PHYSICAL REVIEW RESEARCH 4, 043089 (2022)
Co-Design quan um simula ion o nanoscale NMR
Manuel G. Algaba ,1,*,† Ma io Ponce-Ma inez ,1,2,*,‡ Ca los Munue a-Ja aloy ,3Vicen e Pina-Canelles ,1
Manish J. Thapa ,1B uno G. Take ani ,1Ma in Leib ,1Inés de Vega ,1,2Jo ge Casano a ,3,4and He manni Heimonen5
1IQM Quan um Compu e s, Nymphenbu ge s . 86, 80636 Munich, Ge many
2Depa men o Physics and A nold Somme eld Cen e o Theo e ical Physics, Ludwig-Maximilians-Uni e si ä München,
The esiens asse 37, 80333 Munich, Ge many
3Depa men o Physical Chemis y, Uni e si y o he Basque Coun y UPV/EHU, Apa ado 644, 48080 Bilbao, Spain
4IKERBASQUE, Basque Founda ion o Science, Plaza Euskadi 5, 48009 Bilbao, Spain
5IQM Quan um Compu e s, Keila an a 19, FI-02150 Espoo, Finland
(Recei ed 14 Feb ua y 2022; e ised 5 Augus 2022; accep ed 28 Sep embe 2022; published 8 No embe 2022)
Quan um compu e s ha e he po en ial o e icien ly simula e he dynamics o nanoscale NMR sys ems. In
his wo k, we demons a e ha a noisy in e media e-scale quan um compu e can be used o simula e and p edic
nanoscale NMR esonances. In o de o minimize he equi ed ga e ideli ies, we p opose a supe conduc ing
applica ion-speci ic Co-Design quan um p ocesso ha educes he numbe o SWAP ga es by o e 90% o
chips wi h mo e han 20 qubi s. The p ocesso consis s o ansmon qubi s capaci i ely coupled ia unable
couple s o a cen al co-plana wa eguide esona o wi h a quan um ci cui e ige a o (QCR) o as esona o
ese . The QCR implemen s he nonuni a y quan um ope a ions equi ed o simula e nuclea hype pola iza ion
scena ios.
DOI: 10.1103/PhysRe Resea ch.4.043089
I. INTRODUCTION
Compu e simula ions a e he backbone o scien i ic
esea ch and echnological de elopmen . Quan um compu -
e s p omise in he long e m o enable simula ions o
sys ems ha a e in ac able o e en he la ges supe com-
pu e s [1,2]. Cu en ly, scien is s ha e access o so-called
noisy in e media e-scale quan um (NISQ) compu e s [3], ha
p esen limi ed qubi coun s wi hou e o co ec ion. While
applica ions o e o -co ec ed quan um compu e s a e well
es ablished, use cases whe e NISQ de ices migh achie e
quan um ad an age a e s ill elusi e [4]. In he sea ch o
hese ea ly applica ions, he p oblem mus i he ha dwa e,
and he ha dwa e mus enable implemen a ion wi h minimal
o e heads.
Applica ion-speci ic in eg a ed chips (ASICs) a e highly
specialized p ocesso s op imized o speci ic p oblems when
execu ion speed, powe e iciency, o minia u iza ion is o
u mos impo ance [5]. A p ominen example whe e compu-
a ional speed and ene gy e iciency a e op imised h ough
he use o ASICs is aining o a i icial neu al ne wo ks us-
ing enso p ocessing uni s [6,7]. Building a gene al-pu pose
*Bo h au ho s con ibu ed equally o his wo k.
†Co esponding au ho : [email p o ec ed]
‡Co esponding au ho : [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.
quan um compu e capable o i aling he mos powe ul clas-
sical compu e s has p o en o be a di icul ask, so i is
likely ha he i s de ices eaching use ul quan um ad an-
age will use quan um ASICs, also called Co-Design quan um
compu e s.
A good example o a p oblem wi h sui able s uc u e o
simula ion by quan um compu e s is nanoscale nuclea mag-
ne ic esonance (NMR) [8]. The p oblem can be desc ibed by
a numbe o mu ually in e ac ing spins, which na i ely map o
he qubi s o a quan um compu e , he eby ci cum en ing he
o e heads in mapping he p oblem o qubi s, such as in he
case o e mions [9].
In gene al, as and eliable quan um simula ions o in-
e ac ing spin sys ems would imp o e he in e p e abili y o
solid-s a e NMR and elec on spin esonance (ESR) spec a,
whe e ad anced nume ical echniques p esen e y limi ed
pe o mance [10]. This shows he po en ial o quan um com-
pu e s wi h a mode a e numbe o qubi s o shed ligh on he
dynamics o hese impo an sys ems. A Co-Design quan um
compu e ha minimizes algo i hm implemen a ion o e heads
could be he i s me hod o access hese simula ions. No e
ha , o he NMR p oblems, such as ze o- ield NMR [11] and
Hamil onian lea ning [12], ha e al eady a ac ed esea ch
on how quan um compu e s can be used o ackle hem and
me hods based on Bayesian compu a ion [13] and gene a i e
models [14] ha e been de eloped o compu ing NMR spec a
as well.
NMR echniques ha e a p o ound impac in esea ch a eas
such as ma e ial science, chemis y, biology, and medicine
[15]. Recen ly hey ha e app oached he nanoscale h ough
solid-s a e quan um senso s such as he ni ogen acancy
(NV) cen e in diamond [16]. This is a pa icula ly powe ul
2643-1564/2022/4(4)/043089(20) 043089-1 Published by he Ame ican Physical Socie y
MANUEL G. ALGABA e al. PHYSICAL REVIEW RESEARCH 4, 043089 (2022)
FIG. 1. NV cen e wi h a mic owa e d i e in e ac ing wi h wo
mu ually in e ac ing 13Cnuclei in a magne ic ield
BZ, co esponding
o he Hamil onian in Eq. (1) o M=1andN=2.
quan um de ice, as i enables de ec ion and con ol o nea by
nuclea spins wi h nanoscale esolu ion [17]. Applica ions o
he de ice a e, e.g., he p ecise de e mina ion o he s uc u e
and dynamics o nuclea ensembles such as p o eins [18],
inding in e -label dis ances ( ia, e.g., Bayesian analysis o
he NV esponse) in elec onically labeled biomolecules [19],
and he explo a ion o bespoke mic owa e (MW) sequences
ha e icien ly ans e NV cen e pola iza ion o he nuclea
en i onmen . Hype pola iza ion (i.e., pola iza ion beyond ha
o a he mal s a e in a magne ic ield) o nuclea spins in dia-
mond p esen s he po en ial o de elop new and sa e con as
agen s o magne ic esonance imaging. This p oblem, which
we aim o add ess h ough simula ion by a quan um compu e ,
could lead o imp o ed de ec ion o di e en mal o ma-
ions in issues—such as hea o b ain—wi hou he need
o deli e ionizing adia ion, in con as o o he echniques
[20].
This manusc ip desc ibes a Co-Design p ocess o a
quan um chip able o e icien ly simula e nanoscale NMR
scena ios. I is s uc u ed in h ee main pa s, each o which
is a c ucial s ep in he Co-Design p ocess: (1) iden i ying
he p oblem (Sec. II), which he e is simula ing a nanoscale
NMR sys em o hype pola izing nuclea spins; (2) choosing
an algo i hm o he nanoscale NMR p oblem and showing
ha a s a - opology chip implemen s i wi h minimal o e head
(Sec. III); and (3) Co-Designing he co esponding quan um
chip using a cen al esona o bus (Sec. IV). The sec ions a e
ollowed by esul s and discussions (Sec. V) and an ou look
(Sec. VI).
II. NANOSCALE NMR: HYPERPOLARIZATION
Le us conside a sys em consis ing o Mni ogen- acancy
(NV) cen e s and Nca bon-13 iso opes in he p esence o a
d i ing ield and an ex e nal magne ic ield
BZ. NV cen e s
and nuclei a e all e ec i ely desc ibed as spin-1/2sys ems.
The ep esen a ion o such a sys em o M=1, N=2is
shown in Fig. 1. Fo simplici y, we conside he NV cen e s
aligned wi h he ex e nal magne ic ield, leading o he ol-
lowing Hamil onian:
H=
M
j=1
δjσz
j−
N
k=1
ωc
k·
Ik+
M
j=1
N
k=1
σz
j
2
Ajk ·
Ik
+
N
k>k
gkkIz
kIz
k−1
4(I+
kI−
k+I−
kI+
k)
+
M
j>j
hjjσz
jσz
j−2(σ+
jσ−
j+σ−
jσ+
j)+Hd .(1)
In Eq. (1), we ind he spin ope a o s in he join Hilbe
space C2(M+N)o NV cen e s and nuclei:
σμ
j=1⊗···⊗1⊗
j hpos.
σμ⊗1⊗···⊗1
M ac o s
⊗1⊗...⊗1
N ac o s
,
Iμ
k=1⊗...⊗1
M ac o s ⊗1⊗···⊗1⊗
(M+k) hpos.
1
2σμ⊗1⊗···⊗1
N ac o s
,
whe e (σμ)2×2,μ∈{x,y,z}is he co esponding 2 ×2 Pauli
ma ix on he j h NV cen e and he k h nucleus espec-
i ely, and 1is he 2 ×2 iden i y ma ix. Acco dingly, σ±
j=
σx
j±iσy
j
2(I±
k=Ix
k±iIy
k)a e he j h NV cen e (k h nucleus) lad-
de ope a o s. The e m δjis he de uning o he j h NV cen e
wi h espec o he mic owa e d i e Hd . The hype ine cou-
pling ec o
Ajk ep esen s he coupling be ween he j h NV
cen e and he k h nucleus, while
ωc
k=γc
BZ−1
2M
j=1
Ajk
is he modi ied La mo equency o he k h nucleus wi h
he 13Cgy omagne ic a io γc≈(2π)×10.7MHz/T, gkkis
he coupling be ween he k h and k h nuclei, and hjjis he
coupling be ween he j h and j h NV cen e s.
No e ha , Eq. (1) is exp essed in a o a ing ame wi h
espec o he ee NV Hamil onian, while Hd ep esen s
an ex e nal d i ing uned nea esonance wi h a ce ain NV
ene gy ansi ion. The de i a ion o Eq. (1) can be ound in
Appendix A.
In o de o hype pola ize a diamond sample a oom
empe a u e, he NV cen e s a e i s op ically pola ized em-
ploying lase ligh , and hen hei s a e is ans e ed o he
su ounding nuclei wi h he aid o a ailo ed mic owa e
adia ion scheme. The ini ial s a e o he nuclei in a oom-
empe a u e sample is well desc ibed by a ully mixed s a e
due o he small ene gy spli ing o he nuclea spins. By
eini ializing he NV cen e s and epea ing his p ocedu e, he
pola iza ion ans e ed in o he sample can be ampli ied. In
his pape , we will conside he quan um simula ion o he
pola iza ion ans e mechanism and s udy wo di e en d i -
ing schemes ac ing on he NV cen e s in a oom- empe a u e
diamond.
The i s d i ing scheme is a con inuous d i ing whose
Hamil onian in he o a ing ame men ioned ea lie is Hd =
2σφ, whe e σφ=e−iφ|10|+eiφ|01|=e−iφσ−+eiφσ+,
φa phase, he Rabi equency and he ke s |1and |0
a e he eigen ec o s o he ope a o σzwi h eigen alues ±1
espec i ely. The se {|0,|1} is called he compu a ional
basis o he s a e space o a wo le el sys em, and will be
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CO-DESIGN QUANTUM SIMULATION OF NANOSCALE NMR PHYSICAL REVIEW RESEARCH 4, 043089 (2022)
ou s anda d choice o a basis, |0≡(1,0) and |1≡(0,1) .
NV-nucleus pola iza ion ans e is achie ed when he Rabi
equency ma ches he modi ied nuclea La mo equency
(i.e., when =|
ωc|), leading o he Ha mann-Hahn double-
esonance condi ion [21]. Fo a single NV cen e and nucleus,
he Hamil onian in Eq. (1) educes, in an in e ac ion pic u e,
o HI=A⊥
4(|+−|I++|−+|I−), whe e ±| = 0|±1|,
which shows a pola iza ion ans e mechanism wi h he e -
ec i e ans e a e A⊥
4(a de ailed de i a ion can be ound in
Appendix B).
The second ype o d i ing we conside is a pulsed-d i ing
scheme, Hd =( )
2σφ, whe e ( )isa aino π-pulses, such
as he Ca -Pu cell-Meiboom-Gill sequence [22,23]o he
XY8 sequence [24,25]. We conside pulses wi h a negligible
wid h compa ed o he ime spacing τbe ween he π-pulses.
I τis selec ed such ha τ=nπ
|
ωc|(nbeing an a bi a y in ege
numbe ) and he pulses a e e enly spaced one inds ha , in
an in e ac ion pic u e, o a single nucleus and NV cen e , he
Hamil onian educes o HI=αA⊥σzIx, whe e αis a ac o ha
depends on he in ege n(see Appendix B). A phase imp in ed
on he pulse sequence h ough a ime delay u ns he in e ac-
ion in o HI=αA⊥σzIy. By combining bo h sequences wi h
he app op ia e o a ions o e he NV cen e , he pola iza ion
ans e in e ac ion HI=−αA⊥
4(σ+I−+σ−I+) is achie ed
(see Appendix Band Re . [26] o mo e de ails).
Rega ding common e o sou ces, NV cen e s loca ed a
di e en posi ions in he diamond la ice expe ience s ess
condi ions ha lead o local ene gy de ia ions om he ze o-
ield spli ing. The co esponding e m in Eq. (1)is he
de uning δj. Ano he common ype o impe ec ion appea s
due o una oidable luc ua ions o he Rabi equency o he
d i ing. This luc ua ion can be modelled as an O ns ein-
Uhlenbeck (OU) p ocess [27], which has been shown o be
an accu a e desc ip ion o NV cen e s [28]. I is a Gaussian
p ocess o he ollowing o m [29]:
X( + )=X( )e− /τ +cτ
2(1 −e−2 /τ )1/2
N( ),(2)
whe e is he ime s ep, τ he co ela ion ime, c he
di usion cons an o he p ocess, and N( ) a empo ally
unco ela ed no mally dis ibu ed andom a iable. I is a di-
mensionless e m, which yields an e ec i e Rabi equency
o (1 +X). Nei he o he sys em e o ypes lead o con-
side able o e heads in a simula ion on a quan um compu e .
Finally, 13Cnuclea spin decay is no a ele an e o sou ce
on he ime scale o he p o ocol, since i is o he o de o
seconds [30], while he hype pola iza ion p ocess ope a es in
he o de o mic oseconds.
III. CO-DESIGN ALGORITHM
In his sec ion, we p o ide an in-dep h desc ip ion o ou
Co-Design algo i hm, s a ing wi h he choice o a simula ion
echnique, ollowed by a sho lis ing o ha dwa e assump-
ions ela ed o he allowed qubi ope a ions (ga es and ese s),
as well as he noise and e o s p esen in he physical NMR
sys em and in he quan um compu e . Subsequen ly, he al-
go i hm componen s a e in oduced. We end he sec ion wi h
a discussion on layou and ga e-le el op imiza ion. The high-
le el s uc u e o he simula ion p o ocol is shown in Fig. 2(a).
A. Simula ion echnique
The bes es ablished digi al quan um simula ion echnique
is based on decomposing he ime-e olu ion ope a o in o
single-qubi and wo-qubi ga es h ough he Lie-T o e -
Suzuki o mula [31], known as T o e iza ion. To simula e
ou p oblem on a quan um compu e , we base ou s a egy on
egula T o e iza ion [2] bu we also explo e he andomized
T o e iza ion me hod qDRIFT [32] in Appendix C. O he ,
mo e NISQ-speci ic, simula ion echniques such as he a ia-
ional quan um simula o [33], he quan um assis ed simula o
[34], nume ical quan um ci cui syn hesis [35], and a ple ho a
o o he quan um algo i hms [4] can also be used as simula ion
me hods.
One ad an age o T o e iza ion o e some o hese NISQ
me hods is ha i closely ollows he eal ime e olu ion
o each ime s ep. This is pa icula ly impo an o pulsed-
d i ing schemes, whe e he ee e olu ion in be ween di e en
pulses always s a s wi h a di e en ini ial s a e. Va ia ional
and quan um assis ed me hods would hen equi e ha each
in e pulse e olu ion is sol ed independen ly, making hem
imp ac ical o he p oblem.
A second ad an age o T o e iza ion is ha i s complexi y
and p ecision a e s aigh o wa d o analyze. The T o e iza-
ion p ocedu e can also be expanded o highe o de s, and
symme ized expansions con e ge mo e apidly and educe
he e o wi h espec o he con inuum ime limi [36].
B. Ha dwa e assump ions
1. Na i e ga es
The ha dwa e o he quan um simula ion plays a majo
ole in choosing he op imal quan um algo i hm and i s spe-
ci ic implemen a ion. In ou case, we conside a quan um
compu e based on supe conduc ing qubi s wi h he ollowing
na i e single-qubi ga e se :
Rxy(φ,θ)=e−i(cos φX+sin φY)θ
2and (3)
Rz(θ)=e−iZ θ
2,(4)
whe e X,Y, and Za e Pauli ope a o s on he supe conduc ing
ansmon qubi s. The Rxy(φ,θ) can physically be imple-
men ed h ough a mic owa e d i e [37]. The ga e Rz(θ)on he
o he hand does no need o be implemen ed di ec ly, bu can
be pe o med i ually by uning he phase o he subsequen
ga es applied on he qubi [38]. This educes he numbe o
single-qubi ga es (SQGs) ha need o be implemen ed.
The na i e wo-qubi ga e (TQG) ha a ises om he
supe conduc ing sys em Hamil onian shown in Sec. IV and
Appendix G, is a con inuously pa ame e ized con olled-Z
(CZ) in e ac ion [39], which can be ans o med h ough local
i ual Rz o a ions in o he o m o a ZZ in e ac ion:
UZZ(φ)=⎛
⎜
⎜
⎝
e−iφ00 0
0eiφ00
00eiφ0
000e−iφ
⎞
⎟
⎟
⎠.(5)
043089-3
MANUEL G. ALGABA e al. PHYSICAL REVIEW RESEARCH 4, 043089 (2022)
FIG. 2. (a) Ske ch o he o e all ope a ion o he simula ion algo i hm o one NV cen e and wo nuclei, wi h con inuous d i ing;
(b) co esponding ga e sequence o one T o e s ep on a s a - opology chip o nonin e ac ing nuclei. HSQG e e s o he single-qubi -ga e
componen o he Hamil onian, Ax,y,z
j,1pa ame e s a e he a ious coupling s eng hs o he simula ed sys em, and he X nd ga es e e o X
ga es applied wi h a 50% p obabili y o p epa e an e ec i e ully-mixed s a e. The ini ial s a e p epa a ion can also be pe o med using he
al e na i e andom-phase app oxima ion-inspi ed me hod. NV ini is an ini ial-s a e p epa a ion using single-qubi ga es o he s a e equi ed
by he d i ing scheme. De ails o he ci cui componen s can be ound in Appendix Dalong wi h a igu e ep esen ing he pulsed d i ing case.
E en hough he ZZ-in e ac ion and he con olled-Zin-
e ac ions appea di e en , hei physical implemen a ion is
iden ical since hey a e ela ed h ough i ual Rz o a ions
which come a no addi ional cos . Sec ion IV goes in o mo e
dep h on he wo-qubi -ga e implemen a ion on ou Co-Design
quan um chip.
2. Qubi ese
In he hype pola iza ion p ocess, he s a e o he NV needs
o be e-ini ialized a e each cycle. I is he e o e necessa y
o be able o ese he s a e o he qubi ep esen ing he NV
cen e in he quan um compu e . A qubi ese ope a ion can
be de ined by wo K aus ope a o s:
K ese
1=10
00
,K ese
2=01
00
.(6)
On supe conduc ing ha dwa e his can be ealized h ough
connec ing a quan um ci cui e ige a o (QCR) o each ci -
cui elemen ha needs o be ese [40–43]. Di e en ese
schemes a e discussed in Sec. IV B.
3. Noise and e o s
In his pape , we show ha he simula ion can ole a e
he noise o he quan um p ocessing uni (QPU), and ha
he simula ion does no equi e la ge o e heads o imple-
men impe ec ions p esen in he nanoscale-NMR sys em, as
discussed in Sec. II. We will e e by sys emimpe ec ions o
e ec s in he nanoscale NMR sys em only, while he QPU is
a ec ed by noise, e e ing o he e ec o he en i onmen on
he qubi s, and e o s, e e ing o inaccu acies o ga es.
In ou simula ion o he algo i hm, we use he mos com-
mon noise models o supe conduc ing ansmon qubi s [37],
namely, an ampli ude damping channel modelled by he K aus
ope a o s:
Kamp
1( )=|00|+1−p( )|11|=10
0√1−p( ),
Kamp
2( )=p( )|01|=0√p( )
00
,(7)
wi h p( )=1−exp(− /T1) and T1=60 μs, and a pu e de-
phasing channel ep esen ed by he K aus ope a o s:
Kdeph
1( )=10
0√1−p( ),Kdeph
2( )=10
0√p( ),
(8)
wi h p( )=1−exp(−( )) and ( ) gi en by he exp ession
( )= 2
2∞
0dωI(ω) co anh( βω
2)sinc
2(ω
2) whe e βis he in-
e se empe a u e o he en i onmen . We chose he spec al
unc ion I(ω) o be o he ype 1/ [37], and T2=60 μs.Ad-
di ionally, each ga e ope a ion is assumed o be calib a ed up
o a wo-qubi -ga e (TQG) e o εTQG ∈[10−4,10−2], wi h he
induced e ec i e noise modelled by a depola izing channel
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CO-DESIGN QUANTUM SIMULATION OF NANOSCALE NMR PHYSICAL REVIEW RESEARCH 4, 043089 (2022)
de ined o single-qubi ga es by he K aus ope a o s:
Kdepol
1=1−pI,
Kdepol
2=p/3X,
Kdepol
3=p/3Y,
Kdepol
4=p/3Z,
(9)
and o wo-qubi ga es by an analogous exp ession wi h he
enso p oduc s o wo Pauli ma ices and he coe icien s
√1−p o he iden i y and √p/15 o he o he ope a o s.
Single-qubi -ga e (SQG) e o s εSQG a eassumed obeone
o de o magni ude lowe han TQG e o s.
C. Algo i hm componen s
Ou simula ion o he nanoscale NMR p oblem ollows he
gene al s uc u e shown in Fig. 2(a). I s a s by ini ializing
he s a es o all qubi s, acco ding o whe he hey ep esen
a nucleus o a NV cen e , hen e ol ing hem using T o e
s eps, ollowed by ese and e-ini ializa ion o he qubi s
ep esen ing NV cen e s. The cycle o ime e olu ion and
e-ini ializa ion is hen epea ed as many imes as he p o ocol
calls o . Finally he qubi s a e measu ed, and he pola iza ion
o he NV cen e s and nuclei a e ex ac ed as he expec a ion
alues o he qubi ep esen ing each elemen . Figu e 2(a)
shows he ci cui s o he case o con inuous d i ing, while
he de ails o pulsed d i ing schemes a e shown in Fig. 10(a)
in Appendix B. In he ollowing, we go h ough hese s eps in
mo e de ail o he case o a single NV cen e .
1. Ini ial s a e p epa a ion
To enable he pola iza ion ans e , i is necessa y o p e-
pa e he NV cen e in a speci ic ini ial s a e ha depends on
he d i ing scheme. Fo he con inuous-d i ing scheme, i is
he |+ o |− s a e, and o he pulsed-d i ing scheme, i is
one o he wo compu a ional basis s a es, |0o |1.
Fo a diamond a oom empe a u e, he ini ial s a e o
he nuclea spins is well desc ibed by a ully mixed s a e
ρmixed =1⊗N
2N, whe e 1⊗Nis he 2N×2Niden i y ma ix. The
s a e can be app oxima ed by unning he algo i hm se -
e al imes, each ime wi h a di e en ini ial s a e ob ained
by applying Xga es andomly on he qubi s ep esen ing
nuclei. A as e al e na i e o his sampling is he andom-
phase-app oxima ion-inspi ed me hod, desc ibed in Re . [44],
and in oduced in o quan um compu ing in Re . [45]. In his
me hod, he qubi s a e all p epa ed in an equal supe posi ion
by applying Hadama d ga es, and hen he phases a e an-
domized h ough he applica ion o andom phase ga es. The
me hod e ec i ely educes he p e ac o in he scaling o he
sampling e o [45].
2. Time e olu ion
We choose o implemen he ime e olu ion gene a ed by
he Hamil onian in Eq. (1) h ough T o e iza ion. Fo ha ,
he Hamil onian is ew i en in e ms o qubi Pauli ope a o s
and a anged in o noncommu ing e ms o an op imal T o e
spli ing. The esul ing ci cui , which pe o ms one T o e
s ep o he e olu ion in he con inuous d i ing case, is de-
pic ed in Fig. 2(b). I consis s o a se o ini ial single-qubi
ga es, including he ones co esponding o he d i ing and he
de uning o he NV cen e , ollowed by h ee wo-qubi ga es
pe nucleus. The e a e h ee ypes o in e ac ion e ms, o he
o m XZ,YZ and ZZ, when no in e nuclea in e ac ions a e
conside ed. Wi h in e ac ions he e a e a o al o i e in e -
ac ion e ms. Ou na i e ga e se only includes one ype o
wo-qubi in e ac ion as explained in Sec. III B 1. The e o e
some SQGs need o be applied in o de o con e he in e -
ac ion e ms in o he igh o m, as discussed in Appendix D.
Unde speci ic ci cums ances, some TQGs can be emo ed by
o a ing he Hamil onian in o a mo e sui able basis as shown
in Appendix E.
3. Cycles and ese
The dynamics o he sys em is known o p oduce an ex-
change o pola iza ion be ween he NV cen e and he nuclei.
This exchange is oscilla o y, and he e o e choosing a p ope
s opping ime is impo an in o de o achie e an e ec i e
pola iza ion ans e om he NV cen e o he nuclei. In p ac-
ice, a subop imal ans e ime can su ice, and he p o ocol
is hen epea ed se e al imes by ese ing he NV cen e o i s
ini ial s a e and le ing he sys em e ol e unde he d i e again.
Due o he e-ini ializa ions he ull e olu ion o he sys em
is nonuni a y and a ne gain o pola iza ion o he sys em is
enabled.
This s uc u e is ep esen ed in he quan um ci cui in
Fig. 2(a) by he epea ed T o e e olu ion, ollowed by ese
ope a ions on he qubi ep esen ing he NV cen e , and a
single-qubi ga e o p epa e he ini ial s a e o he d i ing
p o ocol.
D. Layou op imiza ion
When implemen ing a quan um algo i hm on a supe con-
duc ing QPU, he plana qubi connec i i y o ces us o sol e
he qubi - ou ing p oblem by in oducing addi ional SWAP
ga es o connec dis an qubi s. In his sec ion, we s udy he
ad an ages o an op imized chip opology, a s a opology,
o e a squa e-g id a ay o qubi s in e ms o educing he
numbe o SWAP ga es ha mus be inse ed o un he algo-
i hm in Fig. 2on he de ice. Di e en opologies will imply
di e en coun s o SWAPs added on op o he ga es a ising
om he algo i hm i sel , as shown in Fig. 3.OnaNISQ
de ice, his implies di e en compu a ional p ecision o he
same ga e e o magni udes. We choose he SWAP coun
as ou me ic o compa e di e en opologies, as commonly
ga es ha e ideli ies limi ed by calib a ion. The e o s could
be due o c oss alk, leakage, o il e ing causing dis u bances
o he con ol signals. Unde his scena io we wan o mini-
mize he ga e coun . On he o he hand, o a highly uned up
de ice whose ga es a e limi ed by qubi cohe ence imes, i
would be op imal o minimize he ci cui dep h ins ead o he
TQG coun .
Assuming he ga e e o s a e independen , he o al e o
will be bounded by:
εga es =1−(1 −εTQG )NTQG (1 −εSQG)NSQG ,(10)
whe e NTQG is he numbe o wo-qubi ga es, NSQG he
numbe o single-qubi ga es, and εSQG is he SQG e o .
043089-5
MANUEL G. ALGABA e al. PHYSICAL REVIEW RESEARCH 4, 043089 (2022)
FIG. 3. (a) Th ee s eps o he SWAP pa e ns in a i e-qubi linea
chain displayed om op o bo om. G een (blue) a ows ep esen
he SWAP pa e n o he case wi h (wi hou ) in e nuclea in e ac-
ions. The g een pa e n is known as he “odd-e en” SWAP pa e n.
The numbe s a e exp essed acco ding o he blue pa e n, whe e label
0 ep esen s he posi ion o he NV cen e . (b) S a chip opology wi h
he SWAP pa e n o he in e ac ion wi h in e nuclea in e ac ions.
Consequen ly, educing he ga e coun , especially NTQG, has
an exponen ial e ec on he p ecision o he compu a ions,
unde lining he e ec o minimizing he SWAP ga e o e head.
As SWAP ga es a e no na i e o he ha dwa e, bu mus be
compiled ou o h ee CZ ga es, hei e ec i e e o a e is
also much highe han hose o na i e ga es.
1. Squa e g id
A common choice in supe conduc ing quan um chips is
he squa e g id o qubi s. I has high connec i i y and is
sui able o pe o ming he su ace code e o co ec ion when
scaled o la ge enough qubi coun s wi h as measu emen and
eedback [46]. The qubi ou ing p oblem on a squa e g id
can be ackled using a ious nume ical app oaches [47–50].
Howe e , hese me hods a e ine icien . In ou case, a ailo ed
SWAP ou ing me hod, shown in Fig. 3(a), has been chosen
and de eloped in Appendix F ha can be shown o be well
sui ed om wo pe spec i es. Fi s , a compa ison agains he
ci ed nume ical app oaches (shown in Appendix F) e eals
FIG. 4. The ( op) panel shows he pe cen age o SWAP ga es
sa ed by using a s a opology ins ead o a squa e g id o nqubi s
o he cases wi h and wi hou in e nuclea in e ac ions. The (bo om)
panel shows he o al TQG coun agains he qubi coun in he
in e ac ing case o he squa e g id and he s a a chi ec u e.
ha ou ou ing me hod is be e in e ms o numbe o ga es.
Second, i is comple ely de e minis ic and does no ely on
expensi e nume ical op imiza ion me hods. I can also be
shown no o be a om op imal: on a squa e g id each
qubi has a mos ou nea es neighbo s, implying ha any
SWAP ope a ion p o ides a mos h ee new neighbo s. Fo
an all- o-all (ATA) in e ac ing Hamil onian he e a e n2/2
in e ac ions, o leading o de , o a simula ion pe o med on
nqubi s (co esponding o Nnuclei and one NV cen e ). This
implies a lowe bound o a leas n2/6 SWAPs o any SWAP
pa e n on he squa e g id opology. Ou SWAP pa e n wi h
n2/2 SWAPs, discussed in Appendix F, is hus no a om
op imal.
2. S a a chi ec u e
A s a opology allows o implemen he simula ion o
he simpli ied case wi hou in e nuclea in e ac ions di ec ly,
wi hou any SWAP ga es. Wi h in e nuclea in e ac ions con-
side ed, we s ill ind a educ ion in SWAP ga es as compa ed
o he squa e g id opology, as shown in Fig. 3(b). This e-
duc ion comes om he SWAP ou ing we implemen , ha
consis s o making he qubi 0 in Fig. 3(b) in e ac wi h all he
ex e nal qubi s and hen swap i s s a e wi h ha o qubi 1 and
epea his p ocess un il all in e ac ions ha e been pe o med.
This allows us o use only n−1 SWAP ga es. The pe cen age
o SWAP ga es ha can be sa ed can be obse ed in Fig. 4.
Howe e , his imp o emen in he numbe o ga es comes
wi h a p ice o pay in he dep h o he algo i hm. We can only
043089-6
CO-DESIGN QUANTUM SIMULATION OF NANOSCALE NMR PHYSICAL REVIEW RESEARCH 4, 043089 (2022)
TABLE I. O e heads in oduced by he decomposi ion o UZZ(φ)
ga es in o di e en examples o na i e TQGs in supe conduc ing de-
ices. The single-qubi -ga e (SQG) coun includes only Rxy o a ions,
as he Rz o a ions can be implemen ed i ually.
UZZ (φ)UZZ (−π/4) CZ(π)CNOT
TQGs 1 2 2 2
SQGs 0 5 3 1
do one TQG a a ime in he s a chip and we ha e 3
2n(n−1)
TQGs om simula ing he physical in e ac ions and 3(n−2)
TQGs om he SWAPs. This yields a dep h o he TQGs
o 3
2n2+3
2n−6 in a s a chip, while o a squa e g id i is
6n. Such dep h inc ease comes om he educ ion in pa al-
leliza ion, since all ga es now ac ia he cen al qubi . On he
o he hand, less pa alleliza ion educes he ypes o possible
c oss alk e o s. Adding connec ions be ween ex e nal qubi s
educes he dep h o he ci cui , since he main cause o ci cui
dep h is he ac ha he in e ac ion o wo ex e nal qubi s
needs o be done exclusi ely by he cen al qubi . Fu he
s udies a e equi ed o see i he addi ion o mo e ex e nal
laye s o his opology (such as in a spide web) can lead o
be e comp omises be ween dep h and ga e coun , especially
o simula ing sys ems wi h clus e s o s ongly in e ac ing
nuclei.
E. Ga e-le el op imiza ion
The wo-qubi in e ac ions ha appea in he algo i hm a e
he XZ,YZ and ZZ in e ac ions, as shown in Sec. III C 2
and Fig. 2(b). When compiling he algo i hm in o he na i e
ga es o he de ice, all hese in e ac ions mus be implemen ed
in e ms o some a ailable ga e se . We s udy in Table I
he o e head in oduced by decomposing hese in e ac ions
in o di e en examples o na i e TQGs o supe conduc ing
de ices; namely, he pa ame izable and ixed-phase UZZ ga e,
he ixed-phase con olled-Zga e CZ, and he CNOT ga e. The
CNOT ga e is usually pe o med by making use o he c oss-
esonance ga e [37,51], which in oduces an UXZ in e ac ion,
making i equi alen o he UZZ o he pu pose o his algo-
i hm. We assume ha he SQGs ha can be implemen ed a e
he Rxy and he Rzga es. These numbe s can be u he educed
i he i s and las SQGs in oduced by his compila ion a e
combined wi h he adjacen SQGs in he algo i hm.
The conclusion is ha ixed-angle ga es will double he
numbe o TQGs ha need o be physically pe o med. In
Re . [52], he imp o emen s coming om he educ ion o
he ga e coun a e compa ed o he new e o s in oduced by
he in e pola ion o he calib a ed phases. Fo wo ins ances
o a Quan um App oxima e Op imiza ion Algo i hm (QAOA)
[53], i is shown ha he pe o mance is be e when using
pa ame ized TQGs.
The ga e sequences o some o he ga e decomposi ions
a e shown in Fig. 5.
IV. CO-DESIGN HARDWARE
A s a -a chi ec u e chip has undamen al scaling issues
using a ansmon as he cen al qubi as he numbe o neigh-
FIG. 5. (a) Ga e decomposi ion o e−iφZZ in e ms o he ixed-
phase UZZ (π
4) ga e, (b) he CNOT.
bo s g ows. E e y neighbo added o he cen e qubi would
dec ease i s cha ging ene gy Ec. To keep he qubi equency
cons an and anha monici y in he ansmon egime, he a io
o he qubi ’s Josephson ene gy o i s cha ging ene gy, Ej/Ec,
mus emain una ec ed. The e o e we canno a o d o change
i s cha ging ene gy. This leads o a ade-o be ween he
numbe o coupled qubi s and hei coupling s eng h o he
cen al elemen .
The spi i o Co-Design calls o eplacing he cen al
ansmon wi h ano he objec ha enables his scaling in size.
A esona o has no Josephson ene gy Ej,so heEj/Ec a io
is no al e ed by adding mo e capaci i e couplings o he
esona o . Only small co ec ions o i s equency a e in o-
duced by adding coupled qubi s. As a dis ibu ed elemen , a
co-plana wa eguide esona o also has physically mo e space
o couplings han a cen al ansmon qubi . By elonga ing he
esona o and choosing he mode wi h he a ge equency,
he numbe o qubi s coupled o i can u he be inc eased.
These p ope ies make a esona o a a ou able componen in
he cen e o he chip.
In he de ice in Fig. 6(a), he qubi s a e capaci i ely cou-
pled o he esona o ia unable couple s [39,54,55]in he
p oximi y o a ol age maximum o a s anding wa e in he
esona o . As he esona o is elonga ed, we mus use highe
ha monic exci a ions o he esona o o keep he equency
a ound he ope a ional equency o he qubi s. Tunable cou-
ple s a oid he equency c owding issues ela ed o di ec
coupling [56,57], and he linea esona o has highe con-
nec i i y in he cen e han ing esona o s uc u es wi h
quasi-all- o-all connec i i ies [58].
A linea esona o canno in gene al be used as a qubi ,
since a mic owa e d i e on i will no only popula e he
{0|,1|} subspace, bu also highe exci ed s a es. Howe e ,
he e ec i e in e ac ions media ed ia he uneable couple
in Fig. 6(a) a e o he ype a†aZ and (a+a†)X+(a−a†)Y
whe e aand a†[37] a e he esona o c ea ion and annihila-
ion ope a o s. These ypes o in e ac ions conse e exci a ion
numbe , so when a mos one exci a ion is in he qubi -
esona o sys em, he esona o canno be popula ed beyond
i s i s exci ed s a e h ough in e ac ion wi h a qubi medi-
a ed a uneable couple . CZ and iSWAP ga es be ween he
esona o and a qubi can be pe o med using he wo in e -
ac ions, and he heo y is de eloped mo e ully in Sec. IV A.
Then, a esona o oge he wi h an ex e nal qubi can be used
as an e ec i e cen al qubi in he ollowing way.
043089-7
MANUEL G. ALGABA e al. PHYSICAL REVIEW RESEARCH 4, 043089 (2022)
FIG. 6. (a) Cen al λ/4 esona o wi h 6 qubi s coupled ia un-
able couple s. The esona o is also coupled o a quan um ci cui
e ige a o enabling as ese . The de ice ac s e ec i ely as a six
qubi s a -a chi ec u e chip. (b) Elec ical diag am o ansmon qubi
(le ) coupled o a esona o mode ( igh ) ia a unable couple (cen-
e ). The qubi has equency ωq, couple ωc, and esona o ω .The
qubi and esona o ha e a di ec capaci ance Cq and capaci ances
Cqc and C c espec i ely o he couple .
1. P epa e all qubi s and he esona o in hei g ound
s a es
2. Selec one qubi o o m he e ec i e cen al qubi o-
ge he wi h he esona o
3. P epa e an a bi a y s a e in he selec ed qubi
4. Pe o m an iSWAP ope a ion om he selec ed qubi o
he esona o ini ially in he g ound s a e
5. Pe o m CZ ga es be ween he esona o and any o he
qubi s
6. Pe o m an iSWAP ope a ion back om he esona o
o he selec ed qubi o measu emen
The heo e ically mos s aigh o wa d p o ocol would be
o pe o m a SWAP ga e om he qubi o he esona o . The
iSWAP, on he o he hand, is a na i e ga e ha can di ec ly
be implemen ed on he ha dwa e in Fig. 6(b).TheiSWAP
ga e be ween he esona o and he qubi is ep esen ed by he
uni a y ope a o :
UiSWAP =⎛
⎜
⎝
10 00
00−i0
0−i00
00 01
⎞
⎟
⎠.(11)
TABLE II. Pa ame e s o s a -a chi ec u e chip.
Pa ame e Symbol Value
Resona o equency ω 2π×4.3 GHz
Qubi anha monici y αq−2π×0.187 GHz
Couple anha monici y αc−2π×0.110 GHz
Resona o -couple coupling g c 2π×98.5MHz
Qubi -couple coupling gqc 2π×101.8MHz
Resona o -qubi coupling g q 2π×8.9MHz
Resona o elaxa ion T
160 μs
Qubi elaxa ion Tq
160 μs
Couple elaxa ion Tc
130 μs
Resona o dephasing T
260 μs
Qubi dephasing Tq
260 μs
Couple dephasing Tc
230 μs
Since he CZ ga es pe o ming he compu a ion ollowing
he iSWAP a e diagonal in he compu a ional basis, he phase
in oduced by he iSWAP is unin ol ed in he ga e. This
enables subs i u ing he SWAP ga e by an iSWAP ga e in he
p o ocol o u he minimize he ga e coun .
A. Ga e heo y and simula ions
He e we demons a e ha in ou s a a chi ec u e CZ and
iSWAP- ype ga es be ween any o he qubi s and he {0|,1|}
subspace o a chosen esona o mode can be implemen ed.
The ope a ional p inciples o hese ga es a e e y simila
o hose be ween wo qubi s coupled wi h a unable couple
[39,54,55,59]. The main limi a ion o ou a chi ec u e (whe e
one ansmon is eplaced by a esona o ) is ha iSWAP ope a-
ions can only be pe o med in he ze o- and single-exci a ion
subspace o he wo-qubi compu a ional basis.
1. Condi ional-Z ga e
The CZ ope a ion be ween he esona o and he qubi is
desc ibed by he uni a y ope a o :
CZ(φ)=⎛
⎜
⎜
⎝
100 0
010 0
001 0
000e−iφ
⎞
⎟
⎟
⎠.(12)
This ga e is equi alen o he UZZ(φ) ga e in Eq. (5)up
o wo Rz o a ions. To ope a e a CZ ga e, we ini ialize he
esona o -couple -qubi se up shown in Fig. 6(b) a he idling
con igu a ion wi h ze o e ec ing coupling be ween he qubi
and esona o . No e ha he couple is also a ansmon ha
shows a highe sensi i i y o he magne ic lux han egula
qubi s. We nex apply a lux pulse ha lowe s he couple
equency, u ning on he e ec i e coupling be ween he es-
ona o and he qubi . Depending on he lux pulse shape,
he s a e collec s condi ional phase φand possibly expe-
iences popula ion oscilla ions be ween compu a ional and
noncompu a ional s a es, as a unc ion o he ime spen a he
ga e-ope a ion equency. We op imize he pulse ampli ude
and du a ion such ha a e he lux pulse he CZ ga e ideli y
is maximized. De ails o he ga e heo y can be ound in
Appendix Gand he conside ed de ice pa ame e s in Table II.
043089-8
CO-DESIGN QUANTUM SIMULATION OF NANOSCALE NMR PHYSICAL REVIEW RESEARCH 4, 043089 (2022)
FIG. 7. (a) CZ ga e e o landscape a e aged o e andom ini ial
s a es. Con ou s wi h a low e o a e highligh ed wi h a dashed
line. (b) iSWAP ga e e o landscape ob ained by a e aging o e a
numbe o andom ini ial s a es in he ze o- and one-exci a ion man-
i olds. Bo h plo s a e p oduced using sys em pa ame e s shown in
Table II.
In Fig. 7(a), we ope a e ou CZ ga e by uning he couple
equency using a la op Gaussian shaped lux pulse. The
wid h o ou Gaussian il e was ixed a 3 ns. Applying such
a lux pulse o couple esul s in a couple equency shi
by ωshi
c om he idling con igu a ion. Then by app op ia ely
uning ωshi
cand he ga e ime τ, one loca es he op imal pulse
con igu a ion ha minimizes he CZ(π) ga e e o εCZ =1−
( √ρσ√ρ)2, whe e σis he a ge densi y ma ix ob ained
a e p opaga ing some ini ial s a e ||wi h he ideal
uni a y o Eq. (12) and ρ he inal densi y ma ix ob ained
a e p opaga ing ||wi h he Lindbladian co esponding
o ou sys em de ined in Eq. (G1). Fo ou de ice pa ame e s,
he maximal decohe ence limi ed CZ ga e e o a e aged o e
a numbe o andom ini ial s a es is 1.6×10−3. No e ha
he sys em pa ame e s in Table II we e chosen such ha hey
allow o he possibili y o ind a good idling con igu a ion,
whe e he esidual CZ in e ac ion anishes be o e he ga e
ope a ion. In ou simula ions, we ha e included en i onmen al
noise, such as ampli ude damping and pu e dephasing and
ea ed hem using a Lindblad mas e equa ion sol e in QUTIP
[60,61].
2. iSWAP ga e
Jus as he CZ ga e, he iSWAP ga e can be na i ely ealized
in supe conduc ing quan um compu ing a chi ec u e [37].
Wi h ou de ice, we can pe o m high- ideli y iSWAP ga es
be ween ze o- and single-exci a ion compu a ional s a es. The
wo-pho on s a e |1 ⊗|1, whe e |1 deno es he i s ex-
ci ed s a e o he esona o , mus be excluded because i
esonan ly in e ac s wi h he s a e |2 ⊗|0inducing a pop-
ula ion exchange be ween he s a es. Hence he esul ing
ope a ion in his subspace does no ma ch he ac ion o he
a ge ed iSWAP ope a ion.
The capaci i e coupling be ween he elemen s o he elec-
ical ci cui shown in Fig. 6(b) gi es ise o an e ec i e XY
in e ac ion be ween he qubi and esona o unde he o a ing
wa e app oxima ion. Such an in e ac ion conse es exci a ion
numbe . Wi h only he qubi o esona o (o nei he ) ini ially
popula ed, we s ay wi hin he single exci a ion subspace o
he join sys em, he eby minimizing leakage o quan um
popula ion in o he highe exci ed s a es o he esona o . The
XY in e ac ion can be u ned on by i s uning he qubi in
esonance wi h he esona o , and hen applying a lux-pulse
o he couple o u n on he coupling, simila o he CZ ga e
ope a ion.
Figu e 7(b) shows iSWAP ga e e o landscape o he
same de ice pa ame e s (gi en in Table II). The op imal a -
e age iSWAP ga e e o εiSWAP ob ained o ou de ice is
1.7×10−3. This esul is ob ained by a e aging o e a num-
be o andom ini ial s a es wi hin he ze o- and one-exci a ion
mani olds.
The esul s o ou wo-qubi -ga e simula ions demons a e
ha ou s a a chi ec u e suppo s ope a ing ga es wi h simila
ideli ies as egula ansmon qubi s coupled oge he . The
inc eased local connec i i y o he de ice educes he need
o SWAP ga es o simula e he nanoscale NMR p oblem (and
o he s wi h a simila s uc u e) and consequen ly in he end
imp o es simula ion ideli ies.
B. Rese
The hype pola iza ion p o ocol desc ibed in Sec. II needs
egula e-ini ializa ions o he s a e o he NV cen e . The
Co-Design ha dwa e o simula ing he p o ocol mus he e-
o e suppo his ope a ion wi hin qubi li e imes. This is a
ha dwa e challenge, bu one wi h solu ions in sigh . In pa -
icula , he quan um ci cui e ige a o (QCR) has been used
o pe o m he ese in ens o nanoseconds [40–43], which
is a simila imescale o ga e ope a ions. The ad an age o
using a QCR o he ese is he possibili y o ese he cen-
al esona o di ec ly, wi hou he need ans e he esona o
popula ion back o he cen al qubi using an iSWAP ga e.
Al e na i ely, a as ese is possible h ough applying a lux
d i e o a qubi o SWAP i s s a e wi h i s measu emen line
[62]. This scheme has he ad an age o no equi ing any
addi ional ha dwa e no al eady p esen on he chip, bu comes
wi h a small cos in he ci cui dep h, as he s a e o he
esona o mus be anspo ed using an iSWAP ga e in o he
043089-9
MANUEL G. ALGABA e al. PHYSICAL REVIEW RESEARCH 4, 043089 (2022)
(
a
)(
b
)
FIG. 11. (a) Coe icien s ec o s o he i s qubi
A1,
ωc
1be o e he o a ion, wi h p ojec ion o e he h ee axis and (b) coe icien ec o s
o he i s qubi
A o
1,
ωc, o
1a e he o a ion, being
A o
1in he Zaxis.
he con inuous-d i ing case, due o he wo di e en ime-
dependen p ocesses in ol ed in he T o e decomposi ion:
he ee dynamics o he spins and he sequence o pulses.
The mos c ucial poin o be awa e o is he in e play be ween
T o e s eps and in e pulse spacing. The numbe o in e pulse
e olu ions, i.e. numbe o pulses minus one, bounds om be-
low he minimum numbe o T o e s eps o he simula ion.
Clea ly, a leas one T o e s ep is needed o each in e pulse
e olu ion.
Taking his in e play in o accoun , he mos s aigh o -
wa d se up is o choose a equency which will de e mine
he spacing o he pulse sequence, and o iden i y each in-
e pulse e olu ion wi h a single T o e s ep. I he achie ed
p ecision is no high enough, mo e T o e s eps can be added
o each in e pulse e olu ion. Each π-pulse i sel is simply
implemen ed as an Xo Yga e on he qubi ep esen ing he
NV cen e . The OU-dis ibu ed Rabi equency luc ua ions
p esen in nanoscale NMR sys ems a e hen simula ed by
o e - and unde - o a ions o he Xand Yga es.
APPENDIX E: ROTATIONAL OPTIMIZATION
In p inciple, we had a Hamil onian wi h e ms o he ype
ZX,ZY,and ZZ o he case o no in e nuclea in e ac ions.
Howe e , we can o a e he basis so he Hamil onian loses he
ZX and ZY e ms, allowing o educe he numbe o TQGs.
To make up o his o a ion, we need o in oduce di e en
cons an s
A o
i o he p oblem and o a e he ec o s a e we
ob ain a he end be o e measu ing i . The o a ions ha we
will conside a e only one-qubi o a ions on nuclei qubi s and
we a e applying his jus o he case wi h no in e nuclea in e -
ac ions. The e o e we can conside he e ec o his o a ion
on only one qubi ep esen ing an a bi a y nucleus. We will
exempli y his p ocedu e using nucleus 1. I we wan o ob ain
he mean alue o σzac ing on he nucleus:
σz=T (ρ( )σz)=T (U(0, )ρ(0)U†(0, )σz),(E1)
whe e U(0, ) ep esen s he e olu ion ope a o om =0
o = . The densi y ma ix ρ(0) con ains he s a e o he
NV cen e (which is in he |+ s a e a =0) and nucleus
1, i.e., ρ(0) = |++|NV ⊗11
2. Ou in en ion is o ob ain an
exp ession o his mean alue in e ms o he o a ed e olu ion
ope a o s and la e , we will ind he app op ia e o a ion o be
pe o med. Then, aking in o accoun ha he ace is in a ian
unde a o a ion R=1NV ⊗R1, we ge :
σz=T (RU (0, )ρ(0)U†(0, )σzR†)
=T (RU (0, )R†Rρ(0)R†RU †(0, )R†RσzR†).(E2)
This can be exp essed as:
σz=T (U o (0, )ρ o (0)U†
o (0, )RσzR†).(E3)
The densi y ma ix o he nucleus is he iden i y. Thus any
o a ion on nuclei qubi s lea es he densi y ma ix una ec ed,
leading o:
σz=T (U o (0, )ρ(0)U†
o (0, )RσzR†).(E4)
Then we need o o a e he sys em p e ious o he mea-
su emen . By using he in a iance o he ace unde cyclic
pe mu a ions, we ge :
σz=T (R†U o (0, )ρ(0)U†
o (0, )Rσz),(E5)
which is equi alen o in oducing a coun e - o a ion in he
ci cui be o e measu emen .
Now le us ocus on he speci ic o a ion we ha e o im-
plemen . Since he cons an s mul iplying he Pauli ma ices in
he Hamil onian a e
A1
2and
ωc
1=
A1
2−γcBz
ez( o nucleus 1),
we can o a e he basis o ob ain a ep esen a ion in which he
ec o s ha e only zcomponen o
A1and hus, XZ and YZ
e ms a e emo ed. The ec o s be o e and a e he needed
o a ion can be seen in Fig. 11.
To compu e he new ec o s (and hus he new coe icien s
o he ga es o ou algo i hm), we can use Rod igues’ o a ion
o mula o o a e a ec o
an angle θa ound a uni a y
043089-16
CO-DESIGN QUANTUM SIMULATION OF NANOSCALE NMR PHYSICAL REVIEW RESEARCH 4, 043089 (2022)
TABLE III. Ga e coun o one T o e s ep and o one cycle o
di e en opologies wi h and wi hou in e nuclea in e ac ions.
All-To-All S a opology Squa e g id
Nnonin
TQG n−1n−14n−4
Nnonin
SQG
5
2n+25
2n+221
2n−47
2
Nin
TQG
3
2n2−3
2n3
2n2+3
2n−63n2−6n+3
Nin
SQG 4n2−9
2n+7
24n2+7
2n−25
28n2−33
2n+11
2
axis ˆ
k:
o =
cos θ+(ˆ
k×
)sinθ+ˆ
k(ˆ
k·
)(1 −cos θ),(E6)
being in ou case, θ=a ccos (Az
1/|
A1|) and ˆ
k=
(cos(φ),sin(φ),0), wi h φ=−π
2+φxy =−π
2+a c an
(Ay
1/Ax
1).
Fo implemen ing he coun e - o a ion o his in he quan-
um ci cui , we use:
R†
1=eiθ
2(cos(φ)X−sin(φ)Y).(E7)
APPENDIX F: SWAP ROUTING
Ou qubi ou ing me hod consis s o mapping he squa e
g id o a linea chain wi h qubi s labeled om 0 o n. Then, in
he simpli ied case o no in e nuclea in e ac ions, he op imal
SWAP me hod o he one- o-all in e ac ion case on a linea
chain can be used. Fo a single NV cen e he p o ocol goes as
ollows.
(1) Ini ialize he s a e o he NV cen e in he second qubi .
(2) Pe o m in e ac ions wi h he i s and hi d qubi s.
(3) SWAP he NV cen e qubi o he igh .
(4) Pe o m in e ac ion wi h igh qubi .
(5) Repea s eps 3-4 un il all in e ac ions ha e been
achie ed.
The pa e n is seen in Fig. 3(a) deno ed by he in ense
blue a ows. Wi h in e nuclea in e ac ions we need o pe -
o m a swap pa e n ha enables all- o-all in e ac ions. The
so-called odd-e en mapping in Fig. 3(a) is an e icien one
[69] ep esen ed by g een a ows in Fig. 3(a). This consis s
o swapping i s all he e en qubi s wi h hei igh neigh-
bo s and hen swapping all he odd qubi s wi h hei igh
neighbo s. This way, we will ob ain all- o-all in e ac ions wi h
1
2(n−1)(n−2) SWAP ga es and a o al TQG dep h o 6n.A
summa y o he TQG coun s is shown in Table III.
To mo i a e he c ea ion o a chip wi h a s a opology and
he use o an al e na i e linea ized SWAP ou ing o a squa e
g id ins ead o s anda d nume ical app oaches, a compa ison
be ween all he cases is p o ided in Fig. 12. A educ ion
in he numbe o SWAPs can be no iced o bo h he linea
chain app oach and he s a - opology chip agains s anda d
nume ical app oaches o a squa e g id.
APPENDIX G: QUBIT-RESONATOR GATE THEORY
In he ollowing discussion, we conside ga e ope a ion
be ween he esona o and one o he qubi s, and neglec any
e ec s ha a ise om he in e ac ions wi h spec a o qubi s
and o he esona o modes. The ime dynamics in such a
sys em a e de e mined by he Hamil onian:
H=H0+H c +Hqc +H q,(G1)
whe e he uncoupled pa o he o al Hamil onian H0=H +
Hc+Hqis:
H =¯hω b†
b ,
Hc=¯hωcb†
cbc+¯h
2αcb†
cb†
cbcbc,
Hq=¯hωqb†
qbq+¯h
2αqb†
qb†
qbqbq,
(G2)
FIG. 12. (a) Compa ison o he equi ed numbe o SWAPs o simula ing he p oposed sys em wi h no in e nuclea in e ac ions o each
T o e s ep. Nume ical app oaches om e e ences a e applied o a squa e g id. (b) Equi alen compa ison wi h in e nuclea in e ac ions.
Zulehne e al. and Saeedi e al. do no imp o e he linea chain app oach o ew qubi s and a e in ac able o la ge numbe s o qubi s and
hus a e no displayed.
043089-17
MANUEL G. ALGABA e al. PHYSICAL REVIEW RESEARCH 4, 043089 (2022)
whe e bλand ωλa e he annihila ion ope a o and undamen-
al equency o he mode λ={ ,c,q}, espec i ely, and αγ
is he anha monici y o he mode γ={q,c}. The in e ac ion
componen o he Hamil onian is:
Hλμ =−¯hgλμ(b†
λ−bλ)(b†
μ−bμ),(G3)
whe e λμ ={ c,qc, q}, and gλμ deno e esona o -couple ,
qubi -couple and esona o -qubi coupling equencies. Wi h
he Hamil onian o Eq. (G1), we a e now in a posi ion o pe -
o m simula ions o wo-qubi ga es by p opaga ing a sui ably
chosen ini ial s a e.
Be o e he ga e ope a ion, we choose he idling equen-
cies o he qubi , esona o , and he couple such ha he
CZ coupling a e ζis minimized. This CZ coupling a e is
de ined as:
ζ=ω101 −ω100 −ω001 +ω000,(G4)
whe e ωn 0nqco esponds o he eigenene gy o Hamil onian
in Eq. (G1) wi h n exci a ions in esona o and nqexci a ions
in qubi wi h couple being in he g ound s a e. The poin o
minimal |ζ|is also known as he idling con igu a ion, which
we ound o be a [ω ,ω
c,ω
q]/(2π)=[4.30,6.14,4.47] GHz
o he pa ame e s gi en in Table II. The CZ ga e is ope a ed
by sending a lux pulse ha modi ies he couple equency
ωc, which hen in he coupled basis modi ies he equencies
ω101,ω
100,ω
001 and ω000. This makes ζnonze o, so he sys-
em collec s a CZ phase.
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