Digital quantum simulation of fermionic models with a superconducting circuit

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1 Digitl quntum simultion of fermioni models with superonduting iruit R. Brends, 1 L. Lmt, J. Kelly, 3 L. Grí-Álvrez, A. G. Fowler, 1 A. Megrnt, 3, 4 E. Jeffrey, 1 T. C. White, 3 D. Snk, 1 J. Y. Mutus, 1 B. Cmpell, 3 Yu Chen, 1 Z. Chen, 3 B. Chiro, 3 A. Dunsworth, 3 I.-C. Hoi, 3 C. Neill, 3 P. J. J. O Mlley, 3 C. Quintn, 3 P. Roushn, 1 A. Vinsenher, 3 J. Wenner, 3 E. Solno,, 5 nd John M. Mrtinis 1, 3 1 Google In., Snt Brr, CA 93117, USA Deprtment of Physil Chemistry, University of the Bsque Country UPV/EHU, Aprtdo 644, E-488 Bilo, Spin 3 Deprtment of Physis, University of Cliforni, Snt Brr, CA 9316, USA 4 Deprtment of Mterils, University of Cliforni, Snt Brr, CA 9316, USA 5 IKERBASQUE, Bsque Foundtion for Siene, Mri Diz de Hro 3, 4813 Bilo, Spin. Simulting quntum physis with devie whih itself is quntum mehnil, notion Rihrd Feynmn originted [1], would e n unprllelled omputtionl resoure. However, the universl quntum simultion of fermioni systems is dunting due to their prtile sttistis [], nd Feynmn left s n open question whether it ould e done, euse of the need for non-lol ontrol. Here, we implement fermioni intertions with digitl tehniques [3] in superonduting iruit. Fousing on the Hurd model [4, 5], we perform time evolution with onstnt intertions s well s dynmi phse trnsition with up to four fermioni modes enoded in four quits. The implemented digitl pproh is universl nd llows for the effiient simultion of fermions in ritrry sptil dimensions. We use in exess of 3 single-quit nd twoquit gtes, nd reh glol fidelities whih re limited y gte errors. This demonstrtion highlights the fesiility of the digitl pproh nd opens vile route towrds nlog-digitl quntum simultion of interting fermions nd osons in lrge-sle solid stte systems. The key to simultion is mpping model Hmiltonin onto physil system. For fermioni models, suessful pproh hs een to use physil systems whih re ntively fermioni. Cold gses of fermioni toms hve performed hllmrk experiments in the nlog simultion of trnsport properties nd mgnetism [6, 7]. However, nlog systems re limited to speifi lsses of prolems, nd designing the speifi intertions is hllenging. In digitl quntum simultion, intertions n e ritrrily ontrolled; lredy, lol spin Hmiltonin ws simulted in ion trps [8]. Yet, the digitl pproh requires omplex sequenes of logi gtes, espeilly for non-lol ontrol, whih hinge on refully onstruted intertions etween susets of quits in lrger system; prtil ostle for severl pltforms. A digitl fermioni simultion hs therefore remined n open hllenge. With reent dvnes in rhiteture nd ontrol of superonduting quits [9 11], we n explore universlly implementing fermioni models, in first demonstrtion of digitl quntum simultions in the solid stte. Quntum simultion of fermioni models is highly desirle, s omputing the properties of interting prtiles is lssilly diffiult. Determining stti properties with quntum Monte Crlo tehniques is lredy omplited due to the sign prolem [1], rising from ntiommuttion, nd dynmi ehviour is even hrder. Here, we use the Jordn- Wigner trnsformtion [13] to mp fermioni models onto physil quits. In this pproh, the required numer of gtes sles only polynomilly with the numer of modes [5]. Moreover, the model system is not limited to the dimensionlity of the physil system, llowing for the simultion of fermioni models in two nd three sptil dimensions [5, 14]. Furthermore, osoni degrees of freedom, disrete [15] or ontinuous [16], my e lso introdued with n nlogdigitl quntum simultor. At low tempertures, lsses of fermioni systems n e urtely desried y the Hurd model. Here, hopping (strength V ) nd repulsion (strength U) ompete (see Fig. 1), pturing the rih physis of mny-ody intertions suh s insulting nd onduting phses in metls [17, 18]. The generi Hurd Hmiltonin is given y: H = V ( ) i,j i j + j i + U N i=1 n i n i, with the fermioni nnihiltion opertor, nd i, j running over ll djent lttie sites. The first term desries the hopping etween sites nd the lst term the on-site repulsion. It is insightful to look t fermioni two-mode exmple, H = V ( ) U 1 1. (1) We n express the fermioni opertors in terms of Puli nd ldder opertors using the Jordn-Wigner trnsformtion [13]: 1 = I σ+ nd = σ+ σ z, where the σ z term ensures ntiommuttion. In essene, we use non-lol ontrol nd mm mm 1 4 V U 14 3 U 3 = I = I = I = s I I s 1 I + s + s s z 3 + s z s z 4 + s z s z s z FIG. 1. Model nd devie. () Hurd model piture with two sites nd four modes, with hopping strength V nd on-site intertions U. The retion of one exittion from the groundstte is shown for eh mode. () Optil mirogrph of the devie. The oloured ross-shped strutures re the used Xmon trnsmon quits. The onstrution of the fermioni opertors for four modes is shown on the right { 1, } + {, 1 } = = p + + (1 + 1 ) p + + (1 + 1)

2 H= V (σx σx + σy σy ) U + (σz σz + I σz + σz I), () 4 whih n e implemented with seprtely tunle X X, Y Y, nd Z Z intertions. Here, we use the onvention to mp n exited fermioni mode 1i (exited logil quit) onto quit s physil groundstte gi, nd vuum fermioni mode i (ground logil quit) onto quit s physil exited stte ei. Our experiments use superonduting nine-quit multipurpose proessor, see Fig. 1. Devie detils n e found in Ref. [19]. The quits re the ross-shped strutures [] ptterned out of n luminium film on spphire sustrte. They re rrnged in liner hin with nerest-neighour oupling. Quits hve individul ontrol, using mirowve nd frequeny-detuning pulses (top), nd redout is done through dispersive mesurement (ottom) [1]. By frequeny tuning of the quits, intertions etween djent pirs n e seprtely turned on nd off. This system llows for implementing non-lol gtes, s it hs high level of ontrollility, nd is ple of performing high fidelity gtes [9, ]. Importntly, single- nd two-quit gte fidelities re mintined when sling the system to lrger numers of quits. The si element used to generte ll the intertions is simple generliztion of the ontrolled-phse (CZ) entngling gte (Fig. -). We implement stte-dependent frequeny pull y holding one quit stedy in frequeny nd ringing seond quit lose to the voided level rossing of eei nd gf i using n diti trjetory [3]. By tuning this trjetory, we n implement tunle CZφ gte. During this opertion, djent quits re detuned wy in frequeny to minimize prsiti intertions. The prtil rnge for φ is -4. rds; elow this rnge prsiti Z Z intertions with other quits eome relevnt, nd ove this rnge popultion strts to lek into higher energy levels (Supplementry Informtion nd Refs. [9, 19]). Using single quit gtes nd two entngling gtes, we n implement the tunle Z Z intertions, s shown in Fig.. In this gte onstrution, the π-pulses nturlly suppress dephsing [4]. First, we hve experimentlly verified tht the enoded fermioni opertors ntiommute, see Fig. d, y implementing the following ntiommuttion reltion {1, } + {, 1 } =. The ltter n e seprted into two non-trivil Hermitin terms: nd Their ssoited unitry evolution, U = exp( i φ (1 + 1 )) for the first Xp Ap CZf CZf ftul (p) 1.5 CZf gf D ee. d {1,} = {,1} mx U=exp(-i p ( + )) sz sz sz sz g/d Xp ftul (rd) f = Ap Rel Imginry mp lol fermioni Hmiltonin to lol spin Hmiltonin. The quits t s spins, nd rry the fermioni modes (Fig. 1-). A fermioni mode is either oupied or unoupied, nd spinless the spin degree of freedom is implemented here y using four modes to simulte two sites with two spins. We note tht for higher sptil dimensions this pproh is still vile, the only differene is tht the lol fermioni Hmiltonin now mps to nonlol spin Hmiltonin, whih n e effiiently implemented s reently shown [5, 14]. Using the ove trnsformtion, the Hmiltonin eomes U=exp(-i p (1 + 1)) A=Y A=X fidel (rd) =. U=exp(-i p ( )) - min I X Y Z FIG.. Gte onstrution nd opertor ntiommuttion. () Constrution of the gte U = exp( i φ σz σz ) from single quit rottions nd the tunle CZφ entngling gte. To enle smll nd negtive ngles we inlude π pulses round the X-xis (A = X) or Y -xis (A = Y ). The unitry digonls re (1 eiφ eiφ 1). () Tunle CZφ gte, implemented y moving eei (red) lose to gf i (lue). Coupling strength is g/π = 14 MHz, pulse length is 55 ns, nd typilly /π =.7 GHz when idling. () Mesured versus desired phse of the full sequene, determined using quntum stte tomogrphy. (d) Quntum proess tomogrphy of ntiommuttion. The proess mtries re shown for the non-trivil Hermitin terms of the ntiommuttion reltions (left nd enter) s well s the sequene of oth proesses, yielding n identity unitry (right). The signifint mtrix elements (red nd lue) re lose to the idel (trnsprent). one, hs een implemented using gtes with strength φ = π. The mesured proess mtries (χ) for these terms re determined using quntum proess tomogrphy, nd onstrined to e physil (Supplementry Informtion). We find tht the proesses re lose to the idel, with fidelities Tr(χidel χ) =.95,.96. As the Hermitin terms sum up to zero, their unitry evolutions omine to the identity. We find tht the sequene of oth proesses yields in ft the identity, s expeted for ntiommuttion, with fidelity of.91. We now disuss the simultion of fermioni models. We use the Trotter pproximtion P [5] to digitize the evolution of Hmiltonin H = k Hnk : U = exp( iht) ' [exp( ih1 t/n) exp( ih t/n)...], with eh prt implemented using single- nd two-quit gtes (~ = 1). We enhmrk the simultion y ompring the experimentl results to the ext digitl outome. Disretiztion unvoidly leds to devitions, nd the digitl errors re quntified in the Supplementry Informtion. We strt y visulizing the kineti intertions etween two fermioni modes. The onstrution of the Trotter step is shown in Fig. 3 nd diretly follows from the Hmiltonin in Eq.. The step onsists of the X X, Y Y nd Z Z terms, onstruted from Z Z terms nd single quit ro-

3 3 ttions. We simulte the evolution during time t y setting φ xx = φ yy = V t nd φ z = φ zz = U t/, nd using V = U = 1. We evolve the system to time of T = 5., nd inrese the numer of steps ( t = T/n, with n = 1,..., 8). The dt show hllmrk osilltions, Fig. 3, inditing tht the modes intert nd exhnge exittions. We find tht the end stte fidelity [6], tken t the sme simulted time, dereses pproximtely linerly y 54 per step (Fig. 3). The ove exmple shows tht fermioni simultions, lerly pturing the dynmis rising from intertions, n e performed digitlly using single quit gtes nd the tunle CZ φ gte. Moreover, inresing the numer of steps improves the time resolution, ut t the prie of inresing errors. A ruil result is tht the per-step derese in the end stte fidelity is onsistent with the gte fidelities. Using the typil vlues of entngling gte error nd single quit gte error s previously determined for this pltform [9], we rrive t n expeted Trotter step proess error of 7, onsidering the step onsists of six entngling gtes nd 8 single quit gtes (inluding X, Y rottions s well s idles). In ddition, we hve determined the Trotter step gte error in seprte interleved rndomized enhmrking experiment (Supplementry Informtion), nd found proess error of 74, whih is onsistent with the oserved per-step stte error. We find tht the proess fidelity is thus useful estimte, even though the simultion fidelity depends on the stte nd implemented model. Simultions of fermioni models with three nd four modes re shown in Fig. 4. The three-mode Trotter step nd its pulsesequene re shown in Figs. 4-. An implementtion of the Ŷ Ŷ gte is highlighted: the top quit (red) is pssive nd detuned wy, the middle quit (lue) is tuned to n optiml fre- Proility Y p/ f xx s z s z Y p/ Y -p/ Y -p/ two-mode Trotter step X -p/ f yy X -p/ s z s z V s x s s x V s y y X p/ f zz s z s z X p/ idel Time T Z fz Z fz U 4 (s z s z + I s z + s z I) steps= Trotter steps 7 8 End stte fidelity 4 8 Trotter steps FIG. 3. Simultion of two fermioni modes. () Constrution of the two-mode Trotter step, showing the seprte terms of the Hmiltonin (Eq. ). See Supplementry Informtion for the pulse sequene nd gte ount. () Proility of the modes versus simulted time for n = 1,..., 8 steps. Input stte is [( + 1 ) 1 ]/, nd V = U = 1. The idel dependene is shown in the ottom right. () The end-stte fidelity dereses with step y 54, following liner trend. U s z s z Dt Stte fidelity Proility d e Proility CZ f Z fz detuning f zz s z s z XY gte p Z fz Z fz f z = f zz = U 4 Dt p XY gte.75 A p/ A p/ A p/.75 f V s z s z A p/ f V1 s z s z f V1 = V 1 Dt f zz s z s z three-mode Trotter step f yy s y s y Z fu Z fu f xx s x s x f xx = four-mode Trotter step A -p/ A -p/ A -p/ A -p/ f V = V f U s z s z B p/ B p/ B p/ f yy = f V f yy s y s y V Dt f V1 s z s z s z s z B p/ Dt f U = U 4 Dt odd: A=Y, B=X even: A=X, B=Y f xx s x s x 1.5 Time (ms) Trotter steps Trotter steps 11 U= U= Trotter steps f Stte fidelity 3 modes 4 modes B -p/ B -p/ B -p/ B -p/ 1 3 Trotter steps FIG. 4. Fermioni U 14= models U 3=1 with three nd four modes. () Threemode Trotter step, with the Trotter step pulse-sequene in (). The Trotter step onsists of 1 entngling gtes nd 87 single quit gtes (see text). The Ŷ Ŷ intertion is highlighted (dshed). () Simultion results for three modes with nd without on-site intertion. Full symols: experiment. Open symols: idel digitized. Blk symols: popultion of other sttes. Input stte is [ 1 ( )]/, nd V = 1. (d) Constrution of the four mode Trotter step. (e) Four mode simultion results for V 1 = V = 1, U 3 = 1, nd U 14 =. Input stte is [( ) ( )]/. (f) Simultion fidelities. queny for the intertion, nd the ottom quit (green) performs the diti trjetory. π-pulses on the pssive quit suppress dephsing nd prsiti intertions. Fig. 4 shows the simultion results for V = 1, U = (hopping only) nd V = 1, U = 1 (with on-site repulsion). Input stte genertion is shown in the Supplementry Informtion. The simultion dt (losed symols) follows the ext digitl outome (open symols), umulting per-step error of.15 (Fig. 4f) nd grdully populting other sttes (lk symols). The fidelity is the relevnt figure of merit, espeilly for simultions with few steps; the per-step error eing the sme for different models indites tht the simulted time evolutions re distint.

4 4 d Intertion strength Fidelity Proility Proility insulting modes 3 modes U V metlli V 1 V Dt 1 Dt Time FIG. 5. Simultions with time-vrying intertions. () The system is hnged from n insulting stte to onduting phse, y rmping the hopping term V from zero to one. Inset shows the hoie of digitiztion on the rmp for the two-mode simultion. () Twomode simultion showing dynmi ehviour strting t the onset of the V rmp. () Three mode simultion, showing non-trivil dynmis when the hopping term is nonzero. (d) Simultion fidelities. For the four-mode experiment, we simulte n symmetri vrition on the Hurd model. Here, the repulsive intertion is etween the middle modes only (right well in Fig. 1), while the hopping terms re kept equl. Asymmetri models re used in desriing nisotropi fermioni systems [7]. In ddition, the simultion n e optimized: gte ount is redued y the removl of intertion etween the top nd ottom modes, nd the Trotter expnsion n e rewritten in terms of odd nd even steps suh tht the strting nd ending single-quit gtes nel (Supplementry Informtion). The Trotter step is shown in Fig. 4d. The results re plotted in Fig. 4e. We find tht the stte fidelity dereses y.17 for the four mode simultion, see Fig. 4f. The three- nd four-mode experiments underline tht fermioni models n e simulted digitlly with lrge numers of gtes. The three-mode simultion uses in exess of 3 gtes. We perform three Trotter steps, nd per step we use: 1 entngling gtes, 53 mirowve π nd π/ gtes, 19 idle gtes, 3 single-quit phse gtes, nd for the nonprtiipting quit during the entngling opertion: 1 frequeny detuning gtes where phses need to e urtely trked. Using the ove typil errors for gtes, we rrive t n estimted proess error of.16 for the three mode simultion, nd n error of.15 for the four-mode simultion (per four mode Trotter step: 1 entngling gtes nd 98 single quit gtes). The proess errors re lose to the oserved drop in stte fidelity. Importntly, these results strongly suggest tht the simultion errors sle with the numer of gtes, not quits (modes), whih is ruil spet of slly implementing models on our pltform. We now ddress the simultion of fermioni systems with time-dependent intertions. Dt 1 Dt In Fig. 5, we show n experiment where we rmp the hopping term V from to 1 while keeping the on-site repulsion U t 1; essentilly hnging the system from n insulting to metlli phse. This trnsition is simulted for two modes using two Trotter steps, see inset, nd with one step for three modes. For the ltter se, we tke the verge of V over the relevnt time domin. The dt re shown in Figs. 5-, nd lerly mirror the dynmis of the hopping term. At time smller thn, the system is frozen nd the mode proilities re virtully unhnged, refleting the insulting stte. Intertions eome visile when hopping is turned on, effetively melting the system, nd follow the generi fetures of the ext digitl outome (dshed). The simultion fidelities lie round for two modes nd.7-.8 for three modes, see Fig. 5d. These fidelities re round or somewht elow those for time evolution with onstnt intertions, presumly due to ontrol errors relted to prsiti quit intertions, whih lso led to the populting of other sttes (lk symols). The dynmi simultion highlights the possiilities of exploring prmeter spes nd trnsitions with few steps. We hve demonstrted the digitl quntum simultion of fermioni models. Simultion fidelities re lose to the expeted vlues, nd with improvements in gtes nd rhiteture, the onstrution of lrger testeds for fermioni systems ppers vile. Bosoni modes n e elegntly introdued y dding liner resontors to the iruit, estlishing fermionoson nlog-digitl system [15, 16] s distint prdigm for quntum simultion. Methods Summry Experiments re performed in wet dilution refrigertor with se temperture of mk. Quit frequenies re hosen in stggered pttern to minimize unwnted intertion. Typil quit frequenies re 5.5 nd 4.8 GHz. Ext frequenies re optimized sed on the quits e nd f stte spetr long the fully tunle trjetory of the CZ φ -gte, s well s minimizing intertion etween next-nerest neighouring quits. Used quits re Q1-Q4 in Ref. [19]. Dt re orreted for mesurement fidelity. Aknowledgements We thnk A. N. Korotkov for disussions. The uthors knowledge support from Spnish MINECO FIS C3-; Rmón y Cjl Grnt RYC ; UPV/EHU UFI 11/55 nd EHUA14/4; Bsque Government IT47-1; UPV/EHU PhD grnt; PROMISCE nd SCALEQIT EU projets. Devies were mde t the UC Snt Brr Nnofrition Fility, prt of the NSF-funded Ntionl Nnotehnology Infrstruture Network, nd t the NnoStrutures Clenroom Fility. Author Contriutions R.B., L.L., nd L.G.-Á. designed the experiment, with J.M.M. nd E.S. providing supervision. L.G.-Á., L.L., nd E.S. provided the theoretil frmework. R.B. nd L.L. owrote the mnusript with J.M.M. nd E.S. The experiment nd dt were performed nd nlysed y R.B., J.K., L.L., nd L.G.-Á. R.B. nd J.K. designed the devie. J.K., R.B., nd A.M. frited the smple. All uthors ontriuted to the frition proess, experimentl set-up nd

5 5 mnusript preprtion. [1] Feynmn, R. P. Simulting physis with omputers. Int. J. Th. Phys. 1, (198). [] Altlnd, A., nd Simons, B., Condensed Mtter Field Theory (Cmridge University Press, 1). [3] Lloyd, S. Universl Quntum Simultors. Siene 73, 173 (1996). [4] Hurd, J. Eletron Correltions in Nrrow Energy Bnds. Pro. R. So. London Ser. A 76, 38 (1963). [5] Ls Hers, U., Grí-Álvrez, L., Mezzpo A., Solno E., nd Lmt, L. Fermioni Models with Superonduting Ciruits. rxiv: [6] Shneider, U. et l. Fermioni trnsport nd out-of-equilirium dynmis in homogeneous Hurd model with ultrold toms. Nture Phys. 8, p. 13 (1). [7] Greif, D. et l. Short-Rnge Quntum Mgnetism of Ultrold Fermions in n Optil Lttie. Siene 34, p. 137 (13). [8] Lnyon, B. P. et l. Universl Digitl Quntum Simultion with Trpped Ions, Siene 334, p. 57 (11). [9] Brends, R. et l. Superonduting quntum iruits t the surfe ode threshold for fult tolerne. Nture 58, 5-53 (14). [1] Coroles, A. D. et l. Proess verifition of two-quit quntum gtes y rndomized enhmrking. Phys. Rev. A 87, 331(R) (13). [11] Vesterinen, V., Sir, O.-P., Bruno, A., nd DiCrlo, L. Mitigting informtion lekge in rowded spetrum of wekly nhrmoni quits, rxiv: [1] Troyer, M. nd Wiese, U.-J. Computtionl Complexity nd Fundmentl Limittions to Fermioni Quntum Monte Crlo Simultions. Phys. Rev. Lett. 94, 171 (5). [13] Jordn, P. nd Wigner, E. Üer ds Pulishe Äquivlenzverot. Z. Phys. 47, 631 (198). [14] Csnov, J., Mezzpo, A., Lmt, L., nd Solno, E. Quntum Simultion of Interting Fermion Lttie Models in Trpped Ions. Phys. Rev. Lett. 18, 195 (1). [15] Lmt, L., Mezzpo, A., Csnov, J., nd Solno, E. Effiient quntum simultion of fermioni nd osoni models in trpped ions. EPJ Quntum Tehnology 1, 9 (14). [16] Grí-Álvrez, L. et l. Fermion-Fermion Sttering in Quntum Field Theory with Superonduting Ciruits. Phys. Rev. Lett., in press. [17] Jördens, R. et l. A Mott insultor of fermioni toms in n optil lttie, Nture 455, p. 4 (8). [18] Shneider, U. et l. Metlli nd Insulting Phses of Repulsively Interting Fermions in 3D Optil Lttie. Siene 3, p. 15 (8). [19] Kelly, J. et l. Stte preservtion y repetitive error detetion in superonduting quntum iruit, rxiv: [] Brends, R. et l. Coherent Josephson quit suitle for slle quntum integrted iruits. Phys. Rev. Lett. 111, 85 (13). [1] Wllrff, A. et l. Strong oupling of single photon to superonduting quit using iruit quntum eletrodynmis. Nture 431, (4). [] Brends, R. et l. Rolling quntum die with superonduting quit. Phys. Rev. A 9, 333(R) (14). [3] Mrtinis, J. M. nd Geller, M. R. Fst diti quit gtes using only σ z ontrol. Phys. Rev. A 9, 37 (14). [4] O Mlley, P. J. J. et l. Overoming orrelted noise in quntum systems: How mediore loks mke good quits. rxiv: [5] Suzuki, M. Frtl deomposition of exponentil opertors with pplitions to mny-ody theories nd Monte Crlo simultions. Phys. Lett. 46, 319 (199). [6] The stte fidelity is omputed using k Pk,idel P k, whih is equl to Ψ idel Ψ to first order. Here, P k,idel nd P k re proilities nd k runs over the omputtionl sis. The onsisteny with mesured proess fidelities, nd the sling of the simultion fidelity with steps justify this pproh. [7] Dutt, O. et l. Non-stndrd Hurd models in optil ltties: review. rxiv:

6 Supplementry Informtion for Digitl quntum simultion of fermioni models with superonduting iruit R. Brends, 1 L. Lmt, J. Kelly, 3 L. Grí-Álvrez, A. G. Fowler, 1 A. Megrnt, 3, 4 E. Jeffrey, 1 T. C. White, 3 D. Snk, 1 J. Y. Mutus, 1 B. Cmpell, 3 Yu Chen, 1 Z. Chen, 3 B. Chiro, 3 A. Dunsworth, 3 I.-C. Hoi, 3 C. Neill, 3 P. J. J. O Mlley, 3 C. Quintn, 3 P. Roushn, 1 A. Vinsenher, 3 J. Wenner, 3 E. Solno,, 5 nd John M. Mrtinis 1, 3 1 Google In., Snt Brr, CA 93117, USA Deprtment of Physil Chemistry, University of the Bsque Country UPV/EHU, Aprtdo 644, E-488 Bilo, Spin 3 Deprtment of Physis, University of Cliforni, Snt Brr, CA 9316, USA 4 Deprtment of Mterils, University of Cliforni, Snt Brr, CA 9316, USA 5 IKERBASQUE, Bsque Foundtion for Siene, Mri Diz de Hro 3, 4813 Bilo, Spin. I. TROTTER STEP PULSE SEQUENCES A. Pulse sequenes nd gte ounts The two-, three- nd four-mode Trotter step pulse sequenes re shown in Fig. S1. The gte ounts n e found in Tle S1. stte preprtion re suppressed y performing tomogrphy on zero-time idle. The χ mtries for proesses U 1 = exp( i π ( )) nd U = exp( i π ( 1 + 1)) re determined experimen- two-mode Trotter step B. Initiliztion The gte sequenes for the initiliztion of the three- nd four-mode simultion re shown in Fig. S. For the two-mode simultion the input stte is: [( + 1 ) 1 ]/, for three modes: [ 1 ( )]/, nd for four modes: [( ) ( )]/. three-mode Trotter step II. QUANTUM PROCESS TOMOGRAPHY We use quntum proess tomogrphy to determine the χ mtrix. We strt y initilizing the quits into the ground stte, nd prepre input sttes y pplying gtes from {I, X/, Y/, X}. The proess output is reonstruted y pplying gtes from the sme group, essentilly otining the 16 output density mtries. The χ mtrix is then determined using qudrti mximum likelihood estimtion, using the MATLAB pkges SeDuMi nd YALMIP, while onstrining it to e Hermitin, tre-preserving, nd positive semidefinite; the estimtion is overonstrined. Non-idelities in mesurement nd four-mode Trotter step CZ f TABLE S1. Gte ounts for the two-, three-, nd four-mode Trotter step, determined using Fig. S1. We ount idles s hving the sme durtion s the mirowve π nd π/ gtes; this is the relevnt pproh for estimting totl proess fidelities. The gte ounts re for single Trotter step only, nd exlude input stte preprtion. Gtes two-mode three-mode four-mode entngling CZ φ single quit mirowve π nd π/ idle detuning virtul phse 3 detuning p XY gte XY gte p 1.5 Time (ms) FIG. S1. Pulse sequenes for single two-mode (), three-mode (), nd four-mode () Trotter step. Shown re entngling gtes s well s single-quit mirowve, idle nd detuning gtes. The legend is in the ottom right.

7 p tlly, nd the mtrix of proess U U 1 is omputed from the experimentlly otined mtries following Ref. [1]. The used quntum iruits re nd e i π ( ) Y/ Y/ Y/ X/ X/ X/ e i π ( ) Y/ Y/ Y/ X/ X/ X/ III. RANDOMIZED BENCHMARKING OF exp( i π σz σz) AND THE TWO-MODE TROTTER STEP 4 The proess fidelity of the exp( i φ σ z σ z ) gte nd the two-mode Trotter step re determined using interleved Clifford-sed rndomized enhmrking [ 4]. This tehnique is insensitive to mesurement nd stte preprtion error, nd determines the fidelity properly verged over ll input sttes, ut it restrits the gtes to hve unitry whih lies within the group of Cliffords. As representtive ngles we hve therefore used φ = π/, nd φ xx = φ yy = φ zz = π/ for the Trotter step. three-mode init. X p/ X p/ X p/ four-mode init. FIG. S. Initiliztion gte sequene. () Three-mode initiliztion. () Four-mode initiliztion. Sequene fidelity p exp(-i s z s z ) 4 error: Trotter step error: Numer of Cliffords - m referene FIG. S3. Clifford-sed rndomized enhmrking of exp( i π 4 σz σ z) nd the two-mode Trotter step. Sequene fidelity versus numer of Cliffords. Blk: referene. Colour: interleved. X p/ X p/ X p/ X p/ X p/ X p/.75 Proility Proility 1 Step 3 1 Step U= U= Step Stte fidelity 3 modes, U= 3 modes, U=1 4 modes 1 3 Step FIG. S4. Digitl error for the time-independent simultion. () Three mode simultion (U =, U = 1, V = 1). () Four mode simultion (U 3 = 1, U 14 =, V = 1). () Fidelity. Idel evolution (solid lines) nd ext digitl solution (open symols onneted y dshed lines). The dt re shown in Fig. S3. We strt y mesuring the dey in sequene fidelity of sequenes of rndom, twoquit Cliffords (lk symols). When interleving we see n extr derese of sequene fidelity, whih n e linked to the proess fidelity of the interleved gte. We find tht the exp( i π 4 σ z σ z ) gte nd the Trotter step hve errors of nd 74, respetively. We note tht these vlues re onsistent with estimtion y dding individul gte errors (min Letter). IV. DIGITAL ERROR The Trotter expnsion introdues digitl errors due to disretiztion. A full nlysis of the digitl error for the used model n e found in Ref. [5]. For the time-independent model, the two-mode simultion hs zero digitl error. For the three- nd four-mode simultion the full evolution (solid lines), ext digitl solution (open symols onneted y dshed lines), nd fidelities due to digitl error re shown in Fig. S4. For the time-dependent model we find negligile digitl error for two modes, nd signifint error for three, see Fig. S5. The lrge error for three modes rises from hving to pproximte lrger Hmiltonin, s well s using only single step. V. MINIMIZING LEAKAGE OF THE CZ φ GATE The tunle CZ φ gte works y tuning the frequeny of one of the quits to pproh the voided level rossing of the ee nd gf sttes, using n diti trjetory [6]. For lrge phses we need to losely pproh the voided level rossing, induing stte lekge.

8 3 Proility 11 fidelity Time fidelity Time FIG. S5. Digitl error for the time-dependent simultion for two modes, using two Trotter steps () nd three modes, using one Trotter step (). Idel evolution (solid lines), ext digitl solution (dshed lines), nd fidelity (solid lk). To minimize suh lekge we hve hosen to inrese the length of the CZ φ gte from typil 4 ns [7] to 55 ns. However, for lrge phses (> 4. rds), see Fig. S6, we still see onsiderle mount of lekge, see the Fig. S6. By hoosing the leked stte popultion s fitness metri, nd using Nelder-Med optimiztion in similr pproh to Ref. [8] to tune wveform prmeters, see Figs. S6-d, we n signifintly suppress lekge. We note tht this optimiztion took pproximtely one minute in rel time. P lf> phse (rd) Amplitude (GHz) NM optimiztion fter Amplitude (GHz) 1 P lf> 5 efore VI. ASYMMETRIC HUBBARD MODEL Here, we inlude the nlysis of the fermioni symmetri Hurd model for 4 quits employed in the Letter. Firstly, we present the model in terms of spin opertors vi the Jordn- Wigner trnsformtion, nd desrie different limits of the model. Seondly, we nlyse the digitl quntum simultion in terms of Trotter steps involving the optimized gtes (CZ φ ). The symmetri Hurd model (AHM) is vrition of the Hurd model tht desries nisotropi fermioni systems. Here, we re going to onsider this model for two different fermioni speies, tht ould represent spins, interting with eh other y the Coulom term, nd two lttie sites. The opertors for this model hve two indies, A ij, where i nd j indite the site position nd kind of prtile, respetively. Sine the fermions might hve different msses, we hve no reson to ssume tht the hopping terms will e the sme. We n write the Hmiltonin for two sites, x nd y, nd two kinds of fermions, 1 nd, s ) H = V 1 ( x1 y1 + y1 x1 ) V ( x y + y x + U x x1 x1 x x + U y y1 y1 y y, (S1) where mi nd mi re fermioni retion nd nnihiltion opertors of the kind of prtile i for the site m. For the min Letter we use 1,, 3, 4, for x1, y1, y, x. The Jordn-Wigner trnsformtion will e used in our derivtion to relte the fermioni nd ntifermioni opertors d frtionl hnge 1 3 Nelder-Med funtion evlutions 3 l l 1 l l Nelder-Med funtion evlutions FIG. S6. Minimizing lekge of the CZ φ gte. () Tunle phse versus pulse mplitude, determined with quntum stte tomogrphy. () Zoom-in of the mplitude region for lrge phses, showing the f -stte popultion efore (lue) nd fter (red) Nelder-Med optimiztion. () Popultion of f versus Nelder-Med funtion evlution, showing downwrds trend. (d) Optimiztion of the wveform prmeters with Nelder-Med funtion evlution, see Ref. [6] for the definition of these prmeters. with tensor produts of Puli mtries, whih re opertors tht we n simulte in the superonduting iruit setup. This trnsformtion is sed on mpping etween fermioni opertors nd spin-1/ opertors. In this se, the reltions re x1 = I I I σ+ y1 = I I σ+ σ z y = I σ+ σ z σ z x = σ+ σ z σ z σ z. (S) After this mpping, Hmiltonin (S1) is rewritten in terms

9 4 of spin-1/ opertors s H = V 1 (I I σx σ x + I I σ y σ y ) + V (σx σ x I I + σ y σ y I I) + U x ( σ z I I σ z + I I I σ z + σ z I I I ) 4 + U y ( I σ z σ z I + I I σ z I + I σ z I I ), (S3) 4 where the different intertions n e simulted vi digitl tehniques in terms of single quit nd CZ φ gtes. A. Gte deomposition We onsider the digitl quntum simultion of the dynmis of Hmiltonin (S3). The Trotter expnsion onsists of dividing the time t into n time intervls of length t/n, nd pplying sequentilly the evolution opertor of eh term of the Hmiltonin for eh time intervl. In this se the evolution opertors re ssoited with the different summnds of the Hmiltonin. In order to desrie the digitl simultion in terms of Trotter steps involving the optimized gtes (CZ φ ), we will first onsider the Hmiltonin in terms of exp[ i(φ/)σ z σ z ] intertions. We tke into ount the reltions σ x σ x = R y (π/)σ z σ z R y ( π/) σ y σ y = R x ( π/)σ z σ z R x (π/), (S4) where R j (θ) = exp( i θ σj ) is the rottion long the j oordinte of quit. In these expressions the rottions re pplied on the two quits of the produt. The evolution opertor ssoited with Hmiltonin (S3) in terms of exp[ i(φ/)σ z σ z ] intertions is e iht k ( ) n e ih k t n ( R y (π/)e i V 1 I I σ z σ z t n Ry ( π/)r x ( π/)e i V 1 I I σ z σ z t n Rx (π/) R y (π/)e i V σ z σ z I I t n Ry ( π/)r x ( π/)e i V σ z σ z I I t n Rx (π/) Ux i e 4 σz I I σ z t n e i Ux 4 I I I σz t n e i Ux 4 σz I I I t n ) n Uy i e 4 I σz σ z I t n e i Uy 4 I I σz I t n e i Uy 4 I σz I I t n. (S5) Note tht, in priniple, the ordering of the gtes inside Trotter step does not hve sizle effet s fr s there re enough Trotter steps. Here, the numer of Trotter steps is limited (n pproximtely 1) nd different orderings will hve different results. The different vlues in the orderings differ in O(1) onstnt, while the glol digitl error depends on the numer of Trotter steps n s 1/n (the differene in errors due to different orderings does not depend on n). If we onsider the Trotter error, the fidelity ould inrese with n optiml ordering where we group terms of the Hmiltonin tht ommute with eh other. Nevertheless, from the experimentl point of view, the opertors n e rerrnged in more suitle wy in order to optimize the numer of gtes nd eliminte glol phses. In this sense, we must look for the optiml ordering y onsidering oth spets. Here, we simply rerrnge the opertors in order to optimize the numer of gtes. If we onsider tht R j (α) + R j (β) = R j (α + β), then

10 5 n/ ( e iht R y(π/)e i V 1 I I σ z σ z t n R y( π/)r y (π/)e i V σ z σ z I I t n Ry ( π/) i=1 Ux i e 4 σz I I σ z t n e i Ux 4 I I I σz t n e i Ux 4 σz I I I t n Uy i e 4 I σz σ z I t n e i Uy 4 I I σz I t n e i Uy 4 I σz I I t n ) σ z σ z I I t n Rx (π/) R x( π/)e i V 1 I I σ z σ z t n R x(π/)r x ( π/)e i V ( R x( π/)e i V 1 I I σ z σ z t n R x(π/)r x ( π/)e i V σ z σ z I I t n Rx (π/) Ux i e 4 σz I I σ z t n e i Ux 4 I I I σz t n e i Ux 4 σz I I I t n Uy i e 4 I σz σ z I t n e i Uy 4 I I σz I t n e i Uy 4 I σz I I t n R y(π/)e i V 1 I I σ z σ z t n R y( π/)r y (π/)e i V σ z σ z I I t n Ry ( π/) ) i 1, (S6) i where we use the prime nottion in the rottion to distinguish etween gtes pplied on different quits. This deomposition etween even nd odd Trotter steps is suitle in order to simplify rottions in x nd y, nd, therefore, void higher numer of gtes. The sequene of gtes for one odd Trotter step in the digitl simultion of the Hurd model with four quits is Y π/ Y π/ Z x X π/ X π/ Y π/ Y π/ B y Z y X π/ X π/ B x Y π/ A Y π/ Z y X π/ A X π/ Y π/ Y π/ Z x X π/ X π/ nd for one even Trotter step: X π/ X π/ Z x Y π/ Y π/ X π/ X π/ B y Z y Y π/ Y π/ B x X π/ A X π/ Z y Y π/ A Y π/ X π/ X π/ Z x Y π/ Y π/ The gtes A i nd B j re two-quit gtes in terms of the exp[ i(φ/)σ z σ z ] intertions: A i = exp(i Vi σz σ z t n ) nd B j = exp( i Uj 4 σz σ z t n ). The Z i gtes re single quit rottions: Z i = exp( i Ui 4 σz t n ), nd X α nd Y α re rottions long the x nd y xis, respetively. The exp[ i(φ/)σ z σ z ] intertion n e implemented in smll steps with optimized CZ φ gtes. The intertion is 1 e i φ σz σ z = e iφ e iφ. 1 The quntum iruits for simulting this re shown in the min Letter.

11 6 B. Prtiulr se of the model In order to void the gte B x etween the first nd the fourth quit, we n onsider prtiulr se of the symmetri Hurd model, where U x =. In this se, the iruit is the sme ut without the B x nd the Z x gtes. Tht is, for one odd Trotter step: Y π/ Y π/ X π/ X π/ Y π/ Y π/ B y Z y X π/ X π/ Y π/ A Y π/ Z y X π/ A X π/ Y π/ Y π/ X π/ X π/ nd for one even Trotter step: X π/ X π/ Y π/ Y π/ X π/ X π/ B y Z y Y π/ Y π/ X π/ A X π/ Z y Y π/ A Y π/ X π/ X π/ Y π/ Y π/ It is importnt to note tht, for n = Trotter steps, the red gtes nel eh other, nd we redue the numer of gtes tht should e pplied. For n >, the lue gtes lso nel eh other exept in the eginning nd in the end of the quntum simultion. is the following = exp( i V1 σz σ z t n ) Φ = V1 t n A = exp( i V σz σ z t n ) Φ A = V t n B y = exp( i Uy 4 σz σ z t n ) Φ B y = Uy t 4 n Z y = exp( i Uy 4 σz t n ) Φ = Uy t 4 n. C. Digitl quntum simultion of the model The reltion mong the vlues of the prmeters in the numeril simultions nd the vlues of the phses in the gtes Notie tht in the numeril simultions we onsider = 1 for simpliity. In summry, the fermioni symmetri Hurd model with two exittions, one for eh kind of fermion, hs een nlysed nd expressed in terms of simultle spin opertors. We hve onsidered the digitl quntum simultion in terms of Trotter steps involving the optimized gtes (CZ φ ). This is the four-mode system experimentlly simulted in the min Letter. [1] Korotkov, A. N. Error mtries in quntum proess tomogrphy. rxiv: [] Ryn, C. A., Lforest, M., nd Lflmme, R. Rndomized enhmrking of single-nd multi-quit ontrol in liquid-stte

12 7 NMR quntum informtion proessing. New J. Phys. 11, 1334 (9) [3] Brown, K. R. et l. Single-quit-gte error elow 1 4 in trpped ion. Phys. Rev. A. 84, 333 (11). [4] Coroles, A. D. et l. Proess verifition of two-quit quntum gtes y rndomized enhmrking. Phys. Rev. A 87, 331(R) (13). [5] Ls Hers, U., Grí-Álvrez, L., Mezzpo A., Solno E., nd Lmt, L. Fermioni Models with Superonduting Ciruits. rxiv: [6] Mrtinis, J. M. nd Geller, M. R. Fst diti quit gtes using only σ z ontrol. Phys. Rev. A 9, 37 (14). [7] Brends, R. et l. Superonduting quntum iruits t the surfe ode threshold for fult tolerne. Nture 58, 5-53 (14). [8] Kelly, J. et l. Optiml quntum ontrol using rndomized enhmrking, Phys. Rev. Lett. 11, 454 (14).