Co-Design quantum simulation of nanoscale NMR

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 gkkIz kIz k−1 4(I+ kI− k+I− kI+ k) + M  j>j hjjσz jσz j−2(σ+ jσ− j+σ− jσ+ 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 2M 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, gkkis he coupling be ween he k h and k h nuclei, and hjjis 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|10|+eiφ|01|=e−iφσ−+eiφσ+, φa phase,  he Rabi equency and he ke s |1and |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 043089-2 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( )=|00|+1−p( )|11|=10 0√1−p( ), Kamp 2( )=p( )|01|=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 043089-4 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, |0o |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 ⊗|0inducing 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. [1] R. Feynman, Simula ing physics wi h compu e s, in Feynman and Compu a ion (CRC P ess, 2018), pp. 133–153. [2] S. Lloyd, Uni e sal quan um simula o s, Science 273, 1073 (1996). [3] J. P eskill, Quan um compu ing in he NISQ e a and beyond, Quan um 2, 79 (2018). [4] K. Bha i, A. Ce e a-Lie a, T. Kyaw, T. Haug, S. Alpe in-Lea, A. Anand, M. Deg oo e, H. Heimonen, J. Ko mann, T. Menke e al., Noisy in e media e-scale quan um (NISQ) algo i hms, Re . Mod. Phys. 94, 015004 (2022). [5]M.J.S.Smi h,Applica ion-Speci ic In eg a ed Ci cui s,Vol.7 (Addison-Wesley Reading, MA, 1997). [6] K.-C. Hsu and H.-W. Tseng, GPTPU: Accele a ing appli- ca ions using edge enso p ocessing uni s, in P oceedings o he In e na ional Con e ence o High Pe o mance Com- pu ing, Ne wo king, S o age and Analysis (Associa ion o Compu ing Machine y, New Yo k, NY, 2021), pp. 1–4, doi: 10.1145/3458817.3476177. [7] T. Lu, T. Ma in, Y. Zhuo, Y.-F. Chen, and C. Ma, Accele a ing MRI econs uc ion on TPUs, in 2020 IEEE High Pe o mance Ex eme Compu ing Con e ence (HPEC) (IEEE, 2020), pp. 1–9, doi: 10.1109/HPEC43674.2020.9286192. [8] T. S audache , F. Shi, S. Pezzagna, J. Meije , J. Du, C. A. Me iles, F. Reinha d, and J. W ach up, Nuclea magne ic esonance spec oscopy on a (5-Nanome e )3 sample olume, Science 339, 561 (2013). [9] M. Nielsen and I. Chuang, Quan um Compu a ion and Quan um In o ma ion (Camb idge Uni e si y P ess, 2000). [10] H. Hogben, M. K zys yniak, G. Cha nock, P. Ho e, and I. Kup o , Spinach a so wa e lib a y o simula ion o spin dy- namics in la ge spin sys ems, J. Magn. Reson. 208, 179 (2011). [11] K. See ha am, D. Biswas, C. Noel, A. Risinge , D. Zhu, O. Ka z, S. Cha opadhyay, M. Ce ina, C. Mon oe, E. Demle e al., Digi al quan um simula ion o NMR expe imen s, a Xi :2109.13298. [12] T. E. O’B ien, L. B. Io e, Y. Su, D. Fushman, H. Ne en, R. Babbush, and V. Smelyanskiy, Quan um compu- a ion o molecula s uc u e using da a om challenging- o- classically-simula e nuclea magne ic esonance expe imen s, a Xi :2109.02163. [13] D. Sels, H. Dash i, S. Mo a, O. Demle , and E. Demle , Quan um app oxima e Bayesian compu a ion o NMR model in e ence, Na . Mach. In ell 2, 396 (2020). [14] D. Sels and E. Demle , Quan um gene a i e model o sam- pling many-body spec al unc ions, Phys.Re .B103, 014301 (2021). [15] M. H. Le i , Spin Dynamics: Basics o Nuclea Magne ic Res- onance (Wiley, 2013). [16] M. W. Dohe y, N. B. Manson, P. Delaney, F. Jelezko, J. W ach up, and L. C. L. Hollenbe g, The ni ogen- acancy colou cen e in diamond, Phys. Rep. 528, 1 (2013). [17] M. H. Abobeih, J. Randall, C. E. B adley, H. P. Ba ling, M. A. Bakke , M. J. Degen, M. Ma kham, D. J. Twi chen, and T. H. Taminiau, A omic-scale imaging o a 27-nuclea - spin clus e using a quan um senso , Na u e (London) 576, 411 (2019). [18] C. Munue a-Ja aloy, R. Puebla, and J. Casano a, Dynamical decoupling me hods in nanoscale NMR, Eu ophys. Le . 134, 30001 (2021). [19] C. Munue a-Ja aloy, R. Puebla, B. D’Anjou, M. B. Plenio, and J. Casano a, De ec ion o molecula ansi ions wi h ni ogen- acancy cen e s and elec on-spin labels, a Xi :2110.14255. [20] A. Ajoy, K. Liu, R. Naza yan, X. L , P. Zanga a, B. Sa a i, G. Wang, D. A nold, G. Li, A. Lin e al., O ien a ion-independen oom empe a u e op ical 13c hype pola iza ion in powde ed diamond, Sci. Ad . 4, eaa 5492 (2018). [21] S. Ha mann and E. Hahn, Nuclea double esonance in he o a ing ame, Phys. Re . 128, 2042 (1962). [22] H. Ca and E. Pu cell, E ec s o di usion on ee p ecession in nuclea magne ic esonance expe imen s, Phys. Re . 94, 630 (1954). [23] S. Meiboom and D. Gill, Modi ied spin-echo me hod o mea- su ing nuclea elaxa ion imes, Re . Sci. Ins um. 29, 688 (1958). [24] A. Maudsley, Modi ied Ca -Pu cell-Meiboom-Gill sequence o NMR Fou ie imaging applica ions, J. Magn. Reson. (1969) 69, 488 (1986). [25] T. Gullion, D. Bake , and M. Con adi, New, compensa ed Ca - Pu cell sequences, J. Magn. Reson. (1969) 89, 479 (1990). [26] J. Casano a, Z. Y. Wang, and M. B. Plenio, Noise-Resilien Quan um Compu ing wi h a Ni ogen-Vacancy Cen e and Nu- clea Spins, Phys.Re .Le .117, 130502 (2016). [27] G. Uhlenbeck and L. O ns ein, On he heo y o he B ownian mo ion, Phys. Re . 36, 823 (1930). [28] J. Cai, B. Naydeno , R. P ei e , L. McGuinness, K. Jahnke, F. Jelezko, M. Plenio, and A. Re zke , Robus dynamical 043089-18 CO-DESIGN QUANTUM SIMULATION OF NANOSCALE NMR PHYSICAL REVIEW RESEARCH 4, 043089 (2022) decoupling wi h conca ena ed con inuous d i ing, New J. Phys. 14, 113023 (2012). [29] D. T. Gillespie, Exac nume ical simula ion o he O ns ein- Uhlenbeck p ocess and i s in eg al, Phys. Re . E 54, 2084 (1996). [30] A. Ajoy, B. Sa a i, R. Naza yan e al., Hype pola ized elax- ome y based nuclea T1 noise spec oscopy in diamond, Na . Commun. 10, 5160 (2019). [31] M. Suzuki, Gene alized T o e ’s o mula and sys ema ic ap- p oximan s o exponen ial ope a o s and inne de i a ions wi h applica ions o many-body p oblems, Commun. Ma h. Phys. 51, 183 (1976). [32] E. Campbell, Random Compile o Fas Hamil onian Simula- ion, Phys.Re .Le .123, 070503 (2019). [33] X. Yuan, S. Endo, Q. Zhao, Y. Li, and S. Benjamin, The- o y o a ia ional quan um simula ion, Quan um 3, 191 (2019). [34] K. Bha i and T. Haug, Quan um-assis ed simula o , Phys. Re . A104, 042418 (2021). [35] E. Younis, K. Sen, K. Yelick, and C. Iancu, Q as : Con la ing sea ch and nume ical op imiza ion o scalable quan um ci cui syn hesis, a Xi :2103.07093. [36] N. Ha ano and M. Suzuki, Finding Exponen ial P oduc Fo - mulaso Highe O de s(Sp inge , Be lin, Heidelbe g, 2005), pp. 37–68. [37] P. K an z, M. Kjae gaa d, F. Yan, T. P. O lando, S. Gus a sson, and W. D. Oli e , A quan um enginee ’s guide o supe conduc - ing qubi s, Appl. Phys. Re . 6, 021318 (2019). [38] D. C. McKay, C. J. Wood, S. Sheldon, J. M. Chow, and J. M. Gambe a, E icien Z ga es o quan um compu ing, Phys. Re . A96, 022330 (2017). [39] F. Yan, P. K an z, Y. Sung, M. Kjae gaa d, D. L. Campbell, T. P. O lando, S. Gus a sson, and W. D. Oli e , Tunable Coupling Scheme o Implemen ing High-Fideli y Two-Qubi Ga es, Phys. Re . Appl. 10, 054062 (2018). [40] K. Y. Tan, M. Pa anen, R. E. Lake, J. Go enius, S. Masuda, and M. Mö önen, Quan um-ci cui e ige a o , Na . Commun. 8, 15189 (2017). [41] M. Sil e i, H. G abe , S. Masuda, K. Y. Tan, and M. Mö önen, Theo y o quan um-ci cui e ige a ion by pho on-assis ed elec on unneling, Phys. Re . B 96, 094524 (2017). [42] H. Hsu, M. Sil e i, A. Gunyhó, J. Goe z, G. Ca elani, and M. Mö önen, Tunable e ige a o o nonlinea quan um elec ic ci cui s, Phys.Re .B101, 235422 (2020). [43] V. Se iuk, K. Y. Tan, E. Hyyppä, M. Sil e i, M. Pa anen, M. Jenei, S. Masuda, J. Goe z, V. Ves e inen, L. G önbe g e al., Fas con ol o dissipa ion in a supe conduc ing esona o , Appl. Phys. Le . 115, 082601 (2019). [44] M. Celio, New Me hod o Calcula e he Muon Pola iza ion Func ion, Phys. Re . Le . 56, 2720 (1986). [45] S. McA dle, Lea ning om physics expe imen s wi h quan um compu e s: Applica ions in muon spec oscopy, PRX Quan um 2, 020349 (2021). [46] A. G. Fowle , M. Ma ian oni, J. M. Ma inis, and A. N. Cleland, Su ace codes: Towa ds p ac ical la ge-scale quan um compu a- ion, Phys.Re .A86, 032324 (2012). [47] Y. Hi a a, M. Nakanishi, S. Yamashi a, and Y. Nakashima, An e icien me hod o con e a bi a y quan um ci cui s o ones on a linea nea es neighbo a chi ec u e, in 2009 Thi d In e na ional Con e ence on Quan um, Nano and Mic o Technologies, Cancun, Mexico (IEEE, 2009), pp. 26–33, doi: 10.1109/ICQNM.2009.25. [48] G. Li, Y. Ding, and Y. Xie, Tackling he qubi mapping p oblem o NISQ-e a quan um de ices, in P oceedings o he Twen y-Fou h In e na ional Con e ence on A chi ec u al Suppo o P og amming Languages and Ope a ing Sys ems (ACM, New Yo k, NY, USA, 2019), pp. 1001–1014, doi: 10.1145/3297858.3304023. [49] M. Saeedi, R. Wille, and R. D echsle , Syn hesis o quan um ci cui s o linea nea es neighbo a chi ec u es, Quan . In o. P oc. 10, 355 (2011). [50] A. Zulehne , A. Pale , and R. Wille, An e icien me hodology o mapping quan um ci cui s o he IBM QX A chi ec u es, in IEEE ansac ions on compu e -aided design o in eg a ed ci cui s and sys ems (IEEE, 2019), Vol. 38, pp. 1226–1236, doi: 10.1109/TCAD.2018.2846658. [51] E. Magesan and J. M. Gambe a, E ec i e Hamil onian models o he c oss- esonance ga e, Phys.Re .A101, 052308 (2020). [52] N. Lac oix, C. Hellings, C. K. Ande sen, A. D. Paolo, A. Remm, S. Laza , S. K inne , G. J. No is, M. Gabu eac, J. Heinsoo, A. Blais, C. Eichle , and A. Wall a , Imp o ing he pe o mance o deep quan um op imiza ion algo i hms wi h con inuous ga e se s, PRX Quan um 1, 020304 (2020). [53] E. Fa hi, J. Golds one, and S. Gu mann, A quan um app oxi- ma e op imiza ion algo i hm, a Xi :1411.4028. [54] M. Ma ian oni, F. Deppe, A. Ma x, R. G oss, F. K. Wilhelm, and E. Solano, Two- esona o ci cui quan um elec odynamics: A supe conduc ing quan um swi ch, Phys.Re .B78, 104508 (2008). [55] B. Foxen, C. Neill, A. Dunswo h, P. Roushan, B. Chia o, A. Meg an , J. Kelly, Z. Chen, K. Sa zinge , R. Ba ends e al., Demons a ing a Con inuous Se o Two-qubi Ga es o Nea - e m Quan um Algo i hms, Phys. Re . Le . 125, 120504 (2020). [56] C. Song, K. Xu, W. Liu, C. P. Yang, S. B. Zheng, H. Deng, Q. Xie, K. Huang, Q. Guo, L. Zhang, P. Zhang, D. Xu, D. Zheng, X. Zhu, H. Wang, Y. A. Chen, C. Y. Lu, S. Han, and J. W. Pan, 10-Qubi En anglemen and Pa allel Logic Ope a ions wi h a Supe conduc ing Ci cui , Phys.Re .Le .119, 180511 (2017). [57] C. Song, K. Xu, H. Li, Y. Zhang, X. Zhang, W. Liu, Q. Guo, Z. Wang, W. Ren, J. Hao e al., Gene a ion o mul icomponen a omic Sch ödinge ca s a es o up o 20 qubi s, Science 365, 574 (2019). [58] S. Haz a, A. Bha acha jee, M. Chand, K. V. Salunkhe, S. Gopalak ishnan, M. P. Pa anka , and R. Vijay, Ring-Resona o - Based Coupling A chi ec u e o Enhanced Connec i i y in a Supe conduc ing Mul iqubi Ne wo k, Phys. Re . Appl. 16, 024018 (2021). [59] J. Chu and F. Yan, Couple -Assis ed Con olled-Phase Ga e wi h Enhanced Adiaba ici y, Phys. Re . Appl. 16, 054020 (2021). [60] J. Johansson, P. Na ion, and F. No i, Qu ip: An open-sou ce Py hon amewo k o he dynamics o open quan um sys ems, Compu . Phys. Commun. 183, 1760 (2012). [61] J. Johansson, P. Na ion, and F. No i, Qu ip 2: A Py hon ame- wo k o he dynamics o open quan um sys ems, Compu . Phys. Commun. 184, 1234 (2013). [62] Y. Zhou, Z. Zhang, Z. Yin, S. Huai, X. Gu, X. Xu, J. Allcock, F. Liu, G. Xi, Q. Yu e al., Rapid and uncondi ional 043089-19 MANUEL G. ALGABA e al. PHYSICAL REVIEW RESEARCH 4, 043089 (2022) pa ame ic ese p o ocol o unable supe conduc ing qubi s, a Xi :2103.11315. [63] S. Endo, S. C. Benjamin, and Y. Li, P ac ical Quan um E - o Mi iga ion o Nea -Fu u e Applica ions, Phys.Re .X8, 031027 (2018). [64] Z. Cai, Mul i-exponen ial e o ex apola ion and combining e o mi iga ion echniques o NISQ applica ions, npj Quan um In . 7, 80 (2021). [65] M. K ebsbach, B. T auze el, and A. Calzona, Op imiza ion o Richa dson ex apola ion o quan um e o mi iga ion, a Xi :2201.08080. [66] A. M. Childs, A. Os ande , and Y. Su, Fas e quan um simula- ion by andomiza ion, Quan um 3, 182 (2019). [67] C. F. Chen, H. Y. Huang, R. Kueng, and J. A. T opp, Concen- a ion o andom p oduc o mulas, PRX Quan um 2, 040305 (2021). [68] Y. Ouyang, D. R. Whi e, and E. T. Campbell, Compila- ion by s ochas ic Hamil onian spa si ica ion, Quan um 4, 235 (2020). [69] A. Cow an, S. Dilkes, R. Duncan, A. K ajenb ink, W. Simmons, and S. Si a ajah, On he Qubi Rou ing P oblem, in 14 h Con e ence on he Theo y o Quan um Compu a- ion, Communica ion and C yp og aphy (TQC 2019), Leibniz In e na ional P oceedings in In o ma ics (LIPIcs), Vol. 135 (Schloss Dags uhl–Leibniz-Zen um ue In o ma ik, Dags uhl, Ge many, 2019), pp. 51–532, doi: 10.4230/LIPIcs.TQC.2019.5. 043089-20