Entanglement in a quantum annealing processor T. Lanting∗,1 A. J. Przybysz,1 A. Yu. Smirnov,1 F. M. Spedalieri,2,3 M. H. Amin,1,4 A. J. Berkley,1 R. Harris,1 F. Altomare,1 S. Boixo†,2 P. Bunyk,1 N. Dickson‡,1 C. Enderud,1 J. P. Hilton,1 E. Hoskinson,1 M. W. Johnson,1 E. Ladizinsky,1 N. Ladizinsky,1 R. Neufeld,1 T. Oh,1 I. Perminov,1 C. Rich,1 M. C. Thom,1 E. Tolkacheva,1 S. Uchaikin,1,5 A. B. Wilson,1 and G. Rose1 1D-Wave Systems Inc., 3033 Beta Avenue, Burnaby BC Canada V5G 4M9 2Information Sciences Institute, University of Southern California, Los Angeles CA USA 90089 3Center for Quantum Information Science and Technology, University of Southern California 4Department of Physics, Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6 5National Research Tomsk Polytechnic University, 30 Lenin Avenue, Tomsk, 634050, Russia Entanglement lies at the core of quantum algorithms designed to solve problems that are in- tractablebyclassicalapproaches. Onesuchalgorithm,quantumannealing(QA),providesapromis- ingpathtoapracticalquantumprocessor. WehavebuiltaseriesofscalableQAprocessorsconsisting of networks of manufactured interacting spins (qubits). Here, we use qubit tunneling spectroscopy 4 to measure the energy eigenspectrum of two- and eight-qubit systems within one such processor, 1 demonstratingquantumcoherenceinthesesystems. Wepresentexperimentalevidencethat,during 0 acriticalportionofQA,thequbitsbecomeentangledandthatentanglementpersistsevenasthese 2 systems reach equilibrium with a thermal environment. Our results provide an encouraging sign n that QA is a viable technology for large-scale quantum computing. a J 5 I. INTRODUCTION II. QUANTUM ANNEALING 1 QA is designed to find the low energy configurations ] h of systems of interacting spins. A wide variety of opti- p mization problems naturally map onto this physical sys- - t tem [19–22]. A QA algorithm is described by a time- n dependentHamiltonianforasetofN spins,i=1,...,N, a Thelastdecadehasbeenexcitingforthefieldofquan- u tumcomputation. Awiderangeofphysicalimplementa- q tions of architectures that promise to harness quantum 1 (cid:88) [ (s)= (s) ∆(s)σx, (1) mechanicstoperformcomputationhavebeenstudied[1– HS E HP − 2 i i 1 3]. Scaling these architectures to build practical proces- v sorswith manymillions tobillionsof qubitswillbe chal- where the dimensionless P is 0 H lenging [4, 5]. A simpler architecture, designed to imple- (cid:88) (cid:88) 0 = h σz+ J σzσz (2) 5 ment a single quantum algorithm such as quantum an- HP − i i ij i j 3 nealing (QA), provides a more practical approach in the i ik T, there is persistent entanglement at estimates for the energy gaps are derived from the un- B equilibrium(seeRefs.[18,26,28,29,31]andtheSupple- certainty in extracting the centroids from the rate data. mentary Information). We discuss the actual source of the underlying Gaussian We experimentally verified the existence of avoided widths (the observed level broadening) below. For both crossings at multiple values of s in both the two- and the two- and eight-qubit system, we confirmed that the eight-qubitsystemsbyusingQTSacrossarangeofbiases expectationvaluesofσz foralldeviceschangesignasthe h 4,4 . In Fig. 3c we show the measured spectrum systemmovesthroughtheavoidedcrossing(seeFigs. 1-3 i of t∈he{−two-q}ubit system at s = 0.339 up to an energy of of the Supplementary Information and [34]) 6 GHz for a range of bias h . The ground states at the Figures 3e and f show similar measurements of the i far left and far right of the spectrum are the localized spectrum of eight coupled qubits at s = 0.271 and states and , respectively. At h = 0, we observe s = 0.284 for a range of biases h . Again, we observe i i |↓↓(cid:105) |↑↑(cid:105) an avoided crossing between these two states. We mea- an avoided crossing at h = 0. The measured energy i sure an energy gap g at zero bias, h = 0, between the gaps at s = 0.271 and 0.284 are g/h = 2.2 0.08 GHz i ± groundstateandthefirstexcitedstate,g/h=1.75 0.08 and g/h = 1.66 0.06 GHz, respectively. Although the ± ± GHz by fitting a Gaussian profile to the tunneling rate eight qubit gaps in Figs. 3e and f are close to the two dataatthesetwolowestenergylevelsandsubtractingthe qubit gaps in Figs. 3c and d, they are measured at quite centroids. Here h (without any subscript) is the Planck different values of the annealing parameter s. As ex- constant. Figure 3d shows the two-qubit spectrum later pected, the eight-qubit gap is closing earlier in the QA in the QA algorithm, at s = 0.351. The energy gap has algorithm as compared to the two-qubit gap. The solid decreased to g/h=1.21 0.06 GHz. Note that the error curves in Figs. c-f indicate the theoretical energy levels ± 5 predicted by Hamiltonian (1) and measurements of ∆(s) with the avoided crossings shown in Fig. 3, and high oc- and (s). Again, the agreement between the experimen- cupation fraction of the ground state, confirmed early in E tallyobtainedspectraandthetheoreticalspectraisgood. QA by the measurements of P 1 shown in Fig. 4. 1 (cid:39) For the early and intermediate parts of QA, the en- We then performed measurements of all available linear ebrogtyh gthape tgwois- alanrdgeerigthhta-qnubteitmspyesrtaetmurse.,Wge(cid:29)expkeBcTt,thfoart cerxopsesc-tsautsicoenptviabliulietioefsσχzijfo≡rdth(cid:104)eσizit(cid:105)h/dqh˜ujb,itwahnedreh˜(cid:104)σ=iz(cid:105) ishthies i j E j ifweholdthesystemsattheses,thentheonlyeigenstate a bias applied to the jth qubit. The measurements are with significant occupation will be the ground state. We performed at the degeneracy point (in the middle of the confirmed this by using QTS in the limit of long tun- avoidedcrossings)wheretheclassicalcontributiontothe neling times to probe the occupation fractions. Details cross-susceptiblity is zero. are provided in Appendix C. Figures 4a and b show the From these measurements, we calculated as de- χ measured occupation fractions of the ground and first fined in Ref. [34] (see Appendix D for more dWetails). A excited states as a function of s for both the two- and non-zero value of this witness detects ground-state en- eight-qubit systems. The solid curves show the equilib- tanglement, and global entanglement in the case of the rium Boltzmann predictions for T =12.5 mK and are in eight-qubitsystem(meaningeverypossiblebipartitionof good agreement with the data. the eight-qubit system is entangled). Figures 4c and d The width of the measured spectral lines is domi- show for the two- and eight-qubit systems. Note χ W nated by the noise of the probe device used to perform that for two qubits at degeneracy, coincides with χ W QTS [33]. The probe device was operated in a regime in ground-state concurrence. These results indicate that which it is strongly coupled to its environment, whereas the two- and eight-qubit systems are entangled midway the system qubits we studied are in the weak coupling through QA. Note also that a susceptibility-based wit- regime. The measured spectral widths therefore do not ness has a close analogy to susceptibility-based measure- represent the intrinsic width of the two- and eight-qubit ments of nano-magnetic systems that also report strong energy eigenstates. During the intermediate part of QA, non-classical correlations [29, 31]. the ground and first excited states are clearly resolved. TheoccupationfractionmeasurementsshowninFig.4 The ground state is protected by the multi-qubit en- indicatethatmidwaythroughQA,thefirstexcitedstate ergy gap g k T, and these systems are coherent. B ofthesesystemsisoccupiedastheenergygapgbeginsto (cid:29) At the end of the annealing trajectory, the gap between approach k T. The systems are no longer in the ground B the ground state and first excited state shrinks below state, but, rather, in a mixed state. To detect the pres- the probe qubit line width of 0.4 GHz. An analysis of ence of mixed-state entanglement, we need knowledge thespectroscopydata,whichestimatestheintrinsiclevel about the density matrix of these systems. Occupation broadening of the multi-qubit eigenstates, is presented fraction measurements provide measurements of the di- in the Supplementary Information. The analysis shows agonalelementsofthedensitymatrixintheenergybasis. thattheintrinsicenergylevelsremaindistinctuntillater We assume that the density matrix has no off-diagonal in QA. The interactions between the two- and eight- elementsintheenergybasis(theydecayontimescalesof qubit systems and their respective environments repre- severalns). Werelaxthisassumptionbelow. Populations sent small perturbations to Hamiltonian (1), even in the P and P plotted in Figs. 4a and 4b indicate that the 1 2 regime in which entanglement is beginning to fall due to system occupies these states with almost 100% probabil- thermal mixing. ity. This means that the density matrix can be written (cid:80)2 in the form ρ = P ψ ψ where ψ represents i=1 i| i(cid:105)(cid:104) i| | i(cid:105) the ith eigenstate of Hamiltonian (1). VI. ENTANGLEMENT MEASURES AND We use the density matrix to calculate standard en- WITNESSES tanglement measures, Wootters’ concurrence, [18], for C the two-qubit system, and negativity, [16, 35], for the N The tunneling spectroscopy data show that midway two- and eight-qubit system. For the maximally entan- through QA, both the two- and eight-qubit systems had gledtwo-qubitBellstatewenotethat =1and =0.5. C N avoided crossings with the expected gap g kBT and Figure 4c shows as a function of s. Midway through (cid:29) C had ground state occupation P1 1. While observation QAwemeasureapeakconcurrence =0.53 0.05,indi- (cid:39) C ± of an avoided crossing is evidence for the presence of an cating significant entanglement in the two-qubit system. entangledgroundstate(seeRef.[34]andtheSupplemen- Thisvalueof correspondstoanentanglementofforma- C tary Information for details), we can make this observation E =0.388 (see Refs. [16, 18] for definitions). This f tion more quantitative with entanglement measures and is comparable to the level of entanglement, E = 0.378, f witnesses. obtained in Ref. [11], and indeed to the value E =1 for f We begin with a susceptibility-based witness, , the Bell state. Because concurrence is not applicable χ W C which detects ground state entanglement. This witness to more than two qubits, we used negativity to de- N does not require explicit knowledge of Hamiltonian (1), tect entanglement in the eight-qubit system. For N >2, but requires a non-degenerate ground state, confirmed is defined on a particular bipartition of the sys- A,B N 6 A 2QSystem B 8QSystem 1 1 n n o o ati ati ul0.5 ul0.5 p p o o P P1 P P1 P2 P2 0 0 C D 0.8 N N C 0.6 Wχ 0.6 Wχ 0.4 0.4 0.2 0.2 0 0 0.31 0.32 0.33 0.34 0.35 0.36 0.37 0.38 0.39 0.23 0.24 0.25 0.26 0.27 0.28 0.29 0.3 0.31 s s FIG.4: (a),(b)Measurementsoftheoccupationfraction,orpopulation,ofthegroundstate(P )andfirstexcitedstate(P ) 1 2 ofthetwo-qubitandeight-qubitsystem,respectively,versuss. Earlyintheannealingtrajectory,g(cid:29)k T,andthesystemisin B thegroundstatewithP1 <∼1. ThesolidcurvesshowtheequilibriumBoltzmannpredictionsforT =12.5mK.(c)Concurrence C,negativityN andwitnessW versussforthetwo-qubitsystem. EarlyinQA,thequbitsareweaklyinteracting,thusresulting χ in limited entanglement. Entanglement peaks near s=0.37. For larger s, the gap between the ground and first excited state shrinks and thermal occupation of the first excited state rises, thus extinguishing entanglement. Solid curves indicate the expectedtheoreticalvaluesofeachwitnessormeasureusingHamiltonian(1)andBoltzmannstatistics. (d)NegativityN and witness W versus s for the eight-qubit system. For all s shown, the nonzero negativity N and nonzero witness W report χ χ entanglement. Fors>0.39ands>0.312forthetwo-qubitandeight-qubitsystems,respectively,theshadedgreydenotesthe regimeinwhichthegroundandfirstexcitedstatescannotberesolvedviaourspectroscopicmethod. Solidcurvesindicatethe expected theoretical values of each witness or measure using Hamiltonian (1) and Boltzmann statistics. tem into subsystems A and B. We define to be the constraintsonthedensitymatrixofthesystem,ρ(s). We N geometric mean of this quantity across all possible bi- then obtain an upper bound on Tr[ ρ(s)] by search- AB W partitions. A nonzero indicates the presence of global ing over all ρ(s) that satisfy these linear constraints. If N entanglement. Figures 4c and d show the negativity cal- this upper bound is < 0, then we have shown entangle- culated with measured P and P (and with the mea- ment for the bipartition A-B [36]. Figure 5 shows the 1 2 suredHamiltonianparameters∆and J )asafunction upper limit of the witness Tr[ ρ(s)] for the eight- ij AB E W ofsforthetwoandeight-qubitsystems. Theeight-qubit qubitsystem. Weplotdataforthebipartitionthatgives system has nonzero for s<0.315, thus indicating the the median upper limit. The error bars are derived from N presence of mixed-state global entanglement. Both con- a Monte-Carlo analysis wherein we used the experimen- currence and negativity decrease later in QA where tal uncertainties in ∆ and J to estimate the uncertainty C N the first excited state approaches the ground state and in Tr[ ρ]. We also plot data for the two partitions AB W becomesthermallyoccupied. Theexperimentalvaluesof that give the largest and smallest upper limits. For all these entanglement measures are in agreement with the values of the annealing parameter s, except for the last theoretical predictions (solid lines in Fig 4). The error two points, upper limits from all possible bipartitions of bars in Figures 4c and d represent uncertainties in the the eight-qubit system are below zero. In this annealing measurements of occupation fractions, ∆(s) and (s). range, the eight-qubit system is globally entangled. E As stated above, the calculation of and relies on C N theassumptionthattheoff-diagonaltermsinthedensity matrix decay on times scales of several ns. We remove VII. CONCLUSIONS this assumption and demonstrate entanglement through the use of another witness , defined on some bi- To summarize, we have provided experimental evi- AB W partition A-B of the eight-qubit system. The witness, dence for the presence of quantum coherence and en- described in Appendix D, is designed in such a way tanglement within subsets of qubits inside a quantum that Tr[ σ] 0 for all separable states σ. When annealingprocessorduringitsoperation. Ourconclusion AB W ≥ Tr[ ρ(s)] < 0, the state ρ(s) is entangled. Measure- is based on four levels of evidence: a. the observation AB W ments of populations P and P provide a set of linear of two- and eight- qubit avoided crossings with a multi- 1 2 7 Improved designs of this device will allow much larger 0.2 systems to be studied. Our measurements represent an median partition effective approach for exploring the role of quantum me- 0.1 lowest and highest partitions chanics in QA processors and ultimately to understand- ] ρ ing the fundamental power and capability of quantum B A annealing. W 0 [ r T−0.1 ACKNOWLEDGEMENTS n o d We thank C. Williams, P. Love, and J. Whittaker n−0.2 u for useful discussions. We acknowledge F. Cioata and o b P. Spear for the design and maintenance of electron- er−0.3 ics control systems, J. Yao for fabrication support, and p p D. Bruce, P. deBuen, M. Gullen, M. Hager, G. Lamont, u −0.4 L. Paulson, C. Petroff, and A. Tcaciuc for technical support. F.M.S. was supported by DARPA, under contract FA8750-13-2-0035. −0.5 0.24 0.26 0.28 0.3 s Appendix A. QA Processor Description FIG. 5: Upper limit of the quantity Tr[WABρ] versus s for Chip Description several bipartitions A−B of the eight-qubit system. When this quantity is < 0, the system is entangled with respect Theexperimentsdiscussedinhereinwereperformedon to this bipartition. The solid dots show the upper limit on Tr[W ρ] for the median bipartition. The open dots above a sample fabricated with a process consisting of a stan- AB and below these are derived from the two bipartitions that dard Nb/AlOx/Nb trilayer, a TiPt resistor layer, pla- give the highest and lowest upper limits on Tr[W ρ], re- narized SiO dielectric layers and six Nb wiring layers. AB 2 spectively. Forthepointsats>0.3,themeasurementsofP1 Thecircuitdesignrulesincludedaminimumlinewidthof and P do not constrain ρ enough to certify entanglement. 2 0.25 µm and 0.6 µm diameter Josephson junctions. The processor chip is a network of densely connected eight- qubitunitcellswhicharemoresparselyconnectedtoeach qubit energy gap g k T; b. the witness χ, calcu- (cid:29) B W other (see Fig. 1 for photographs of the processor). We lated with measured cross-susceptibilities and coupling report measurements made on qubits from one of these energies,whichreportsgroundstateentanglementofthe unitcells. Thechipwasmountedonthemixingchamber two-andeight-qubitsystem. Noticethatthesetwolevels of a dilution refrigerator inside an Al superconducting of evidence do not require explicit knowledge of Hamil- shield and temperature controlled at 12.5 mK. tonian (1); c. the measurements of energy eigenspectra and equibrium occupation fractions during QA, which allow us to use Hamiltonian (1) to reconstruct the den- Qubit Parameters sity matrix, with some weak assumptions, and calculate concurrence and negativity. These standard measures The processor facilitates quantum annealing (QA) of entanglement report non-classical correlations in the of compound-compound Josephson junction rf SQUID two- and eight-qubit systems; d. the entanglement wit- (radio-frequency superconducting quantum interference ness ,whichiscalculatedwiththemeasuredHamil- AB device) flux qubits [37]. The qubits are controlled via W tonian and with constraints provided by the measured the external flux biases Φx and Φx which allow us to qi ccjj populations of the ground and the first excited states. treat them as effective spins (see Fig. 1). Pairs of qubits This witness reports global entanglement of the eight- interact through tunable inductive couplings [25]. The qubit system midway through the QA algorithm. system can be described with the time-dependent QA The observed entanglement is persistent at thermal Hamiltonian, equilibrium,anencouragingresultasanypracticalhard-   ware designed to run a quantum algorithm will be in- N N (cid:88) (cid:88) 1 (cid:88) evitably coupled to a thermal environment. The ex- HS(s)=E(s)− hiσiz+ Jijσizσjz−2∆(s) σix, perimental techniques that we have discussed provide i i 1 terswithon-chipcontrolasdescribedin[8]. Thisallowed GHz[33]. Theredlineshowsthetheoreticalpredictionforan us to match Ip and ∆ across all qubits throughout the | q| rf SQUID model employing the median qubit parameters of annealing trajectory. Table I shows the median qubit the eight devices. The vertical black line separates coherent parameters for the devices studied here. (left)andincoherent(right)evolutionasestimatedbyanaly- eigFhitguqureb6itss.ho∆wswmaseamseuarseumreendtwsoitfh∆sianngdle|Iqqpu|bvits.Lsanfodraaul-l ssihsowofmsineagsleurqeumbeinttsspfeocrtraalll leiingehtshqaupbeist.s s(tbu)di|eIqdp|(ins)tvhsissw.oWrke. Weusedatwo-qubitcoupledfluxmeasurementwiththeinter- Zenermeasurementsfroms=0.515tos=0.658[38]and qubitcouplingelementsetto1.37pH[8]. Theredlineshows withqubittunnelingspectroscopy(QTS)froms=0.121 the theoretical prediction for an rf SQUID model employing tos=0.407[33]. Theresolutionlimitofqubittunneling the median qubit parameters of the eight devices. spectroscopy and the bandwidth of our external control lines during the Landau-Zener measurements prevented us from characterizing ∆ between s = 0.4 and s = 0.5, respectively. Ip was measured by coupling a second | q| probequbittothequbitq withacouplingofM =1.37 i eff pH and measuring the the flux M Ip(s) as a function eff| qi | of s. Ip is matched between qubits to within 3% and | q| ∆(s) is matched between qubits to within 8% across the annealing region explored in this study. 9 Appendix B. Qubit Tunneling Spectroscopy (QTS) QTSallowsonetomeasuretheeigenspectrumofanN- qubit system governed by Hamiltonian . Details on S H the measurement technique are presented elsewhere [33]. For convenience in comparing with this reference, we define a qubit energy bias (cid:15) 2h˜. Measurements are per- i i ≡ formed by coupling an additional probe qubit q , with P qubit tunneling amplitude ∆ ∆, J˜, to one of the N P (cid:28) | | qubits of the system under study, for example q . When 1 we use a coupling strength J˜ between q and q and P P 1 apply a compensating bias (cid:15) = 2J˜ to q , the resulting 1 P 1 system + probe Hamiltonian becomes H =H [J˜ σz (1/2)(cid:15) ](1 σz). (5) S+P S − P 1 − P − P Foroneofthelocalizedstatesoftheprobequbit, , FIG. 7: Typical waveforms during QTS. We prepare the |↑(cid:105)P for which an eigenvalue of σz is equal to +1 (i.e. the initialstatebyannealingprobeandsystemqubitsfroms=0 P to s=1 in the presence of a large polarization bias (cid:15) . We probe qubit in the right well), the contribution of the pol thenbiasthesystemqubitq (towhichtheprobeisattached) probe qubit is exactly canceled, leading to H =H , 1 S+P S to a bias (cid:15) and the probe qubit to a bias (cid:15) . With these with composite eigenstates n, = n and eigen- 1 P | ↑(cid:105) | (cid:105)⊗|↑(cid:105)P biases asserted, we then adjust the system qubits’ annealing values EnR = En, which are identical to those of the parametertoanintermediatepoints∗ andtheprobequbitto original system without the presence of the probe qubit. a point s and dwell for a time τ. Finally, we complete the P Here, n is an eigenstate of the Hamiltonian H (n = anneal s→1 and read out the state of the qubits. S | (cid:105) 1,2,...,2N). For the other localized state of the probe qubit, , when this qubit is in the left well, the ground sta|t↓e(cid:105)Pof qubit. Figure 7 summarizes these waveforms during a HELS++P(cid:15)is,|wψh0Le,r↓e(cid:105) ψ=L|ψi0sL(cid:105)th⊗e|g↓r(cid:105)oPu,nwdistthateeigoefnHvalue2E(cid:101)J˜0Lσ=z typWicealpQerTfoSrmmetahsisurmemeaesnutr.ement for a range of τ which an0d ELPis its eig|en0v(cid:105)alue. We choose J˜ Sk−T sPuch1 allows us to measure an initial rate of tunneling Γ from that th0e state |ψ0L,↓(cid:105) is well separate|dPf|r(cid:29)om tBhe next r|ψan0Lg,e↓(cid:105)oftoth|ψe,p↑r(cid:105)o.bWe qeurbeiptebaitasth(cid:15)is.mPeeaaskusreimneΓntcoorfreΓspfoornda excited state for ferromagnetically coupled systems, and P toresonancesbetweentheinitiallypreparedstateandthe thus system + probe can be initialized in this state to state n, , thus allowing us to map the eigenspectrum high fidelity. | ↑(cid:105) Introducing a small transverse term, 1∆ σx, to of the N-qubit system. −2 P P For the plots in the main paper, measurements of Γ Hamiltonian (5) results in incoherent tunneling from the initial state ψL, to any of the available n, are normalized to [0,1] by dividing the maximum value | 0 ↓(cid:105) | ↑(cid:105) across a vertical slice to give a visually interpretable re- states [39]. A bias on the probe qubit, (cid:15) , changes the P sult. Figure 8b shows a typical raw result in units of energy difference between the probe and mani- folds. We can thus bring ψL, into|↓re(cid:105)Psonanc|e↑(cid:105)wPith any µs−1. | 0 ↓(cid:105) We posed ferromagnetically coupled instances of the of n, states (when E(cid:101)L = ER) allowing resonant tun- | ↑(cid:105) 0 n form neling between the two states. The rate of tunneling out tohfethloecaintiitoinasllyofpnre,par.ed state |ψ0L,↓(cid:105) is thus peaked at HP =−(cid:88)hiσiz+(cid:88)Jijσizσjz (6) | ↑(cid:105) i i

Entanglement in a quantum annealing processor

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