means ha he inal e olu ion will be he sum o all
he Hamil onian pa hs weigh ed by hei espec i e e olu ion
ime. This las s a emen is summa ized in he equa ion

i
ei iHPi(gj)=eii iHPi(gj),(1)
whe e HPi(gj) is a Hamil onian ha desc ibes a ZZ in e ac-
ion which has a g aph ep esen a ion o a Hamil onian pa h
wi h weigh s gj. This is unde s ood in he g aph ep esen a ion
as ha ing as inal g aph he sum o all he weigh ed Hamil o-
nian pa hs.
Ou i s ask is hen o spli he comple e g aph, which
ep esen s he ATA Hamil onian, in o a se o Hamil onian
=++
(b) (c) (d)(a)
1
2
3
6
5
4
1
2
3
6
5
4
1
2
3
6
5
4
1
2
3
6
5
4
FIG. 2. An example o how o ill exac ly a comple e g aph o six
e ices using Hamil onian pa hs. Panel (a) ep esen s he comple e
g aph we a e ying o ob ain. Panels (b), (c), and (d) co espond o
he Hamil onian pa hs we use. Panel (b) is buil by s a ing in he
i s node, going o wa d o he nex node, hen wo nodes backwa d,
h ee o wa d, e c. The o he wo Hamil onian pa hs a e ob ained by
o a ing he i s one. F om his cons uc ion echnique, we ob ain
Hamil onian pa hs de ined by he pe mu a ions o Eq. (2). The
pe mu a ions ha de ine panels (b), (c), and (d) a e espec i ely P1
6=
[1,2,6,3,5,4], P2
6=[2,3,1,4,6,5], and P3
6=[3,4,2,5,1,6].
pa hs ha will be la e simula ed using ou esou ce ( he NN
Hamil onian). This will allow us o e icien ly decompose he
ATA e olu ion in e ms o Hamil onian pa hs.
Pa i ioning a comple e g aph in o a se o Hamil onian
pa hs esembles he Hamil onian decomposi ion p oblem,
which is abou pa i ioning a comple e g aph in o a se o
Hamil onian cycles. This las p oblem was sol ed by Walecki
[19,20] in 1890, who used a cons uc ion in which one Hamil-
onian cycle is o a ed o ge all he cycles ha compose he
comple e g aph. Using a simila decomposi ion schema ized
o six qubi s in Fig. 2, we can decompose he comple e g aph
in o a se o disjoin Hamil onian pa hs. These Hamil onian
pa hs a e cha ac e ized by he e ex pe mu a ion
Pk
L(j)=k−1+j
2mod L+1,i je en,
k−1−j−1
2mod L+1,i jodd, (2)
whe e Pk
L∈SL,SLis he symme ic g oup o Lelemen s, L
is he numbe o qubi s, and k∈Zsuch ha 1 ⩽k⩽L/2. j
ep esen s he j h posi ion o he e ex pe mu a ion. No e ha
kis jus a label o he Hamil onian pa h.
I is also no ewo hy o men ion ha his pa i ion is only
alid o an e en numbe o qubi s. When dealing wi h an odd
numbe o qubi s, and hence, and odd numbe o e ex in
he g aph ep esen a ion, we use he no ion o single pe ec
ma ching. This in ol es using he same Hamil onian pa h
cons uc ion o Eq. (2), bu allowing one Hamil onian pa h
o o e lap wi h he es . This will pose no p oblem in ou la e
cons uc ion, since we will be able o selec i ely u n o he
desi ed in e ac ions.
In Fig. 2, we show an example o six qubi s whe e we
also depic ed he co esponding Hamil onian pa hs. Hence,
he p oblem now consis s on e icien ly simula ing hese
Hamil onian pa hs o ob ain he ATA e olu ion.
III. SWAPPING GATES
The nex s ep is o ob ain each o he Hamil onian pa h
connec ions using a NN Hamil onian as a esou ce. Fo ha ,
we will change he connec ions using a SWAP-like ga e U ha
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ENHANCED CONNECTIVITY OF QUANTUM HARDWARE … PHYSICAL REVIEW RESEARCH 2, 033103 (2020)
pe o ms he ope a ions
U(σz⊗I)U†=I⊗σz,
U(I⊗σz)U†=σz⊗I.(3)
The ga e ha changes he ac ion o an a bi a y ope a o in
a qubi o ano he qubi is he SWAP ga e, de ined as
USWAP =eiπ
4(X1X2+Y1Y2+Z1Z2),(4)
whe e he supe indices 1 and 2 e e o he qubi on which he
ga e ac s. Howe e , as we only need o change Zga es, we
ha e some deg ees o eedom a ailable. Mo e p ecisely, he
mos gene al uni a y ga e ha ul ils Eq. (3)is
U(α,β, γ )=R1
zπγ−α
2+1
2+β
×eiπ
4(X1X2+Y1Y2+(γ+α)Z1Z2)
×R1
zπγ−α
2−1
2−β,(5)
whe e R1
z[θ]=eiσ1
z
θ
2and α,β, γ ∈R.
Since we will la e build his ga e using inhomogeneous
Ising Hamil onians om he DAQC pe spec i e, we will se
α=γ=0 and β=−
1
2, ob aining he so-called iSWAP ga e.
The choice o pa ame e s will minimize he amoun o single-
qubi ga es and analog blocks equi ed, since
UiSWAP =eiπ
4(X1X2+Y1Y2).(6)
We will ocus now on ob aining a sequence o iSWAP ga es
ac ing on adjacen qubi s ha e icien ly ans o ms a sys em
wi h NN connec ions in o a sys em wi h he desi ed Hamil-
onian pa h connec ions. The eason o es ic ou sel es o
adjacen iSWAP ga es is ha we wan o decompose hem using
NN Ising Hamil onians as a esou ce.
Fo he sake o cla i y, we will use Uij o ep esen he
iSWAP ga e be ween qubi s iand j. We will i s show how
his ope a ion ans o ms a sys em wi h NN couplings. The
esul will be a sys em ep esen ed by a Hamil onian pa h in
i s g aph ep esen a ion.
I we sandwich σk
zσl
z, a ga e ha applies he Zga e o he
qubi s kand l, wi h he ga e Uij, we ob ain
Uijσk
zσl
zU†
ij =στij(k)
zστij(l)
z,(7)
whe e we ha e de ined he unc ion τij as a pe mu a ion o he
indices iand j, ha is, a ansposi ion. Mo e p ecisely, i k=
i,j,τij(k)=k, o he wise, τij(i)=jand τij(j)=i. Basically,
he Uij ga e changes σzga es ac ing on qubi i o ac on
qubi j.
Because o he uni a i y o he iSWAP ope a ion, i s ac-
ion on a NN Hamil onian e olu ion can be w i en as
Uijei H(g)NNU†
ij =ei UijH(g)NNU†
ij. The e o e, he inal Hamil o-
nian esul s in
UijHNN(g)U†
ij =
L−1

k=1
gUijσk
zσk+1
zU†
ij
=
L−1

k=1
gστij(k)
zστij(k+1)
z.(8)
The ini ial e ex pe mu a ion de ining he sys em’s NN
coupling was P =[1,2,3, ..., L]. A e he iSWAP ope -
a ion, he pe mu a ion ha de ines ou sys em is P=
[τij(1),τ
ij(2),τ
ij(3), ..., τij(L)]. This app oach is s aigh o -
wa dly gene alized o a sys em wi h a bi a y connec ions.
This means ha , a e sandwiching wi h iSWAP ga es a sys em
wi h ce ain connec ions, i in e changes he ones ha had he
wo qubi s being a ec ed by he iSWAP ga e. In ou case, since
we will be dealing wi h sys ems ep esen ed by a Hamil onian
pa h, he iSWAP ope a ion educes o a ansposi ion in he
co esponding e ex pe mu a ion.
As he se o ansposi ions ha ou NN esou ce can
implemen , {τii+1} o i∈[1,L−1], is a gene a o o he
symme ic g oup SL, we can ob ain any desi ed pe mu a ion
using he co ec ansposi ions. This means ha we can sim-
ula e he e olu ion o any sys em wi h couplings ep esen ed
by a Hamil onian pa h using as esou ce he sys em wi h NN
couplings. Fo example, he Ising Hamil onian ha ep esen s
Fig. 1(c), which we will call H1, is jus ob ained using he
ollowing ans o ma ions H1(g)=U34U23HNN(g)U†
23U†
34.
Fo he sake o simplici y, we will de ine a sequence o
ansposi ions o be
Si→j=τii+1τi+2i+3...τj−1ji i<j
Ii i⩾j.(9)
The pe mu a ions de ined in Eq. (2) can be composed in
e ms o wo g oups o sequences,
G1(k)=S2k−2
1→2S2k−3
2→3···
×S4
1→2k−4S3
2→2k−3S2
1→2k−2S1
2→2k−1,(10)
G2(k,L)=SL−2k−1
L−1→LSL−2k−2
L−2→L−1···
×S4
2k+4→L−1S3
2k+3→LS2
2k+2→L−1S1
2k+1→L,(11)
whe e kand L e e o he pe mu a ion Pk
Lwe a e building and
he sequences ha e been labeled o make clea he o de o
applica ion. Tha is, Pk
L=G1(k)G2(k,L). In Appendix A,we
p o e ha hese a e indeed he g oups o sequences needed
o ob ain he desi ed pe mu a ions o Eq. (2). No e ha bo h
g oups o sequences commu e be ween hem.
Le us now show an example o he case o six qubi s. In
o de o ob ain he pe mu a ion P3
6 om Fig. 3, we need o
apply he ansposi ions de ined in Eq. (11), which a e G1(3)
and G2(3,6). Howe e , in he pa icula case o G2(3,6),
since 2 ×k+1=7 and L=6, G2(3,6) =I. The sequences
applied a e G1(3) =S4
1→2S3
2→3S2
1→4S1
2→5. The ci cui ha im-
plemen s his se o ansposi ions using iSWAP ga es is shown
in Fig. 3, whe e each column o iSWAP ga es ep esen s one
sequence o ansposi ions.
To sum up, in his sec ion we ha e shown an algo i hm
ha , using adjacen iSWAP ga es and he NN Ising Hamil onian
as esou ce, is able o simula e he e olu ion o an ATA
Hamil onian o he case o an e en numbe o qubi s. Le
us now simpli y he ci cui so ha i equi es he smalles
possible amoun o iSWAP ga es.
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GALICIA, RAMON, SOLANO, AND SANZ PHYSICAL REVIEW RESEARCH 2, 033103 (2020)
1
2
3
6
5
4
1
2
3
6
5
4
FIG. 3. Ci cui ha simula es he e olu ion o a Hamil onian pa h wi h e ex pe mu a ion P3
6. Each o he su ounding columns o iSWAP
ga es is ela ed wi h one o he sequences belonging o he g oups de ined in Eq. (11).
IV. SIMPLIFICATION OF THE CIRCUIT
In his sec ion, we will show an op imized e sion o he
p e iously discussed ci cui . Besides, we will also gi e a
decomposi ion o he ci cui in e ms o ZZ ga es, which will
be use ul la e .
Since we now need o combine he se o Hamil onian pa hs
{Pk
L} o ob ain he desi ed ATA e olu ion, i is possible o
simpli y he o al numbe o iSWAP (U) ga es needed. The o al
ci cui is desc ibed by he se o ga es,
F(k,L)=
2k+1

i=2,ie en
Ui,i+1
L−1

i=2k+1,ie en
U†
i,i+1
×
2k

i=1,iodd
Ui,i+1
L−1

i=2k+1,iodd
U†
i,i+1,
×k∈1,L
2−1,F(0,L)
=
j=1

j=L
2
⎡
⎣
L−1

i=2j−1,iodd
Ui,i+1
L−2

i=2j−2,ie en
Ui,i+1⎤
⎦,
FL
2,L=
j=L
2

j=1L−2j

i=1,iodd
U†
i,i+1
L+1−2j

i=2,ie en
U†
i,i+1.(12)
The inal ATA e olu ion will be desc ibed as
ei HATA =⎡
⎣
L
2

k=1
F(k,L)ei HNN ⎤
⎦F(0,L).(13)
In Appendix C, we show he demons a ion leading o his
simpli ied ci cui . In Fig. 4, we show an example o he
simpli ica ion o he case o six qubi s.
Using Eq. (5), we can decompose he ga es F(k,L)in o
a se o SQR and ZZi,j(φ=gj ) ga es ac ing on adjacen
qubi s. G ouping pa allel ZZij ga es na u ally leads o a ci cui
which can be implemen ed using he DAQC p o ocol, as
explained in he ollowing sec ion. An example o he case
o six qubi s is shown in Fig. 5.
We now ake in o accoun ha wo consecu i e pa allel se s
o iSWAP ga es can be decomposed in o h ee analog blocks
plus some SQR (as will be p o en in he nex sec ion). Since
he o al numbe o F(k,L) o k∈[1,L
2−1] ga es is L
2−1
and each one equi es wo pa allel se s o iSWAP ga es, we
equi e a o al o 3L
2−3 analog blocks o gene a e hese ga es.
We also need a mos 3(L
2−2) analog ga es o implemen he
F(0,L) se o ga es and 3(L
2−1) analog ga es o implemen
F(L
2,L). Mo eo e , we need L
2analog blocks mo e e ol ing
du ing a ime . The o al amoun o inhomogeneous analog
blocks needed is hen 5L−12.
In he ollowing sec ion, gi en a NN inhomogeneous
Hamil onian wi h ixed couplings, we will de i e an algo-
i hm o simula e he e olu ion o an a bi a y NN inho-
mogeneous Hamil onian e icien ly, bo h in ime and in he
numbe o analog blocks. This is he las s ep be o e ob aining
an algo i hm o simula e an a bi a y inhomogeneous ATA
Hamil onian.
V. SIMULATING AN ARBITRARY
INHOMOGENEOUS HAMILTONIAN
In his sec ion, we will show an algo i hm o simula e
he e olu ion o an a bi a y inhomogeneous NN Hamil onian
unde he pa adigm o DAQC, employing a simila algo i hm
o he one shown in Re . [15]. In his case, ou esou ce will
be a ixed inhomogeneous NN Hamil onian. The algo i hm we
use has he ollowing h ee ad an ages o e he one shown in
Re . [15]: (i) I wo ks o an a bi a y numbe o qubi s, (ii)
i equi es he minimum amoun o analog blocks, and (iii) i
op imizes he ime equi ed o he simula ion.
We will i s need o no ice ha i is possible o selec i ely
change he sign o any desi ed combina ion o couplings. This
is done by su ounding some o he qubi s wi h Xga es.
We ep esen he ac ion o he Xga es by colou ing he
co esponding qubi s in he g aph ep esen a ion (see Fig. 6).
In o de o change he sign o he desi ed combina ion o
couplings, i su ices o colo di e en ly he qubi s connec ed
o he desi ed couplings. In Appendix B, we p o e ha his
can be done o a NN chain wi h an a bi a y leng h. In Fig. 6,
we change he sign o all he couplings in Fig. 6(a) and he
sign o jus one coupling in Fig. 6(b).
We will now decompose a HNN(g
j) e olu ion du ing a ime
in o a se o HNN(gj) e olu ions ha ha e been e ol ing
033103-4
ENHANCED CONNECTIVITY OF QUANTUM HARDWARE … PHYSICAL REVIEW RESEARCH 2, 033103 (2020)
(a)
(b)
(c)
(d)
1
2
3
6
5
4
1
2
3
6
5
4
1
2
3
6
5
4
1
2
3
6
5
4
1
2
3
6
5
4
1
2
3
6
5
4
1
2
3
6
5
4
1
2
3
6
5
4
1
2
3
6
5
4
1
2
3
6
5
4
1
2
3
6
5
4
1
2
3
6
5
4
FIG. 4. Simpli ied ci cui simula ing an ATA Hamil onian e olu ion. Panel (a) shows he h ee Hamil onian pa hs ha o m he ATA
e olu ion. Panel (b) is ob ained by in oducing he iSWAP ga es ha ans o m a NN Hamil onian in he desi ed Hamil onian pa hs. This ga es
a e desc ibed by he g oup o ansposi ions de ined in Eqs. (10)and(11). The se o ansposi ions ob ained wi h his equa ion is ansla ed
in o a se o iSWAP/iSWAP†ga es su ounding he NN Hamil onian as discussed in Sec. III. In panel (c), we simpli y he ci cui by making use
o iSWAP iSWAP†=I. The esul ing ga es a e he se o iSWAP ga es de ined in Eq. (12), which lead o he ci cui depic ed in panel (d).
du ing a ime neach:
HNN(g
j)=
L−1

j=1
g
jσj
zσj+1
z
=
L−1

j=1
L−1

n=1
ngj(−1) n(j)+ n(j+1)σj
zσj+1
z
=
L−1

j=1
L−1

n=1
nL−1

k=1
X n(k)
kgjσj
zσj+1
zL−1

k=1
X n(k)
k
=
L−1

n=1
nL−1

k=1
X n(k)
kHNN(gj)L−1

k=1
X n(k)
k,(14)
whe e Xj
iis an Xga e applied on he i h qubi j imes and
n(k) is a bina y unc ion ha de e mines whe he an Xga e
is being applied in he k h qubi du ing he n h analog block,
ye o be de e mined. In he i s s ep, we make use o
bj=g
j
gj
=
L−1

n=1
Mnj
n
,(15)
whe e Mnj =(−1) n(j)+ n(j+1). This de ines he ollowing sys-
em o linea equa ions,
b=M
.(16)
The ma ix Mhas only ±1 en ies. The in e p e a ion o
Mnj =−1 is ha du ing he n h analog block, he j h coupling
changes he sign. F om now on, we will only ocus on he M
ma ix ins ead o he n(j) unc ions.
033103-5

GALICIA, RAMON, SOLANO, AND SANZ PHYSICAL REVIEW RESEARCH 2, 033103 (2020)
1
2
3
6
5
4
1
2
3
6
5
4
1
2
3
6
5
4
FIG. 5. This ci cui is ob ained by eplacing each iSWAP in Fig. 4by he exp ession shown in Eq. (6), whe e each o he exponen ials
ha e been mul iplied by he app op ia e single-qubi ga es o ans o m hem in o ZZ in e ac ions. Then, hese in e ac ions a e ga he ed
and exp essed as e olu ions o NN Ising Hamil onians, cha ac e ized by he di e en couplings gand hei e olu ion ime . Each o hese
inhomogeneous Hamil onian e olu ions can be implemen ed using he DAQC ci cui discussed in Sec. V. The SQR employed in his ci cui
a e he Hadama d ga e (H) and he ga e R,de inedasR=HSH,whe eSis he phase ga e.
We will now assume wi hou loss o gene ali y ha he
ollowing condi ions hold ∀j:
bj⩾bj+1,(17)
|b1|⩾|bj|,(18)
bj>0.(19)
Wi hou loss o gene ali y, bjcan always be elabeled and
ha e hei signs changed o hold hese inequali ies. Unde
hese condi ions, we p opose he ollowing Mma ix:
M=
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
1111··· 1111
−1111··· 1111
−1−111 1111
−1−1−11
...1111
.
.
..
.
..
.
...........
.
..
.
..
.
.
−1−1−1−1...1111
−1−1−1−1...−1111
−1−1−1−1··· −1−111
−1−1−1−1··· −1−1−11
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
.
(20)
A e in e ing his ma ix (see Appendix D), Eq. (16) leads
o he ime in e als
k
=bk−bk+1
2,(21)
L−1
=b1+bL−1
2,(22)
wi h k∈[1,L−2]. Recall ha k=0 means ha we do no
need he k h analog block o he simula ion.
Le us now p o e ha hese solu ions o nimply he
minimum amoun o analog blocks and ha he ime equi ed
o he simula ion is minimal. As, h ough Eqs. (21) and (22),
we a e mapping he se o imes { n}L−1
n=1 o he alues {bj}L−1
j=1,
we need a leas as much di e en nas bj. Indeed, suppose
ha bj=bj. We can elabel hem such ha j=j+1. Then,
om Eq. (21), we ge ha j=0, so he o al numbe o analog
blocks is educed by 1. Ano he pa icula ly ele an case is
when bj=0 o kdi e en alues, o which he numbe o
analog blocks needed is educed by k−1.
The ime equi ed o simula e he desi ed HNN(g
j)is
de ined as sim =L−1
n=1| n|. No e ha we do no ake in o
accoun he ime equi ed o implemen he Xga es since
we a e supposing ha hey a e ideal digi al blocks, ins an-
aneous. Howe e , we belie e ha he ci cui will s ill be
minimum in ime as long as we can pa allelize he ap-
plica ion o hese ga es. In Appendix E, we p o e ha ,
(a)
(b)
FIG. 6. Colo ed g aphs ep esen ing he su ounding o Xga es
in he e olu ion o a NN sys em o i e qubi s. A colo ed node
co esponds o a qubi su ounded by Xga es. The do ed lines
ep esen he change o sign in he coupling. Panel (a) ep esen s
he in e sion o all couplings, which is he same as in e ing he ime
e olu ion o he sys em. Panel (b) ep esen s he in e sion o only
one o he couplings.
033103-6
ENHANCED CONNECTIVITY OF QUANTUM HARDWARE … PHYSICAL REVIEW RESEARCH 2, 033103 (2020)
(a
)
(b
)
(c
)
FIG. 7. Implemen a ion o he ci cui discussed in Sec. V. The digi al block implemen ed is used in he ci cui o Fig. 5 o gene a e an
iSWAP ga e be ween qubi s 2 and 3, in pa allel wi h an iSWAP†ga e be ween qubi s 4 and 5. The analog blocks shown in he igh -hand side
o panel (a) ep esen he e olu ion o a NN sys em, whe e he sign o some o he couplings ha e been in e ed acco ding o Eq. (20). The
di e en e olu ion imes a e de e mined by Eqs. (21)and(22). In o de o mee he cons ain s imposed by Eqs. (17)–(19), he coe icien s
b1,b2,b3,b4,andb5a e equal o g
2
g2,g
4
g4,g
1
g1,g
3
g3,andg
5
g5, espec i ely (we made he assump ion ha g4>g2). Since b2<0, we need o change
g4o sign in all he analog blocks, which is achie ed by applying he Xga es o panel (a). The dashed lines in he analog block ep esen a
coupling changed o sign. These dashed lines connec s wo qubi s wi h di e en colo , whe eas a solid line connec s wo qubi s wi h he same
colo . The change o sign o he i h coupling (bi) is gi en by he i h column o he ma ix de ined in Eq. (20). Fo example, he i s column
o he ma ix shows ha all he signs o bibelonging o he i s block mus be in e ed excep om b1, which is ela ed o g2. Consequen ly,
all he signs o he couplings mus be in e ed, excep om g2. In panel (b), we sandwiched each analog block wi h Xga es in he qubi s ha
we e colo ed, ep esen ing in ha way he same e olu ion o panel (a). We also show he ime e olu ion co esponding o each analog block.
Las ly, in panel (c) we simpli ied he p e ious ci cui bo h by elimina ing all he analog blocks wi h ze o ime e olu ion, and all unnecessa y
Xga es, aking in o accoun ha XX =I.
unde he cons ains o Eqs. (17)–(19), min( sim)≡ min =
|b1| . We also p o e in Appendix E ha ou ci cui e-
qui es a ime min o pe o m he simula ion o an a bi a y
inhomogenous Hamil onian, which is he minimum ime
possible.
As an example, in Fig. 7we ep esen he implemen a ion
o one o he analog blocks shown in Fig. 5, equi ed o a se
o iSWAP ga es. Mo e p ecisely, he depic ed block is necessa y
o an iSWAP ga e be ween he qubi s 2 and 3 and an iSWAP†
ga e be ween he qubi s 4 and 5. I is no ewo hy o men ion
ha we equi e a leas L−1 analog blocks o simula e he
e olu ion o an a bi a y inhomogeneous NN Hamil onian.
Un il now, we ha e shown an algo i hm ha simula es
he e olu ion o an homogeneous ATA Hamil onian using as
esou ce an inhomogeneous NN Hamil onian. Fu he mo e,
i is a ained wi h O(5L2) analog blocks, since we need
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GALICIA, RAMON, SOLANO, AND SANZ PHYSICAL REVIEW RESEARCH 2, 033103 (2020)
O(5L) inhomogeneous analog blocks, each o which can be
simula ed using O(L) homogeneous analog esou ces.
I is s aigh o wa d o modi y he ci cui in o de o
simula e an inhomogeneous ATA Hamil onian wi h negligible
impac on he pe o mance. I su ices o change he analog
blocks o Fig. 4 o he necessa y inhomogeneous NN Ising
Hamil onian.
In o de o implemen his ci cui o an odd numbe o
qubi s, L, we can use he same se o Hamil onian pa hs, Pk
L,
o Eq. (2). In his case, k∈[1,L+1
2] and, in o de o ob ain
an homogeneous ATA Hamil onian, we need o se o ze o
some o he couplings used o he Hamil onian pa h e olu ion
ep esen ing P
L+1
2
L. I should be no ed ha he numbe o
analog blocks will s ill be O(5L2). E en hough we do no
discuss he e how o ob ain he iSWAP ga es o his case,
echniques simila o hose discussed in Appendix Acan be
employed.
VI. CONCLUSIONS
We ha e shown ha , wi hin he DAQC pa adigm, na u ally
a ising e olu ions can be u ilized o simula e any inhomoge-
neous ATA Ising Hamil onian along wi h SQR. In pa icula ,
we ha e designed an algo i hm based on a NN Ising Hamil-
onian wi h O(5L2) analog blocks, whe e Lis he numbe o
qubi s in he chip. Fo his, we also discussed bo h a digi al
app oach ha simula es an ATA sys em while ha ing NN-like
connec ions and an algo i hm ha simula es unde he DAQC
pa adigm he e olu ion o an inhomogeneous Hamil onian.
This las algo i hm has been p o en o be e icien in he
numbe o analog blocks and in he simula ion ime equi ed,
as long as we ea SQR as ideal ga es. This p o ocol can be
ex ended o pla o ms desc ibed by di e en Hamil onians,
such us he XX +YY NN Ising Hamil onian.
ACKNOWLEDGMENTS
The au ho s acknowledge suppo om Spanish Go e n-
men PGC2018-095113-B-I00 (MCIU/AEI/FEDER, UE) and
Basque Go e nmen IT986-16. The au ho s also acknowledge
suppo om he p ojec s QMiCS (820505) and OpenSupe Q
(820363) o he EU Flagship on Quan um Technologies,
as well as om he EU FET Open p ojec Qu omo phic
(828826). This ma e ial is also based upon wo k suppo ed by
he U.S. Depa men o Ene gy, O ice o Science, O ice o
Ad ance Scien i ic Compu ing Resea ch (ASCR), unde ield
wo k P oposal No. ERKJ333.
APPENDIX A: GROUP DEMONSTRATION
In his Appendix, we p o e ha he combina ion o se-
quences G1(k) and G2(k,L), de ined in Eq. (11), decom-
pose Pk
Lin o a se o adjacen ansposi ion, ha is, Pk
L=
G1(k)G2(k,L). Fo ha , we will i s b ie ly discuss how o
s a e he p oblem in e ms o he ma ix ep esen a ion o he
pe mu a ion g oup [21]. Fo he sake o cla i y, we will deno e
by Tk he ma ix ep esen a ion o a ansposi ion τk.
The p oblem we a e sol ing can be s a ed as ob aining a
ini e se o ansposi ions {Tk}M
k=1 ha ans o ms he ec o b
wi h componen s biin o he ec o bwi h componen s bPk
L(i).
Tha is, M
k=1Tkb=b. Howe e , since T−1=T, his se o
ansposi ions will hold ha 1
k=MTkb=b. The decompo-
si ion o PL
kis hen Pk
L=τ1◦τ2◦···◦τk, whe e we use
b◦a=b(a) o deno e he o de o he ope a ions.
The only es ic ion we will impose in he a ailable ans-
posi ions is ha hey need o be adjacen . Deno ing by T(i,j)
he ansposi ion τi,j ha ansposes he elemen s iand j,
we no e ha T(i,j)b ansposes he elemen s o bin he
posi ions iand j. Hence, we can use algo i hms, such as he
Bubble Sho algo i hm o i s pa allelized e sion, o di ec ly
ob ain an op imized se o ansposi ions Tk ha sho s a gi en
ec o b. Indeed, he se o ansposi ions G1and G2ha e
been ob ained using hose echniques, hough we conside ed
i necessa y o gi e a closed o mula, which we now p o e o
be co ec .
We will now de ine ci,las he ope a ion ha ul ills
ci,lPk
L(j)=⎧
⎨
⎩
Pk
L(j)i j= i,l
Pk
L(i)i j=l
Pk
L(l)i j=i.
(A1)
Tha is, i changes he posi ion o he en y iwi h he en y
l. We will de ine g1and g2as G1and G2in Eqs. (10) and
(11) bu wi h all he τij ope a ions eplaced by cij.F om he
ep esen a ion we see ha eplacing all he ansposi ions, τij
o cij, makes he new g oup o ope a ions g1and g2 o ul ill
he equa ion
g1◦g2◦Pk
L=1,(A2)
whe e 1 s ands o he iden i y pe mu a ion. This again e-
sembles o sho ing and a ay de ined by Pk
Lwhe e he ope -
a ions cij a e he changes in posi ions made o ha a ay. We
con inue by ealizing ha
Pk
L(j)=Pk
2k(j)i j⩽2k,
PL−2k
L−2k(j−2k)+2ki j>2k,(A3)
ha is, we ob ain wo commu ing pe mu a ions. This is why
wo commu ing g oups o sequences, G1(k) and G2(k,L),
a ise. No e ha he second pe mu a ion, which can be e-
ga ded as PL
Lwi h L=L−2k, is ob ained om
G2(L)=SL−1
L−1→L◦SL−2
L−2→L−1◦···
×···◦S4
4→L−1◦S3
3→L◦S2
2→L−1◦S1
1→L.(A4)
G2(L) di e s om G2(k,L) by a ac o o 2k ha appea s in
all he sequences. This ex a ac o in G2(k,L) comes om
Eq. (A3), whe e i appea s adding o he pe mu a ion PL−2k
L−2k.I
has he e ec o changing he ac ion o all ansposi ions om
τij o τi+2kj+2k.
We will only p o e how o sho he pe mu a ion ha
ul ills g1(k)◦Pk
2k=1 because p o ing ha g
2(L)◦PL
L=1
equi es he same s eps. We will p o e his by induc ion.
Since g1(1) is he iden i y ope a ion and P1
2is he iden i y
pe mu a ion, i is clea ha g1(1) ◦P1
2=1. We now sup-
pose ha g1(k−1) ◦Pk−1
2k−2=1 and we p o e ha , wi h his
condi ion, g1(k)◦Pk
2k=1. Since g1(k)◦Pk
2k=g1(k−1) ◦
s1→2k−2◦s2→2k−1◦Pk
2k, i su ices o p o e ha s1→2k−2◦
s2→2k−1◦Pk
2k=Pk−1
2k−2, whe e si→jis de ined in Eq. (9)bu
wi h all he τij ope a ions eplaced by cij ope a ions. No ice
ha s2→2k−1has he e ec o changing he posi ion o all
033103-8
ENHANCED CONNECTIVITY OF QUANTUM HARDWARE … PHYSICAL REVIEW RESEARCH 2, 033103 (2020)
he en ies in Pk
Lexcep o he las and he i s one. All he
numbe s in an odd posi ion change o hei le posi ion and all
he numbe in a e en posi ion change o hei igh posi ion.
Hence, he pe mu a ion ob ained om π1=s2→2k−1◦Pk
2k
esul s in
π1(j)=⎧
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎩
Pk
2k(j+1) i je en
Pk
2k(j−1) i jodd
Pk
2k(1) i j=1
Pk
2k(2k)i j=2k
=⎧
⎪
⎪
⎨
⎪
⎪
⎩
Pk
2k(j+1) i je en
Pk
2k(j−1) i jodd
2ki j=2k
,
whe eweusedPk
2k(0) =Pk
2k(1).
We now compu e he ope a ion π2=s1→2k−2◦π1, which
changes he posi ion o all he en ies excep om he las wo.
Hence,
π2(j)=⎧
⎪
⎨
⎪
⎩
π1(j−1) i je en
π1(j+1) i jodd
π1(2k−1) i j=2k−1
π1(2k)i j=2k
=⎧
⎪
⎪
⎨
⎪
⎪
⎩
Pk
2k(j−2) i je en
Pk
2k(j+2) i jodd
2k−1i j=2k−1
2ki j=2k
=⎧
⎨
⎩
Pk−1
2k−2(j)i j<2k−1
2k−1i j=2k−1
2ki j=2k
.
Since g1(k−1) does no a ec o he posi ions 2kand 2k−1,
g1(k−1) ◦π1(j)
=⎧
⎨
⎩
g1(k−1) ◦Pk−1
2k−2(j)i j<2k−1
2k−1i j=2k−1
2ki j=2k
=1.
This comple es he p oo .
APPENDIX B: COLORED GRAPHS DEMONSTRATION
In o de o p o e ha we can selec i ely change he sign
o a coupling in an a bi a y leng h NN chain, we will use an
induc ion p ocess. I Lis he numbe o nodes in a NN chain,
hen k=L−1 is he numbe o couplings.
In e ing he couplings o he k=1 case is i ial. Sup-
posing ha we can selec i ely change he coupling sign o he
k=k−1, we will p o e ha we can do i o he case k=k.
The cons uc ion elies in he ac ha , in o de o change he
sign o he new coupling, i su ices o change he colo o
he newly added node. In his case, he colo o he new node
mus be di e en o m i s neighbo ’s colo , which can always
be achie ed in he NN case. I we do no wan o change he
sign o he kcoupling, we jus need o colo he new node as
i s neighbo .
APPENDIX C: DEMONSTRATION OF EQ. (12)
In his Appendix, we show how o ob ain he ga es de ined
in Eq. (12). Fo ha , we i s de ine a ious se o ga es ha
will simpli y he no a ion. We de ine ˜
S o be he se o iSWAP†
ga es ha a e ob ained om subs i u ing τij →iSWAP†
ij in
Eq. (9). A he same ime, ˜
Gis de ined o be he se o iSWAP†
ga es ha a e ob ained om subs i u ing S→˜
Sin Eqs. (10)
and (11). Hence, i holds ha
HPPk
L=˜
G1†˜
G2†HNN ˜
G2˜
G1,(C1)
whe e HNN is he NN Hamil onian and HP(Pk
L)is heZZ
in e ac ion Hamil onian desc ibed by he Hamil onian Pa h
Pk
L.
F(k,L) desc ibes he ga es be ween he NN Hamil onians
ha will be used o implemen he Hamil onian pa hs wi h
e ex pe mu a ion Pk
Land Pk=1
L. Hence,
F(k,L)
=⎧
⎪
⎨
⎪
⎩
˜
G†
1(k+1) ˜
G†
2(k+1,L)˜
G1(k)˜
G2(k,L)k∈1,L
2−1,
˜
G†
2(1,L)k=0,
˜
G1L
2k=L
2,
(C2)
whe e i is s aigh o wa d o p o e ha F(k,L) has he o m
desc ibed in Eq. (12) o k=0 and k=L
2.Fo k∈[1,L
2−
1], we will p o e ha he se o iSWAP ga es can be u he
simpli ied o ob ain Eq. (12).
We i s no e ha he ollowing equa ions hold om he
de ini ion o ˜
G,
˜
G1(k+1) =˜
G1(k)˜
S1→2k˜
S2→2k+1,(C3)
˜
G2(k,L)=˜
G2(k+1,L)˜
S2k+2→L−1˜
S2k+1→L.(C4)
Hence, i ollows ha
˜
G†
1(k+1) ˜
G†
2(k+1,L)˜
G1(k)˜
G2(k,L)
=˜
S†
2→2k+1˜
S†
1→2k˜
G†
1(k)˜
G†
2(k+1,L)˜
G1(k)˜
G2(k,L)
=˜
S†
2→2k+1˜
S†
1→2k˜
G†
2(k+1,L)˜
G2(k,L)
=˜
S†
2→2k+1˜
S†
1→2k˜
G†
2(k+1,L)˜
G2
×(k+1,L)˜
S2k+2→L−1˜
S2k+1→L
=˜
S†
2→2k+1˜
S†
1→2k˜
S2k+2→L−1˜
S2k+1→L,
whe e we used ha [ ˜
G†
1(k),˜
G†
2(k+1,L)] =0 and
˜
G†
1(k)˜
G1(k)=˜
G†
2(k+1,L)˜
G2(k+1,L)=I.
We now ha e ha
F(k,L)
=⎧
⎪
⎨
⎪
⎩
˜
S†
2→2k+1˜
S†
1→2k˜
S2k+2→L−1˜
S2k+1→Lk∈1,L
2−1,
˜
G†
2(1,L)k=0,
˜
G1L
2k=L
2,
(C5)
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