Entanglement in a Quantum Annealing Processor - T. Lanting

8/13/2019 Entanglement in a Quantum Annealing Processor - T. Lanting

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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 4M92Information Sciences Institute, University of Southern California, Los Angeles CA USA 900893Center for Quantum Information Science and Technology, University of Southern California

4

Department of Physics, Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S65National 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-tractable by classical approaches. One such algorithm, quantum annealing (QA), provides a promis-ing path to a practical quantum processor. We have built a series of scalable QA processors consistingof networks of manufactured interacting spins (qubits). Here, we use qubit tunneling spectroscopyto measure the energy eigenspectrum of two- and eight-qubit systems within one such processor,demonstrating quantum coherence in these systems. We present experimental evidence that, duringa critical portion of QA, the qubits become entangled and that entanglement persists even as thesesystems reach equilibrium with a thermal environment. Our results provide an encouraging signthat QA is a viable technology for large-scale quantum computing.

I. INTRODUCTION

The last decade has been exciting for the field of quan-tum computation. A wide range of physical implementa-tions of architectures that promise to harness quantummechanics to perform computation have been studied [13]. Scaling these architectures to build practical proces-sors with many millions to billions of qubits will be chal-lenging[4,5]. A simpler architecture, designed to imple-

ment a single quantum algorithm such as quantum an-nealing (QA), provides a more practical approach in thenear-term[6,7]. However, one of the main features thatmakes such an architecture scalable, namely a limitednumber of low bandwidth external control lines [8], pro-hibits many typical characterization measurements usedin studying prototype universal quantum computers [914]. These constraints make it challenging to experimen-tally determine whether a scalable QA architecture, onethat is inevitably coupled to a thermal environment, iscapable of generating entangled states [1518]. A demon-stration of entanglement is considered to be a criticalmilestone for any approach to building a quantum com-

puting technology. Herein, we demonstrate an experi-mental method to detect entanglement in subsections ofa quantum annealing processor to address this fundamen-tal question.

Electronic address: [email protected] at Google, 340 Main St, Venice, California 90291currently at Side Effects Software, 1401-123 Front Street West,Toronto, Ontario, Canada

II. QUANTUM ANNEALING

QA is designed to find the low energy configurationsof systems of interacting spins. A wide variety of opti-mization problems naturally map onto this physical sys-tem [1922]. A QA algorithm is described by a time-dependent Hamiltonian for a set ofNspins,i = 1, . . . , N ,

HS(s) = E(s) HP 12

i

(s) xi, (1)

where the dimensionlessHP is

HP =

i hiz

i

+i


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FIG. 1: (a) Photograph of the QA processor used in thisstudy. We report measurements performed on the eight-qubitunit cell indicated. The bodies of the qubits are extendedloops of Nb wiring (highlighted with red rectangles). Inter-qubit couplers are located at the intersections of the qubitbodies. (b) Electron micrograph showing the cross-sectionof a typical portion of the processor circuitry (described inmore detail in Appendix A).(c) Schematic diagram of a pairof coupled superconducting flux qubits with external controlbiases xqi and

xccjj and with flux through the body of the

ith qubit denoted as qi. An inductive coupling between thequbits is tuned with the bias xco,ij. (d) Energy scales (s)and

E(s) in Hamiltonian (1)calculated from an rf-SQUID (Su-

perconducting Quantum Interference Device) model based onthe median of independently measured device parameters forthese eight qubits. See Appendix A for more details. (e),(f)The two and eight-qubit systems studied were programmedto have the topologies shown. Qubits are represented as goldspheres and inter-qubit couplers, set to J =2.5, are repre-sented as silver lines.

24]. Figure1a shows a photograph of the processor. Fig-

FIG. 2: An illustration of entanglement between two qubitsduring QA with hi = 0 and J < 0. We plot calculations ofthe two-qubit ground state wave function modulus squaredin the basis of q1 and q2, the flux through the bodies ofq1 andq2, respectively. The color scale encodes the probabil-ity density with red corresponding to high probability densityand blue corresponding to low probability density. We used

Hamiltonian (1) and the energies in Fig 1d for the calcula-tion. The four quadrants represent the four possible statesof the two-qubit system in the computation basis. We alsoplot the single qubit potential energy (U versus q1) cal-culated from measured device parameters. (a) At s = 0( 2|J|E 0), the qubits weakly interact and are eachin their ground state 1

2(| + |), which is delocalized in

the computation basis. The wavefunction shows no correla-tion betweenq1 andq2 and therefore their wavefunctions areseparable. (b) At intermediate s ( 2|Jij|E), the qubitsare entangled. The state of one qubit is not separable fromthe state of the other, as the ground state of the system isapproximately|+ (| + |)/2. A clear correlationis seen between q1 and q2. (c) As s 1, 2|Jij|E andthe ground state of the system approaches

|+

. However, the

energy gapg between the ground state (|+) and the first ex-cited state (|) is closing. When the qubits are coupled toa bath with temperature T and g < kBT, the system is in amixed state of |+ and | and entanglement is extinguished.

ure1c shows the circuit schematic of a pair of flux qubitswith the magnetic flux controls xqi and

xccjj. The an-

nealing parameter s is controlled with the global biasxccjj(t) (see Appendix A for the mapping between s andxccjj and a description of how

xqi is provided for each

qubit). The strength and sign of the inductive couplingbetween pairs of qubits is controlled with magnetic flux

xco,ij that is provided by an individual on-chip digital-to-analog converter for each coupler [8]. The parame-tershi andJij are thus in situtunable, thereby allowingthe encoding of a vast number of problems. The time-dependent energy scales (s) andE(s) are calculatedfrom measured qubit parameters and plotted in Fig. 1d.We calibrated and corrected the individual flux qubit pa-rameters in our processor to ensure that every qubit hada close to identical andE (the energy gap is bal-anced to better than 8% between qubits andEto better


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than 5%). See Appendix A for measurements of theseenergy scales. The interqubit couplers were calibratedas described in Ref. [25]. The processor studied here wasmounted on the mixing chamber of a dilution refrigeratorheld at temperatureT= 12.5 mK.

IV. FERROMAGNETICALLY COUPLEDINSTANCES

The experiments reported herein focused on one of theeight-qubit unit cells of the larger QA processor as indi-cated in Fig. 1a. The unit cell was isolated by settingall couplings outside of that subsection to Jij = 0 forall experiments. We then posed specificHP instanceswith strong ferromagnetic (FM) coupling Jij = 2.5 andhi = 0 to that unit cell as illustrated in Figs. 1e and f.These configurations produced coupled two- and eight-qubit systems, respectively. Hamiltonian (1) describesthe behaviour of these systems during QA.

Typical observations of entanglement in the quantumcomputing literature involve applying interactions be-

tween qubits, removing these interactions, and then per-forming measurements. Such an approach is well suitedto gate-model architectures (e.g. Ref. [11]). During QA,however, the interaction between qubits is determinedby the particular instance ofHP, in this case a stronglyferromagnetic instance, and cannot be removed. In thisway, systems of qubits undergoing QA have much more incommon with condensed matter systems, such as quan-tum magnets, for which interactions cannot be turnedoff. Indeed, a growing body of recent theoretical andexperimental work suggests that entanglement plays acentral role in many of the macroscopic properties of con-densed matter systems [2632]. Here we introduce otherapproaches to quantifying entanglement that are suitedto QA processors. We establish experimentally that thetwo- and eight-qubit systems, comprising macroscopic su-perconducting flux qubits coupled to a thermal bath at12.5 mK, become entangled during the QA algorithm.

To illustrate the evolution of the ground state of theseinstances during QA, a sequence of wave functions forthe ground state of the two-qubit system is shown inFig. 2. A similar sequence could be envisioned for theeight-qubit system. We consider these systems subjectto zero biases, hi = 0. For small s, 2|Jij|E, andthe ground state of the system can be expressed as a

product of the ground states of the individual qubits:Ni=1 12(|i + |i) where N = 2, 8 (see Fig. 2a). Forintermediate s,


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FIG. 3: Spectroscopic data for two- and eight-qubit systems plotted in false colour (colour indicates normalized qubit tunnelspectroscopy rates). A non-zero measurement (false colour) indicates the presence of an eigenstate of the probed system at agiven energy (ordinate) ands (abscissa). Panel(a) shows the measured eigenspectrum for the two-qubit system as a function ofs. Panel(b) shows a similar set of measurements for the eight-qubit system. The ground state energy E1 has been subtractedfrom the data to aid in visualization. The solid curves indicate the theoretical expectations for the energy eigenvalues usingindependently calibrated qubit parameters and Hamiltonian (1). We emphasize that the solid curves are not a fit, but rather aprediction based on Hamiltonian(1) and measurements of andE. The slight differences between the high-energy spectrumprediction and measurements are due to the additional states in the rf-SQUID flux qubits. A full rf-SQUID model that is inagreement with the measured high energy spectrum is explored in the Supplementary Information. Panel (c) and (d) showmeasured eigenspectra of the two-qubit system vs. h1 = h2 hi for two values of annealing parameter s, s = 0.339 ands = 0.351 from left to right, respectively. Notice the avoided crossing at hi = 0. Panel(e) and (f ) show analogous measuredeigenspectra for the eight-qubit system with (with h1 = . . .= h8 hi). Because the eight-qubit gap closes earlier in QA forthis system, we show measurements for smaller s,s = 0.271 and s = 0.284 from left to right, respectively.

large gaps, g > kBT, there is persistent entanglement atequilibrium (see Refs. [18,26, 28, 29,31]and the Supple-mentary Information).

We experimentally verified the existence of avoidedcrossings at multiple values of s in both the two- andeight-qubit systems by using QTS across a range of biaseshi {4, 4}. In Fig.3c we show the measured spectrumof the two-qubit system at s = 0.339 up to an energy of6 GHz for a range of bias hi. The ground states at thefar left and far right of the spectrum are the localized

states| and|, respectively. Athi = 0, we observean avoided crossing between these two states. We mea-sure an energy gap g at zero bias, hi = 0, between theground state and the first excited state, g/h = 1.750.08GHz by fitting a Gaussian profile to the tunneling ratedata at these two lowest energy levels and subtracting thecentroids. Hereh (without any subscript) is the Planckconstant. Figure3d shows the two-qubit spectrum laterin the QA algorithm, at s = 0.351. The energy gap hasdecreased tog/h= 1.21 0.06 GHz. Note that the error

estimates for the energy gaps are derived from the un-certainty in extracting the centroids from the rate data.We discuss the actual source of the underlying Gaussianwidths (the observed level broadening) below. For boththe two- and eight-qubit system, we confirmed that theexpectation values ofz for all devices change sign as thesystem moves through the avoided crossing (see Figs. 1-3of the Supplementary Information and [34])

Figures 3e and f show similar measurements of thespectrum of eight coupled qubits at s = 0.271 and

s = 0.284 for a range of biases hi. Again, we observean avoided crossing at hi = 0. The measured energygaps at s = 0.271 and 0.284 are g/h = 2.2 0.08 GHzand g/h = 1.66 0.06 GHz, respectively. Although theeight qubit gaps in Figs. 3e and f are close to the twoqubit gaps in Figs.3c and d, they are measured at quitedifferent values of the annealing parameter s. As ex-pected, the eight-qubit gap is closing earlier in the QAalgorithm as compared to the two-qubit gap. The solidcurves in Figs. c-f indicate the theoretical energy levels


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predicted by Hamiltonian (1) and measurements of (s)andE(s). Again, the agreement between the experimen-tally obtained spectra and the theoretical spectra is good.

For the early and intermediate parts of QA, the en-ergy gap g is larger than temperature, g kBT, forboth the two- and eight-qubit systems. We expect thatif we hold the systems at these s, then the only eigenstatewith significant occupation will be the ground state. Weconfirmed this by using QTS in the limit of long tun-

neling times to probe the occupation fractions. Detailsare provided in Appendix C. Figures4a and b show themeasured occupation fractions of the ground and firstexcited states as a function of s for both the two- andeight-qubit systems. The solid curves show the equilib-rium Boltzmann predictions forT= 12.5 mK and are ingood agreement with the data.

The width of the measured spectral lines is domi-nated by the noise of the probe device used to performQTS[33]. The probe device was operated in a regime inwhich it is strongly coupled to its environment, whereasthe system qubits we studied are in the weak couplingregime. The measured spectral widths therefore do not

represent the intrinsic width of the two- and eight-qubitenergy eigenstates. During the intermediate part of QA,the ground and first excited states are clearly resolved.The ground state is protected by the multi-qubit en-ergy gap g kBT, and these systems are coherent.At the end of the annealing trajectory, the gap betweenthe ground state and first excited state shrinks belowthe probe qubit line width of 0.4 GHz. An analysis ofthe spectroscopy data, which estimates the intrinsic levelbroadening of the multi-qubit eigenstates, is presentedin the Supplementary Information. The analysis showsthat the intrinsic energy levels remain distinct until laterin QA. The interactions between the two- and eight-

qubit systems and their respective environments repre-sent small perturbations to Hamiltonian (1), even in theregime in which entanglement is beginning to fall due tothermal mixing.

VI. ENTANGLEMENT MEASURES ANDWITNESSES

The tunneling spectroscopy data show that midwaythrough QA, both the two- and eight-qubit systems hadavoided crossings with the expected gap g kBT andhad ground state occupation P1 1. While observationof an avoided crossing is evidence for the presence of anentangled ground state (see Ref.[34]and the Supplemen-tary Information for details), we can make this observa-tion more quantitative with entanglement measures andwitnesses.

We begin with a susceptibility-based witness, W,which detects ground state entanglement. This witnessdoes not require explicit knowledge of Hamiltonian (1),but requires a non-degenerate ground state, confirmed

with the avoided crossings shown in Fig.3, and high oc-cupation fraction of the ground state, confirmed early inQA by the measurements of P1 1 shown in Fig. 4.We then performed measurements of all available linearcross-susceptibilitiesij d zi /dhj , wherezi is theexpectation value ofzi for theith qubit and

hj = Ehj isa bias applied to the jth qubit. The measurements areperformed at the degeneracy point (in the middle of theavoided crossings) where the classical contribution to the

cross-susceptiblity is zero.From these measurements, we calculatedW as de-

fined in Ref. [34] (see Appendix D for more details). Anon-zero value of this witness detects ground-state en-tanglement, and global entanglement in the case of theeight-qubit system (meaning every possible bipartition ofthe eight-qubit system is entangled). Figures 4c and dshowW for the two- and eight-qubit systems. Notethat for two qubits at degeneracy,W coincides withground-state concurrence. These results indicate thatthe two- and eight-qubit systems are entangled midwaythrough QA. Note also that a susceptibility-based wit-ness has a close analogy to susceptibility-based measure-

ments of nano-magnetic systems that also report strongnon-classical correlations [29, 31].

The occupation fraction measurements shown in Fig. 4indicate that midway through QA, the first excited stateof these systems is occupied as the energy gapg begins toapproach kBT. The systems are no longer in the groundstate, but, rather, in a mixed state. To detect the pres-ence of mixed-state entanglement, we need knowledgeabout the density matrix of these systems. Occupationfraction measurements provide measurements of the di-agonal elements of the density matrix in the energy basis.We assume that the density matrix has no off-diagonal

elements in the energy basis (they decay on timescales ofseveral ns). We relax this assumption below. PopulationsP1 and P2 plotted in Figs. 4a and 4b indicate that thesystem occupies these states with almost 100% probabil-ity. This means that the density matrix can be writtenin the form =

2i=1 Pi |i i| where|i represents

theith eigenstate of Hamiltonian (1).

We use the density matrix to calculate standard en-tanglement measures, Wootters concurrence,C [18], forthe two-qubit system, and negativity,N [16, 35], for thetwo- and eight-qubit system. For the maximally entan-gled two-qubit Bell state we note that C = 1 andN = 0.5.Figure 4c shows

Cas a function ofs. Midway through

QA we measure a peak concurrence C = 0.530.05, indi-cating significant entanglement in the two-qubit system.This value ofCcorresponds to an entanglement of forma-tionEf = 0.388 (see Refs. [16,18] for definitions). Thisis comparable to the level of entanglement, Ef = 0.378,obtained in Ref.[11], and indeed to the value Ef= 1 forthe Bell state. Because concurrenceC is not applicableto more than two qubits, we used negativity N to de-tect entanglement in the eight-qubit system. ForN >2,

NA,B is defined on a particular bipartition of the sys-


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FIG. 4: (a),(b)Measurements of the occupation fraction, or population, of the ground state (P1) and first excited state (P2)of the two-qubit and eight-qubit system, respectively, versuss. Early in the annealing trajectory,g kBT, and the system is inthe ground state with P1 0.39 ands >0.312 for the two-qubit and eight-qubit systems, respectively, the shaded grey denotes theregime in which the ground and first excited states cannot be resolved via our spectroscopic method. Solid curves indicate theexpected theoretical values of each witness or measure using Hamiltonian(1) and Boltzmann statistics.

tem into subsystems A and B. We defineN to be thegeometric mean of this quantity across all possible bi-partitions. A nonzeroN indicates the presence of globalentanglement. Figures4c and d show the negativity cal-culated with measured P1 and P2 (and with the mea-sured Hamiltonian parameters and EJij) as a functionofs for the two and eight-qubit systems. The eight-qubitsystem has nonzeroN fors


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0.24 0.26 0.28 0.30.5

0.4

0.3

0.2

0.1

0

0.1

0.2

s

upperbound

on

Tr[WAB

]

median partition

lowest and highest partitions

FIG. 5: Upper limit of the quantity Tr[W

AB] versus s forseveral bipartitions A B of the eight-qubit system. Whenthis quantity is < 0, the system is entangled with respectto this bipartition. The solid dots show the upper limit onTr[WAB] for the median bipartition. The open dots aboveand below these are derived from the two bipartitions thatgive the highest and lowest upper limits on Tr[WAB], re-spectively. For the points at s >0.3, the measurements ofP1and P2 do not constrain enough to certify entanglement.

qubit energy gap g kBT; b. the witnessW, calcu-lated with measured cross-susceptibilities and couplingenergies, which reports ground state entanglement of the

two- and eight-qubit system. Notice that these two levelsof evidence do not require explicit knowledge of Hamil-tonian (1); c. the measurements of energy eigenspectraand equibrium occupation fractions during QA, whichallow us to use Hamiltonian (1)to reconstruct the den-sity matrix, with some weak assumptions, and calculateconcurrence and negativity. These standard measuresof entanglement report non-classical correlations in thetwo- and eight-qubit systems; d. the entanglement wit-ness WAB, which is calculated with the measured Hamil-tonian and with constraints provided by the measuredpopulations of the ground and the first excited states.This witness reports global entanglement of the eight-

qubit system midway through the QA algorithm.The observed entanglement is persistent at thermal

equilibrium, an encouraging result as any practical hard-ware designed to run a quantum algorithm will be in-evitably coupled to a thermal environment. The ex-perimental techniques that we have discussed providemeasurements of energy levels, and their populations,for arbitrary configurations of Hamiltonian parameters, hi, Jij during the QA algorithm. The main limitationof the technique is the spectral width of the probe device.

Improved designs of this device will allow much largersystems to be studied. Our measurements represent aneffective approach for exploring the role of quantum me-chanics in QA processors and ultimately to understand-ing the fundamental power and capability of quantumannealing.

ACKNOWLEDGEMENTS

We thank C. Williams, P. Love, and J. Whittakerfor useful discussions. We acknowledge F. Cioata andP. Spear for the design and maintenance of electron-ics control systems, J. Yao for fabrication support, andD. Bruce, P. deBuen, M. Gullen, M. Hager, G. Lamont,L. Paulson, C. Petroff, and A. Tcaciuc for technical sup-port. F.M.S. was supported by DARPA, under contractFA8750-13-2-0035.

Appendix A. QA Processor Description

Chip Description

The experiments discussed in herein were performed ona sample fabricated with a process consisting of a stan-dard Nb/AlOx/Nb trilayer, a TiPt resistor layer, pla-narized SiO2 dielectric layers and six Nb wiring layers.The circuit design rules included a minimum linewidth of0.25m and 0.6m diameter Josephson junctions. Theprocessor chip is a network of densely connected eight-qubit unit cells which are more sparsely connected to eachother (see Fig. 1 for photographs of the processor). Wereport measurements made on qubits from one of theseunit cells. The chip was mounted on the mixing chamber

of a dilution refrigerator inside an Al superconductingshield and temperature controlled at 12.5 mK.

Qubit Parameters

The processor facilitates quantum annealing (QA)of compound-compound Josephson junction rf SQUID(radio-frequency superconducting quantum interferencedevice) flux qubits [37]. The qubits are controlled viathe external flux biases xqi and

xccjj which allow us to

treat them as effective spins (see Fig. 1). Pairs of qubitsinteract through tunable inductive couplings [25]. The

system can be described with the time-dependent QAHamiltonian,

HS(s) = E(s) N

i

hizi +i


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qubit parameter median measured value

critical current,Ic 2.89A

qubit inductance, Lq 344 pH

qubit capacitance, Cq 110 fF

TABLE I: Qubit Parameters.

the unitless biases hi and couplings Jij encode a par-

ticular optimization problem. We define hi E

hi andJij EJij . We have mapped the annealing parameters for this particular chip to a range of xccjj with therelation

s (xccjj(t)xccjj,initial)/(xccjj,finalxccjj,initial) = t/tf,(4)

wheretfis the total anneal time. We implement QA forthis processor by ramping the external control xccjj(t)from xccjj,initial = 0.596 0 (s = 0) a t t = 0 t oxccjj,final = 0.666 0 (s= 1) at t = tf. The energy scale

E Meff|Ipq (s)|2 is set by thes-dependent persistent cur-rent of the qubit |Ipq (s)| and the maximum mutual induc-tance between qubitsM

eff= 1.37 pH [8]. The transverse

term in Hamiltonian (3), (s), is the energy gap betweenthe ground and first excited state of an isolated rf SQUIDat zero bias. also changes with annealing parameter s.xqi(t) is provided by a global external magnetic flux biasalong with local in situtunable digital-to-analog convert-ers (DAC) that tune the coupling strength of this globalbias into individual qubits and thus allow us to specifyindividual biases hi. The coupling energy between theith andj th qubit is set with a local in situtunable DACthat controls xco,ij.

The main quantities associated with a flux qubit, and |Ipq |, primarily depend on macroscopic rf SQUID pa-rameters: junction critical current Ic, qubit inductanceLq, and qubit capacitanceCq. We calibrated all of theseparameters on this chip as described in [6, 8]. We cali-brated all inter-qubit coupling elements across their avail-able tuning range from 1.37 pH to3.7 pH as describedin Ref. [25]. We corrected for variations in qubit parame-ters with on-chip control as described in [8]. This allowedus to match|Ipq | and across all qubits throughout theannealing trajectory. Table I shows the median qubitparameters for the devices studied here.

Figure6shows measurements of and |Ipq | vs. sfor alleight qubits. was measured with single qubit Landau-Zener measurements froms= 0.515 tos= 0.658[38] and

with qubit tunneling spectroscopy (QTS) froms = 0.121tos = 0.407 [33]. The resolution limit of qubit tunnelingspectroscopy and the bandwidth of our external controllines during the Landau-Zener measurements preventedus from characterizing between s = 0.4 and s = 0.5,respectively. |Ipq | was measured by coupling a secondprobe qubit to the qubitqiwith a coupling ofMeff= 1.37pH and measuring the the flux Meff|Ipqi(s)| as a functionofs. |Ipq | is matched between qubits to within 3% and(s) is matched between qubits to within 8% across the

0.2 0.3 0.4 0.5 0.610

5

106

107

108

109

1010

/

h

(Hz)

s

Coherent Incoherent

q1

q2

q3

q4

q5

q6

q7

q8

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

s

|

Ip q

|

(A)

q1

q2

q3

q4

q5

q6

q7

q8

FIG. 6: (a) (s) vs s. We show measurements for all eightqubits studied in this work. We used a single qubit Landau-Zener experiment to measure /h < 100 MHz[38]. We usedqubit tunneling spectroscopy (QTS) to measure /h > 1GHz[33]. The red line shows the theoretical prediction for anrf SQUID model employing the median qubit parameters ofthe eight devices. The vertical black line separates coherent(left) and incoherent (right) evolution as estimated by analy-sis of single qubit spectral line shapes. (b)|Ipq |(s) vs s. Weshow measurements for all eight qubits studied in this work.We used a two-qubit coupled flux measurement with the inter-qubit coupling element set to 1.37 pH[8]. The red line showsthe theoretical prediction for an rf SQUID model employingthe median qubit parameters of the eight devices.

annealing region explored in this study.


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Appendix B. Qubit Tunneling Spectroscopy (QTS)

QTS allows one to measure the eigenspectrum of an N-qubit system governed by HamiltonianHS. Details onthe measurement technique are presented elsewhere[33].For convenience in comparing with this reference, we de-fine a qubit energy biasi 2hi. Measurements are per-formed by coupling an additional probe qubit qP, withqubit tunneling amplitude P , |J|, to one of the Nqubits of the system under study, for exampleq1. Whenwe use a coupling strength JP between qP and q1 andapply a compensating bias 1 = 2JP to q1, the resultingsystem + probeHamiltonian becomes

HS+P =HS [ JPz1 (1/2) P](1 zP). (5)For one of the localized states of the probe qubit, |P,

for which an eigenvalue of zP is equal to +1 (i.e. theprobe qubit in the right well), the contribution of theprobe qubit is exactly canceled, leading to HS+P =HS,with composite eigenstates|n, = |n|P and eigen-values ERn = En, which are identical to those of theoriginal system without the presence of the probe qubit.Here,|n is an eigenstate of the Hamiltonian HS (n =1, 2,..., 2N).

For the other localized state of the probe qubit,|P,when this qubit is in the left well, the ground state ofHS+P is|L0, =|L0 | P, with eigenvalueEL0 =EL0 + P, where|L0 is the ground state ofHS 2JPz1and EL0 is its eigenvalue. We choose| JP| kBT suchthat the state|L0, is well separated from the nextexcited state for ferromagnetically coupled systems, andthus system + probe can be initialized in this state tohigh fidelity.

Introducing a small transverse term, 1

2Px

P, toHamiltonian (5) results in incoherent tunneling fromthe initial state |L0, to any of the available |n, states[39]. A bias on the probe qubit, P, changes theenergy difference between the probe |P and |P mani-folds. We can thus bring|L0, into resonance with anyof|n, states (whenEL0 = ERn) allowing resonant tun-neling between the two states. The rate of tunneling outof the initially prepared state|L0, is thus peaked atthe locations of|n, .

The measurement of the eigenspectrum of an N-qubitsystem thus proceeds as follows. We couple an additionalprobe qubit to one of theN-qubits (say, to q1) with cou-

pling constant JP. We prepare theN+1-qubit system inthe state|L0, by annealing from s = 0 to s = 1 in thepresence of large biaspol< 0 on all the system and probequbits. We then adjusts for the N-qubit system to anintermediate points [0, 1] such that kBT /h ands for the probe qubit to sP = 0.612 such that P/h 1MHz (here h is the Planck constant). We assert a com-pensating bias1 = 2JP to this qubit. We dwell at thispoint for a time , complete the anneal s 1 for thesystem+probe, and then read out the state of the probe

FIG. 7: Typical waveforms during QTS. We prepare theinitial state by annealing probe and system qubits from s = 0tos = 1 in the presence of a large polarization bias pol. Wethen bias the system qubit q1(to which the probe is attached)to a bias 1 and the probe qubit to a bias P. With thesebiases asserted, we then adjust the system qubits annealingparameter to an intermediate points and the probe qubit toa point sP and dwell for a time . Finally, we complete theanneals 1 and read out the state of the qubits.

qubit. Figure 7 summarizes these waveforms during atypical QTS measurement.

We perform this measurement for a range of whichallows us to measure an initial rate of tunneling from|L0, to|, . We repeat this measurement of for arange of the probe qubit bias P. Peaks in correspondto resonances between the initially prepared state and thestate|n, , thus allowing us to map the eigenspectrumof theN-qubit system.

For the plots in the main paper, measurements of are normalized to [0, 1] by dividing the maximum valueacross a vertical slice to give a visually interpretable re-sult. Figure 8b shows a typical raw result in units ofs1.

We posed ferromagnetically coupled instances of theform

HP = i

hizi +i


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centering

FIG. 8: Spectroscopy data for two FM coupled qubits at JP < 0. (a) Measurements of tunneling rate for three values ofh1 = h2 hi. These data were taken ats = 0.339. Peaks in reveal the energy eigenstates of the two-qubit system. (b)Multiple scans of for different values ofhi assembled into a two-dimensional color plot. For better interpretability, we havesubtracted off a baseline energy with respect to (a) such that the ground and first excited levels are symmetric about zero.Notice the avoided crossing at hi = 0. The peak tunneling rate |PL0 | n|2 [33]. The solid black and white curves plotthe theoretical expectations for the energy eigenvalues using independent measurements shown in Figure 6and Hamiltonian (1).

are in the limit| JP| kBTsuch that there is only oneaccessible state in the |P manifold:|

L0 |P. As de-

scribed above, the other available states in the system arethe composite eigenstates |n|Pin the |P manifoldwhere |n is an eigenstate of theN-qubit system withoutthe probe qubit attached. Energy levelsERn of the|Pmanifold coincide with the energy levels En of the sys-tem,ERn =En, even in the presence of coupling betweenthe probe qubit and the system. We make the assump-tion that the population of an eigenstate depends only onits energy. Degenerate states have the same population.

Let PL represent the probability of finding theprobe+systemin the state|L0 |P andPRn representthe probability of finding the probe+system in the state

|n | P. At any point in the probe+systemevolutionwe expect:

PL +

2Ni=1

PRi = 1 (7)

As described in the previous section, we can alter theenergy of|L0 |Pwith the probe bias P. Based onthe spectroscopic measurements of the N-qubit eigen-

spectrum, we can choose an P such that|L0 | Pand|n | P are degenerate. Since the occupation ofthe state depends on its energy, we expect that, after longevolution times, these two degenerate states are occupiedwith equal probability, PL(P=En) =P

Rn . Aligning the

state |L0 |Pwith all possible 2N states |n|P weobtain a set of relative probabilities PRn . These relativeprobabilities characterize the population distribution inthe system since they are uniquely determined by theenergy spectrum En. However, as follows from Eq. (7),the set PRn is not properly normalized. The probabilitydistribution of the system itself is given by:

Pn(En) = PRn

2Ni=1 PRi, (8)

where2N

n=1 Pn = 1. At every eigenenergy, P = En,the denominator of Eq. (8) can be found from Eq. (7),so that the population distribution of the system Pn hasthe form

Pn= PRn1 PL =

PL(P=En)

1 PL(P=En) . (9)

Thus, the probabilityPnto find the system ofNqubits in


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the state with energyEn can be estimated by measuringPL atP =En and using Equation (9).

Measurements ofPL proceed as they do for the spec-troscopy measurements. The system+probe is preparedin|L0, . We then adjust P = En, and an annealingparameters for the N-qubit system to some intermedi-ate point, and also sP = 0.612 for the probe qubit suchthat P/h1 MHz. We dwell at this point for a time 1/, complete the anneal s 1, and then read outthe state of the probe qubit. We typically investigate arange of to ensure that we are in the long evolutiontime limit in which PL is independent of. We usePL

measured with = 7041 s to estimate P1 and P2. TheSupplementary Information contains typical data usedfor these estimates.

Appendix D.

Susceptibility-based entanglement witnessW

For a bipartion of the system into two parts, A and B,

we define a witnessRAB asRAB= 1

4NAB|iA

jB

Jijij|, (10)

whereij is a cross-susceptibility, Jij = EJij , andNABisa number of non-zero couplings, Jij= 0, between qubitsfrom the subset A and the subset B (see Ref. [34] andthe Supplementary Information). We note that at lowtemperature, T = 12.5 mK, the measured susceptibilityij(T) almost coincides with the ground-state suscepti-bilityij(T= 0) since contributions of excited states toij(T) are proportional to their populations,Pn

1, for

n >1. We analyze a deviation of the measured suscepti-bility from its ground-state value in the SupplementaryInformation. To characterize global entanglement in thesystem ofNqubits we introduce a witnessW,

W= (RAB)1/Np

1 + (RAB)1/Np , (11)

which is given by a bounded geometrical mean of wit-nessesRAB calculated for all possible partitions of thewhole system into two subsystems. HereNp is a num-ber of such bipartitions, in particular, Np = 127 for the

eight-qubit ring.

Entanglement witnessWAB

Consider Hamiltonian (1) with measured parameters.This Hamiltonian describes a transverse Ising model hav-ing N qubits. The ground state|1 of this model isentangled with respect to some bipartitionA B of theN-qubit system. We can form an operator|11|TA

whereTA is a partial transposition operator with respectto the Asubsystem [16]. Let| be the eigenstate of|11|TA with the most negative eigenvalue. We canform a new operatorWAB =||TA . This operatorcan serve as an entanglement witness (it is trivially pos-itive on all separable states).

Let (s) be the density matrix associated with the stateof the system at the annealing point s. If we have ex-perimental measurements of the occupation fraction of

the ground state and first excited state, P1(s) P1 andP2(s)P2, respectively, we can place a set of linearconstraints on(s):

Tr[(s)|11|] P1(s) P1Tr[(s)|11|] P1(s) + P1Tr[(s)|22|] P2(s) P2Tr[(s)|22|] P2(s) + P2

We now search over all possible (s) that satisfy the lin-ear constraints provided by the experimental data. Thegoal is to maximize the witness Tr[

WAB(s)] in order

to establish an upper limit for this quantity. Maximizingthis quantity can be cast as a semidefinite program [36], aclass of convex optimization problems for which efficientalgorithms exist. When this upper limit is less than zero,entanglement is certified for the bipartitionA B.

We tested the robustness of this result with uncertain-ties in the parameters of the Hamiltonian. To do this,we have repeated the analysis at several points duringthe QA algorithm when adding random perturbationson the measured Hamiltonian that correspond to the un-certainty on these measured quantities. We sampled 104

perturbed Hamiltonians and, for every perturbation, the

optimization resulted in Tr[WAB(s)]< 0.

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Entanglement in a Quantum Annealing Processor - T. Lanting