LETTER. Quantum annealing with manufactured spins

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1 doi:.38/nture2 Quntum nneling with mnufctured spins M. W. Johnson, M. H. S. Amin, S. Gildert, T. Lnting, F. Hmze, N. Dickson,R.Hrris, A. J. Berkley, J. Johnsson 2, P. Bunyk, E. M. Chpple, C. Enderud, J. P. Hilton, K. Krimi, E. Ldizinsky, N. Ldizinsky,T.Oh, I. Perminov, C. Rich, M. C. Thom, E. Tolkchev, C. J. S. Truncik 3, S. Uchikin, J. Wng, B. Wilson & G. Rose Mny interesting ut prcticlly intrctle prolems cn e reduced to tht of finding the ground stte of system of intercting spins; however, finding such ground stte remins computtionlly difficult. It is elieved tht the ground stte of some nturlly occurring spin systems cn e effectively ttined through process clled quntum nneling 2,3. If it could e hrnessed, quntum nneling might improve on known methods for solving certin types of prolem 4,5. However, physicl investigtion of quntum nneling hs een lrgely confined to microscopic spins in condensed-mtter systems 6 2. Here we use quntum nneling to find the ground stte of n rtificil Ising spin system comprising n rry of eight superconducting flux quntum its with progrmmle spin spin couplings. We oserve cler signture of quntum nneling, distinguishle from clssicl therml nneling through the temperture dependence of the time t which the system dynmics freezes. Our implementtion cn e configured in situ to relize wide vriety of different spin networks, ech of which cn e monitored s it moves towrds low-energy configurtion 3,4. This progrmmle rtificil spin network ridges the gp etween the theoreticl study of idel isolted spin networks nd the experimentl investigtion of ulk mgnetic smples. Moreover, with n incresed numer of spins, such system my provide prcticl physicl mens to implement quntum lgorithm, possily llowing more-effective pproches to solving certin clsses of hrd comintoril optimiztion prolems. Physiclly interesting in their own right, systems of intercting spins lso hve prcticl importnce for quntum computtion 5. One widely studied exmple is the Ising spin model, where spins my tke on one of two possile vlues: up or down long preferred xis. Mny seemingly unrelted yet importnt hrd prolems, in fields rnging from rtificil intelligence 6 to zoology 7, cn e reformulted s the prolem of finding the lowest energy configurtion, or ground stte, of n Ising spin system. Quntum nneling hs een proposed s n effective wy for finding such ground stte 2 5. To implement processor tht uses quntum nneling to help solve difficult prolems, we would need progrmmle quntum spin system in which we could control individul spins nd their couplings, perform quntum nneling nd then determine the stte of ech spin. Until recently, physicl investigtion of quntum nneling hs een confined to configurtions chievle in condensed-mtter systems, such s moleculr nnomgnets 6 or ulk solids with quntum criticl ehviour,2. Unfortuntely, these systems cnnot e controlled or mesured t the level of individul spins, nd re typiclly investigted through the mesurement of ulk properties. They re not progrmmle. Nucler mgnetic resonnce techniques hve een used to demonstrte quntum nneling lgorithm on three quntum spins 8. Recently, three trpped ions were used to perform quntum simultion of smll, frustrted Ising spin system 9. One possile implementtion of n rtificil Ising spin system involves superconducting flux quntum its 2 28 (quits). We hve implemented such spin system, interconnected s iprtite grph, using n in situ reconfigurle rry of coupled superconducting flux quits 4. The device friction is discussed in Methods nd in Supplementry Informtion. The simplified schemtic in Fig. shows two superconducting loops in the quit, ech suject to n externl flux is W x or W 2x, respectively. The device dynmics cn e modelled s quntum mechnicl doule-well potentil with respect to the flux, W, in loop (Fig. ). The rrier height, du, is controlled y W 2x. The energy difference etween the two minim, 2h, is controlled y W x. The two lowest energy sttes of the system, corresponding to clockwise or nticlockwise circulting current in loop, re lelled j#æ nd j"æ, with flux loclized in the left- or the right-hnd well (Fig. ), respectively. If we consider only these two sttes ( vlid restriction t low temperture), the quit dynmics is equivlent to those of n Ising spin, nd we tret the quits s such in wht follows. Quits (spins) re Φ 2x Φ x Josephson junction U Φ Φ 2 ω p Spin-down circulting current δu Spin-up circulting current Figure Superconducting flux quit., Simplified schemtic of superconducting flux quit cting s quntum mechnicl spin. Circulting current in the quit loop gives rise to flux inside, encoding two distinct spin sttes tht cn exist in superposition., Doule-well potentil energy digrm nd the lowest quntum energy levels corresponding to the quit. Sttes "æ nd #æ re the lowest two levels, respectively. The intr-well energy spcing is v p. The mesurement detects mgnetiztion, nd does not distinguish etween, sy, "æ nd excited sttes within the right-hnd well. In prctice, these excittions re exceedingly improle t the time the stte is mesured. 2h ωp Φ D-Wve Systems Inc., -44 Still Creek Drive, Burny, British Columi V5C 6G9, Cnd. 2 Deprtment of Nturl Sciences, University of Agder, Post Box 422, NO-464 Kristinsnd, Norwy. 3 Deprtment of Physics, Simon Frser University, Burny, British Columi V5A S6, Cnd. 94 NATURE VOL MAY 2

2 RESEARCH coupled together using progrmmle coupling elements 29 which provide spin spin coupling energy tht is continuously tunle etween ferromgnetic nd ntiferromgnetic coupling. This llows spins to fvour lignment or nti-lignment, respectively. The ehviour of this system cn e descried with n Ising model Hmiltonin H P ~ XN i~ h i s z i z XN i,j~ J ij s z i sz j where for spin i s z i is the Puli spin mtrix with eigenvectors {j"æ, j#æ} nd 2h i is the energy is; nd 2J ij is the coupling energy etween the spins i nd j. Our implementtion llows ech J ij nd h i to e progrmmed independently within the constrints of the connectivity of our devices. The quntum mechnicl properties of the individul devices hve een well chrcterized 3, ut we re interested in wht hppens when severl of them re coupled together. It is resonle to sk whether this mnufctured, mcroscopic (, mm) system of rtificil spins ehves quntum mechniclly. We report here on n experiment tht demonstrtes signture of quntum nneling in coupled set of eight rtificil Ising spins. Wheres therml nneling uses progressively weker therml fluctutions to llow system to explore its energy lndscpe nd rrive t low-energy configurtion, quntum nneling uses progressively weker quntum fluctutions, medited y tunnelling. In oth therml nd quntum nneling, system strts with mixture of ll possile sttes: clssicl mixed stte in the former nd coherent superposition in the ltter. Quntum nneling cn e performed y slowly chnging the system Hmiltonin H(t)~C(t) XN i~ D i s x i zl(t)h P where C decreses from one to zero nd L increses from zero to one monotoniclly with time, nd D i prmeterizes quntum mechnicl tunnelling etween j"æ nd j#æ. At the eginning of the nneling, C 5, L 5 nd the system is fully chrcterized y the trnsverse terms, P N i~ D is x i. The ground stte of this is superposition of ll sttes in the s z sis. It is strightforwrd to initilize the system in this stte. During quntum nneling, the trnsverse term is grdully turned off (C R ) nd the weight of the Ising Hmiltonin, H P, is incresed (L R ) (Fig. 2). If this nneling is done slowly enough, the system should remin in the ground stte t ll times, thus ending up in the ground stte of H P (ref. 4). The ove description of quntum nneling is in the lnguge of n idel Ising spin system. Let us look more closely t wht this mens for n individul flux quit. During nneling, the energy rrier, du(t), etween the two wells is grdully rised (Fig. 2). If therml fluctutions re dominnt, then the quit dynmics my e viewed s therml ctivtion over the rrier with rte tht is proportionl to e {du=kbt t temperture T (k B, Boltzmnn s constnt). This suggests tht the dynmics stops when du?k B T. Becuse du is incresing with time, this freezing out hppens t t<tfreeze TA, where du(tta freeze )<k BT. Within the relevnt regime, du is nerly liner in time, therefore clssiclly we expect tfreeze TA to e linerly dependent on T. If, however, the dominnt fluctutions re quntum mechnicl, then the quit my tunnel etween the two wells, tht is, etween sttes j#æ nd j"æ. Rising the rrier, y incresing du, reduces this tunnelling until t some point it ecomes negligile. In this picture, we expect to find quntum freeze-out time, t QA freeze, tht is independent of (or t lest very wekly dependent on) T. By mesuring the T dependence of t freeze, the time t which the system cn no longer respond to chnges in its energy lndscpe, we cn determine whether clssicl therml ðþ Clssicl therml nneling Quntum nneling.5 c h d Γ t/t finl h t Erly t/t finl ctivtion or quntum tunnelling is the dominnt effect governing quit dynmics. Here we modify this nneling procedure to perform specilized experiment tht permits us to distinguish etween these two cses, y llowing the h i in eqution () to e time dependent. We mesure the step response of the system to rpid chnges in h (rpid y comprison with chnges in C nd L) t different stges during the nneling process. In this wy, we re le to mesure t freeze. By mesuring t freeze s function of T, we cn infer whether the system dynmics is dominted y therml or quntum fluctutions. We ruptly increse h from zero to level h t t dely time t d during nneling s shown in Fig. 2c, nd then mesure the proility of the spin eing in either configurtion t the end of nneling. If h is switched on very erly in the nneling process, while the rrier du is still smll in comprison with the therml or quntum trnsition Lte Λ. Erly Lte Figure 2 Quntum nneling., Anneling is performed y grdully rising the energy rrier etween sttes. In therml nneling, when the rrier ecomes much lrger thn k B T therml excittion over the rrier eventully ceses, t some time tfreeze TA. In quntum nneling, tunnelling etween sttes lso will eventully cese, t time t QA freeze., The vlue of the prmeters C nd L during nneling re not independent of ech other in the flux quit. The nneling ends t t finl 5 48 ms. c, Chnging the vlue of h(t) (see d) t vrious points during nneling cn e used to proe the freeze-out time, t freeze. d, Doule-well potentil during nneling. If h is turned on erly enough (lue line), the system follows the ground stte through nneling nd reches the finl ground stte of eqution () with high proility. If h(t) is turned on too lte (red line), the stte proilities re determined y the erlier Hmiltonin, for which h MAY 2 VOL 473 NATURE 95

3 RESEARCH LETTER energy scles, then the quit will quickly respond nd will e le to evolve into the lower energy well, such tht P ", the proility of the spin eing in stte j"æ, is greter thn /2. The vlue of P " will depend on oth h t nd T, s the system will strive to chieve Boltzmnn distriution of its popultion sttistics etween j#æ nd j"æ. However, if h is not turned on until fter the rrier hs een rised sufficiently high (t d. t freeze ), the system will not e le to follow it nd will e eqully likely to settle into either potentil energy well, such tht P " < /2. These two situtions re illustrted in Fig. 2d. For intermedite vlues of t d, the quit will only prtly succeed in responding to the sudden ppliction of the is h. Exmple plots of mesured P " vlues versus t d for single quit t different tempertures re shown in Fig. 3. In this cse, h t GHz nd D GHz. Experimentl prmeters controlling the nneling process re discussed in Supplementry Informtion. As expected, P " shows n initil (t d <) T dependence nd then converges to /2 t lte dely times. These curves were numericlly fitted to extrct t freeze, the time t the middle of this trnsition region, which in turn is plotted versus T in Fig. 3. The curve used for the fitting ws otined y numericl simultions of this P t freeze (μs) 5 mk 22 mk Erly P.9 9 mk Dely time (μs) Model Mesured Temperture (mk) 9 Quntum model 2-level Quntum model 4-level Clssicl model Mesurement Temperture (mk) Figure 3 Single-quit results., Mesured finl ground-stte proility, P ", in single quit versus the dely time, t d, of step h t GHz in energy is, for T 5 22 mk (lue), 5 mk (green) nd 9 mk (red). The solid lines re the result of fits used to extrct the freeze-out time, t freeze. Inset, mesured nd simulted (four-level quntum model) T dependence of P " for t d <., Mesured t freeze versus T (red points). We lso show simulted plots of t freeze from two-level (dshed lue) nd four-level (solid lue) quntum mechnicl models nd from clssicl model of the quit (lck). Error rs, s s.e. process using quntum mechnicl model s discussed in the Supplementry Informtion. In ddition to the experimentl results, in Fig. 3 we show the results of three different numericl simultions. In ll three cses, the model prmeters were independently mesured for the individul devices, leving no free prmeters. A simultion, sed on clssicl model, treted the flux in the two superconducting loops s the coordintes of discrete prticle inside the two-dimensionl flux quit potentil, nd then coupled tht prticle to therml th. The dynmics ws simulted y numericlly solving the Lngevin eqution, s descried in Supplementry Informtion. The clssiclly simulted t freeze vlue vries linerly with T, s expected. The other two simultions involved solving quntum mechnicl model of flux quit coupled to therml th in which only the two or, respectively, four lowest-lying energy levels of the flux quit were kept. The dynmics ws simulted y numericlly solving non-mrkovin density mtrix eqution of motion (Supplementry Informtion). These two models will e referred to here s the two-level nd four-level quntum models. The experimentl results clerly show sturtion of t freeze elow 45 mk, in greement with oth the two-level nd the four-level quntum models nd in disgreement with the clssicl model. The experimentl dt devite from the two-level model ove 45 mk, s the upper energy levels in the flux quit strt to ecome thermlly occupied. The fourlevel quntum model descries the ehviour of the system well up to 8 mk, where more energy levels strt to e occupied. The experimentl dt symptoticlly pproch the clssicl simultion results t higher tempertures. We propose tht if the quntum mechnicl modelling were extended y keeping even more energy levels, then it would reproduce the dt to ever higher tempertures. Both the mesured nd the simulted (four-level quntum model) T dependence of P " for t d < re shown in the inset of Fig. 3. Becuse this proility hs strong T dependence for t d, t freeze, its mesurement provides us with n independent check on the effective temperture of the spin system in this regime. Moreover, ecuse the proility does not sturte t 45 mk, where t freeze sturtes, it is cler indiction tht sturtion of t freeze is not result of sturtion of quit temperture. The key conclusion we drw from Fig. 3 is tht our quit dynmics is est chrcterized s eing quntum mechnicl in nture for T=8 mk. The system evolves to its ground stte through process of quntum nneling. But so fr we hve shown this only for n individul quit. It remins to e shown whether quntum nneling cn e performed on severl spins coupled together. To investigte this, we now configure our rry into chin of eight ferromgneticlly coupled rtificil spins (Fig. 4), with J i,i 52J for i 5, 2,, 7 long the chin nd J ij 5 otherwise. In our experiment, we used J GHz, which is ner the mximum ville for the couplers. The lowest-energy configurtions of this system correspond to the two ferromgnetic sttes j""""""""æ nd j########æ. Applying strong ut opposing ises, h B 562J, to the ends of the chin introduces frustrtion into the system, nd the lowest-energy configurtion will hve rek in the ferromgnetic order; this is known s domin wll (where the spins chnge direction). For exmple, we depict the stte j""""####æ in Fig. 4, for which the domin wll is the middle of the chin. In our step response experiment with the spins configured s chin, ll six spins internl to the chin egin nneling with h 5. In this sitution, it is energeticlly equivlent for the domin wll to e etween ny djcent pir of spins, nd ech such stte should occur with proility P 5 /7. If we leve h 5 for too long (t d?t freeze ), we expect to oserve this distriution of single-domin-wll sttes. At t 5 t d, we pply uniform is, h t 5.J, to the six intermedite spins. Now the ground stte is j"""""""#æ, with the domin wll t the righthnd end of Fig. 4. More system energy is required for the domin wll to occupy positions to the left in Fig. 4. If t d =t freeze, we should oserve stte j"""""""#æ occurring with proility P. /7. As in the single-quit cse, we mesure t freeze y finding the trnsition point 96 NATURE VOL MAY 2

4 RESEARCH +h B 2h t J Domin wll h t h B P mk 5 mk Erly P.6.4 Model Mesured Temperture (mk) 9 mk 2h t.2 Figure 4 Eight-quit ferromgnetic chin., Chin of eight ferromgneticlly coupled quits with uniform coupling coefficient J i,i 52J, fori 5, 2,, 7. The two end quits re ised in opposite directions with h B 562J,suchtht domin wll hs to form within the chin. All middle quits re ised with trget h t 5.J. The configurtion depicted is n excited stte. The fint grey rrows indicte the spin ises h i., Effective energy of the spin stte corresponding to there eing domin wll t ech position long the chin t finite h t. The ground stte is the rightmost site. etween these two stte distriutions (Fig. 5). As efore, we re le to determine the dominnt mechnism (therml or quntum nneling) y mesuring the T dependence of t freeze. A summry of the experimentl results for the eight-quit chin is shown in Fig. 5. As with the single-quit cse, the experimentlly determined t freeze vlues show sturtion t low T nd crossover to ner-liner T dependence for T>45 mk. In this cse, the clssicl model trets the fluxes of ll eight quits s coordintes of discrete prticle in sixteen-dimensionl potentil. The clssicl model does not cpture the ehviour oserved t low T. However, the quntum models quntittively gree with the experimentl results for T=5 mk. At higher tempertures, the clssicl model nd the four-level quntum model re oth in qulittive greement with the experimentl results. The sturtion of t freeze t low T for the single-spin nd eight-spin systems is cler signture of quntum nneling. It cnnot e explined y n experimentl filure to rech lower T, sp """""""#, for t d =t freeze, follows its expected temperture dependence t low T (Fig. 5, inset). Nor cn it e explined y clssicl therml ctivtion processes, ecuse for these lowering T would lwys decrese the rte of therml ctivtion. This mens tht, clssiclly, freeze-out should hppen erlier in the evolution, where the rrier is smller, tht is, sturtion is not possile. This qulittive rgument is independent of the detiled model used to descrie clssicl dynmics. The lowtemperture ehviour of t freeze in this system of eight coupled rtificil spins cnnot e explined y therml ctivtion ut is nturlly explined y quntum tunnelling. This mesurement nd its result re reminiscent of the T-dependent escpe rte mesurements in the pioneering works on mcroscopic quntum tunnelling 3,3, which demonstrted cler signture of quntum tunnelling in currentised Josephson junctions. This rings us to our min conclusion: progrmmle rtificil spin system mnufctured s n integrted circuit cn e used to implement quntum lgorithm. The experiments presented here constitute step etween understnding single-quit nneling nd understnding the multi-quit processes tht could e used to find low-energy configurtions in relistic ditic quntum processor. In ddition to its prolem-solving potentil, system such s this lso provides n interesting test ed for investigting the physics of intercting quntum spins, nd is n importnt step in n ongoing investigtion into much more complex spin systems relized using this type of rchitecture. Although our mnufctured spin system is not yet t freeze (μs) Dely time (μs) 9 Quntum model 2-level Quntum model 4-level Clssicl model Mesurement Temperture (mk) Figure 5 Results for the eight-quit ferromgnetic chin., Mesured finl ground-stte proility, P """""""#, in the eight-quit chin versus t d for h t 5.J nd T 5 22 mk (lue), 5 mk (green) nd 9 mk (red). The solid lines re the result of fits used to extrct the freeze-out time, t freeze. Inset, mesured nd simulted (four-level quntum model) T dependence of P """""""# for t d <., Mesured t freeze versus T (red points). We lso show simulted plots of t freeze from two-level (dshed lue) nd four-level (solid lue) quntum mechnicl models nd from clssicl model of the quits (lck). Error rs, s s.e. universl quntum computer 5, y dding new type of coupler etween the quits, universl quntum computtion would ecome possile 32. METHODS SUMMARY Smple friction. We fricted smples in four-nioium-lyer superconducting integrted circuit process using stndrd N/AlO x /N trilyer, TiPt resistor lyer nd plnrized SiO 2 dielectric lyers pplied y plsm-enhnced chemicl vpour deposition. Design rules included.25-mm lines nd spces for wiring lyers nd minimum junction dimeter of.6 mm. Circuit detils re discussed in ref. 3. Thermometry. We mesured the effective device temperture ttined during these experiments in two wys. The first is sed on nlysis of the single-quit mcroscopic resonnt tunnelling. The second is sed on mesurement of P " versus W x t equilirium nd t fixed rrier height. Both methods re descried in ref. 33 nd in Supplementry Informtion. Both mesurements generlly greed with the ruthenium oxide thermometer on the dilution refrigertor mixing chmer to within few millikelvin over the rnge of tempertures used in the experiment. Confirmtion of the thermometry comes from the greement etween the mesured T dependence of P : (t d =t freeze ) nd tht predicted y the four-level quntum model (insets of Fig. 3 nd Fig. 5). This is discussed further in Supplementry Informtion. 2 MAY 2 VOL 473 NATURE 97

5 RESEARCH LETTER Anneling. Anneling ws performed y sweeping W 2x (Fig. ) from.592w to.652w linerly over period of 48 ms, where W is the mgnetic flux quntum. These vlues rcket the point t which the quit ecomes istle. The devices used re those nlysed in ref. 4. (See Supplementry Informtion for more detils.) Received 3 June 2; ccepted 5 Mrch 2.. Brhon, F. On the computtionl complexity of Ising spin glss models. J. Phys. Mth. Gen. 5, (982). 2. Kdowki, T. & Nishimori, H. Quntum nneling in the trnsverse Ising model. Phys. Rev. E 58, (998). 3. Finnil, A. B., Gomez, M. A., Seenik, C., Stenson, C. & Doll, J. D. Quntum nneling: new method for minimizing multidimensionl functions. Chem. Phys. Lett. 29, (994). 4. Frhi, E. et l. A quntum ditic evolution lgorithm pplied to rndom instnces of n NP-complete prolem. Science 292, (2). 5. Hogg, T. Quntum serch heuristics. Phys. Rev. A 6, 523 (2). 6. Wernsdorfer, W. Moleculr nnomgnets: towrds moleculr spintronics. Int. J. Nnotechnol. 7, (2). 7. 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Mcroscopic quntum tunneling in -mm N Josephson junctions. Phys. Rev. Lett. 47, (98). 3. Devoret, M. H., Mrtinis, J. M. & Clrke, J. Mesurements of mcroscopic quntum tunneling out of the zero-voltge stte of current-ised josephson junction. Phys. Rev. Lett. 55, 98 9 (985). 32. Bimonte, J. D. & Love, P. J. Relizle Hmiltonins for universl ditic quntum computers. Phys. Rev. A 78, 2352 (28). 33. Hrris, R. et l. Proing noise in flux quits vi mcroscopic resonnt tunneling. Phys. Rev. Lett., 73 (28). Supplementry Informtion is linked to the online version of the pper t Acknowledgements We would like to thnk J. Preskill, A. Kitev, D. A. Lidr, F. Wilhelm, A. Lupşcu, A. Blis, T. A. Brun, P. Smith, F. Altomre, E. Hoskinson, T. Przyysz, T. Mhon nd R. Neufeld for discussions. We re grteful to the volunteers of the AQUA@home BOINC project for their help in running the clssicl simultions. Author Contriutions M.H.S.A. nd M.W.J. developed the ide for the experiment; M.W.J. conducted the experiment; T.L., R.H., M.W.J. nd J.W. conducted supporting experiments; M.H.S.A. developed the theory; N.D., F.H. nd M.H.S.A. developed simultion code; N.D., M.H.S.A., M.W.J., F.H. nd C.J.S.T. performed simultions nd nlysed results; M.W.J., M.H.S.A., S.G. nd R.H. wrote the rticle; M.W.J., S.G., M.H.S.A. nd N.D. generted the figures; A.J.B., R.H., J.J., M.W.J., T.L., I.P., E.M.C. nd B.W. developed mesurement lgorithms nd testing softwre; C.R., S.U. nd M.C.T. chieved the low-mgnetic-field environment for the device; C.E. nd C.R. mounted the smple, P.B., E.T., A.J.B., R.H., J.J., M.W.J. nd T.L. designed the devices; E.L., N.L. ndt.o. fricted the devices; M.C.T. nd S.U. developed the testing pprtus; K.K. llowed use of BOINC for clssicl simultions; J.P.H. nd G.R. provided logisticl support; nd J.P.H. selected the chip. Author Informtion Reprints nd permissions informtion is ville t The uthors declre competing finncil interests: detils ccompny the full-text HTML version of the pper t Reders re welcome to comment on the online version of this rticle t Correspondence nd requests for mterils should e ddressed to M.W.J. (mwjohnson@dwvesys.com). 98 NATURE VOL MAY 2