QUANTUM INFORMATION

A blueprint for demonstratingquantum supremacy withsuperconducting qubitsC. Neill,1*† P. Roushan,2* K. Kechedzhi,3,4 S. Boixo,2 S. V. Isakov,2 V. Smelyanskiy,2

A. Megrant,2 B. Chiaro,1 A. Dunsworth,1 K. Arya,2 R. Barends,2 B. Burkett,2 Y. Chen,2

Z. Chen,1 A. Fowler,2 B. Foxen,1 M. Giustina,2 R. Graff,2 E. Jeffrey,2 T. Huang,2

J. Kelly,2 P. Klimov,2 E. Lucero,2 J. Mutus,2 M. Neeley,2 C. Quintana,1 D. Sank,2

A. Vainsencher,2 J. Wenner,1 T. C. White,2 H. Neven,2 J. M. Martinis1,2†

A key step toward demonstrating a quantum system that can address difficult problemsin physics and chemistry will be performing a computation beyond the capabilities ofany classical computer, thus achieving so-called quantum supremacy. In this study, we usednine superconducting qubits to demonstrate a promising path toward quantum supremacy.By individually tuning the qubit parameters, we were able to generate thousands of distinctHamiltonian evolutions and probe the output probabilities.The measured probabilities obey auniversal distribution, consistent with uniformly sampling the full Hilbert space. As thenumber of qubits increases, the system continues to explore the exponentially growingnumber of states. Extending these results to a system of 50 qubits has the potential toaddress scientific questions that are beyond the capabilities of any classical computer.

Aprogrammable quantum system consist-ing ofmerely 50 to 100 qubits could have amarked impact on scientific research. Al-though such a platform is naturally suitedto address problems in quantum chemistry

and materials science (1–4), applications extendto fields as diverse as classical dynamics (5) andcomputer science (6–9). An important milestoneon the path toward realizing these applicationswill be the demonstration of an algorithm thatexceeds the capabilities of any classical computer,thus achievingquantumsupremacy (10). Samplingproblems are an iconic example of algorithmsdesigned specifically for this purpose (11–14). Asuccessful demonstration of quantum supremacywould prove that engineered quantum systems,although still in their infancy, can outperform themost advanced classical computers.Consider a system of coupled qubits whose

dynamics uniformly explore all accessible statesover time. The complexity of simulating this evo-lution on a classical computer is easy to under-stand and quantify. Because every state is equallyimportant, it is not possible to simplify the prob-lem by using a smaller truncated state-space.The complexity is then simply given by howmuchclassical memory it takes to store the state vector.Storing the state of a 46-qubit system requiresnearly a petabyte of memory and is at the limitof themost powerful computers (14, 15). Sampling

from the output probabilities of such a systemwould therefore constitute a clear demonstrationof quantum supremacy. Note that this is an upperboundononly thenumberof qubits required—otherconstraints, such as computation time,may placepractical limitations on even smaller system sizes.In this study, we experimentally illustrate a

blueprint for demonstrating quantum supremacy.We present data characterizing two basic ingre-dients required for any supremacy experiment:complexity and fidelity. First, we show that thequbits can quasi-uniformly explore the Hilbertspace, providing an experimental indication ofalgorithm complexity [see (16) for a formal dis-cussion of computational complexity]. Next, wecompare the measurement results with the ex-pected behavior and show that the algorithm canbe implemented with high fidelity. Experimentsprobing complexity and fidelity provide a founda-tion for demonstrating quantum supremacy.The more control a quantum platform offers,

the easier it is to embed diverse applications. Forthis reason, we have developed superconductinggmon qubits, which are based on transmon qubitsbut have tunable frequencies and tunable inter-actions (Fig. 1A). The nine-qubit device consists ofthree distinct sections: control (bottom), qubits(center) and readout (top). A detailed circuit dia-gram is provided in (16).Each of our gmon qubits can be thought of as a

nonlinear oscillator. The Hamiltonian for thedevice is given by

H ¼X9

i¼1

dini þ hi2niðni � 1Þ þ

X8

i¼1

giðai†aiþ1 þ ai aiþ1† Þ ð1Þ

where n is the number operator and a† (a) is theraising (lowering) operator. The qubit frequencysets the coefficient di, the nonlinearity sets hi,and the nearest-neighbor coupling sets gi. Thetwo lowest energy levels (j0i and j1i) form thequbit subspace. The higher energy levels ofthe qubits, although only virtually occupied, sub-stantially modify the dynamics. In the absence ofhigher levels, this model maps to free particlesand can be simulated efficiently (16). The inclusionof higher levels effectively introduces an inter-action and allows for the occurrence of complexdynamics.In Fig. 1, B and C, we outline the experimental

procedure and provide two instances of theraw output data. Figure 1B shows a five-qubitexample of the pulses used to control the qubits.First, the system is initialized (red) by placingtwo of the qubits in the excited state; e.g., j00101i.The dynamics result from fixing the qubit fre-quencies (orange) and simultaneously rampingall of the nearest-neighbor interactions on andthen off (green). The shape of the coupling pulseis chosen to minimize leakage out of the qubitsubspace (17). After the evolution, we simulta-neously measure the state of every qubit. Eachmeasurement results in a single output state,such as j10010i; the experiment is repeated manytimes to estimate the probability of every pos-sible output state. We then carry out this pro-cedure for randomly chosen values of the qubitfrequencies, the coupler pulse lengths, and thecoupler pulse heights. The probabilities of thevarious output states are shown in Fig. 1C fortwo instances of the evolution after 10 couplerpulses (cycles). The height of each bar representthe probability with which that output state ap-peared in the experiments.The Hamiltonian in Eq. 1 conserves the total

number of excitations. This means that if westart in a state with half of the qubits excited,we should also end in a state with half of thequbits excited. However, most experimental errorsdonot obey this symmetry, allowing us to identifyand remove erroneous outcomes. Although thissymmetry helps to reduce the impact of errors, itslightly reduces the size of the Hilbert space. ForN qubits, the number of states is given by thepermutations of N/2 excitations in N qubitsand is approximately 2N=

ffiffiffiffiN

p. As an example, a

64-qubit system would access ~261 states underour protocol.Although the measured probabilities appear

largely random, they provide important insightinto the quantum dynamics of the system. A keyfeature of these data sets are the rare, taller-than-average peaks, which are analogous to the high-intensity regions of a laser’s speckle pattern. Thesehighly likely states serve as a fingerprint of theunderlying evolution and provide a means forverifying that the desired evolution was properlygenerated. The distribution of these probabilitiesprovides evidence that the dynamics coherentlyand uniformly explore the Hilbert space.In Fig. 2, we use the measured probabilities to

show that the dynamics uniformly explore theHilbert space for experiments carried out with

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Neill et al., Science 360, 195–199 (2018) 13 April 2018 1 of 4

1Department of Physics, University of California, SantaBarbara (UCSB), Santa Barbara, CA 93106, USA. 2Google,Santa Barbara, CA 93117, USA. 3Quantum ArtificialIntelligence Laboratory (QuAIL), NASA Ames ResearchCenter, Moffett Field, CA 94035, USA. 4Universities SpaceResearch Association, Mountain View, CA 94043, USA.*These authors contributed equally to this work.†Corresponding author. Email: cneill@google.com (C.N.);martinis@physics.ucsb.edu (J.M.M.)

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five to nine qubits. We begin by measuring theoutput probabilities after five cycles for between500 and 5000 distinct instances. To compare ex-periments with different numbers of qubits, theprobabilities areweighted by the number of statesin theHilbert space. Figure 2A shows a histogramof theweighted probabilitieswherewe findnearlyuniversal behavior. Small probabilities (<1/Nstates)appear most often, and probabilities as large as4/Nstates showupwith a frequency of ~1%. In starkcontrast to this, we observe a tall, narrow peak cen-tered at 1.0 for longer evolutions whose duration iscomparable to the coherence time of the qubits.A quantum system that uniformly explores all

states is expected to have an exponential dis-tribution of weighted probabilities. The solid line inFig. 2A corresponds to such a distribution andis simply given by e�probability�Nstates; this is also re-ferred to as a Porter-Thomas distribution (14, 18).Although, in principle, approaching the univer-sal form of the distribution takes exponentialtime, the nonuniversal deviations become smallon a much shorter time scale that is linear inthe number of qubits (14, 16). Note that the de-

viations from a purely exponential distributionare consistent with decoherence. The deviationsscalewith the number of qubits, and the histogramappears to be converging to the incoherent dis-tribution shown in green.A measure of algorithm complexity is a key in-

gredient for demonstrating quantum supremacy.We argue that evolution under the Hamiltonian inEq. 1 cannot be efficiently simulated on a classicalcomputer under plausible assumptions (16). Theexperimental results in Fig. 2A suggest that we cancoherently evolve the system for long enough torealize this computationally notable regime.In Fig. 2B, we illustrate the number of cycles

necessary for the system to uniformly explore allstates by comparing the measured probabilitiesto an exponential distribution. After each cycle,we compare the measured histogram to anexponential decay. The distance between thesetwo distributions is measured using theKullback-Leibler divergence DKL

DKL ¼ Sðrmeasured; rexponentialÞ�SðrmeasuredÞ ð2Þ

where the first term is the cross-entropy betweenthe measured distribution rmeasured and an ex-ponential distribution rexponential, and the secondterm is the self-entropy of themeasured distribu-tion. The entropy of a set of probabilities is given

by SðPÞ ¼ �X

i

pilogðpiÞ and the cross-entropy

of two sets of probabilities is given by SðP;QÞ ¼�X

i

pilogðqiÞ. Their difference, the Kullback-

Leibler divergence, is zero if and only if the twodistributions are equivalent.We find that the experimental probabilities

closely resemble an exponential distribution afterjust two cycles. For longer evolutions, decoherencereduces this overlap. These results suggest thatwe can generate very complex dynamics withonly two pulses, a surprisingly small number.However, rather than breaking up the evolu-tion into two-qubit gates, we allow the entiresystem to interact at once. Therefore, one ofour pulses corresponds to roughly eight simul-taneous two-qubit gates. Additionally, each ofour pulses lasts long enough to effectively imple-ment five square-root-of-swap gates. So, althoughthe evolution is only two cycles, this translates to~80 two-qubit gates.In addition to demonstrating an exponential

scaling of complexity, it is necessary to charac-terize the algorithm fidelity. Determining thefidelity requires a means for comparing the mea-sured probabilities (Pmeasured) with the proba-bilities expected from the desired evolution(Pexpected). On the basis of the proposal outlinedin (14), we use the cross-entropy to quantify thefidelity

SðPincoherent;PexpectedÞ � SðPmeasured;PexpectedÞSðPincoherent;PexpectedÞ � SðPexpectedÞ ð3Þ

where Pincoherent represents an incoherent mix-turewith each output state given equal likelihood—this is the behavior that we observe after manycycles.When the distances between themeasuredand expected probabilities are small, the fidelityapproaches 1. When the measured probabilitiesapproach an incoherent mixture, the fidelity ap-proaches 0.In Fig. 3A, we show that the desired evolution

can be implemented with high fidelity. We findthat at short times the fidelity decays linearlywith an increasing number of cycles (fits to thedata are shown as dashed lines). The slope ofthese linesmeasures the error per cycle; this slopeis shown in the inset for each number of qubits.We find that the error scales with the numberof qubits at a rate of ~0.4% error per qubit percycle. If such an error rate extends to larger sys-tems, we will be able to perform 60-qubit experi-ments of depth = 2 with a fidelity >50%. Theseresults provide promising evidence that quan-tum supremacy may be achievable with the useof existing technology.Predicting the expected probabilities is a ma-

jor challenge. First, substantial effort has beentaken to accurately map the control currents toHamiltonian parameters; the detailed procedure

Neill et al., Science 360, 195–199 (2018) 13 April 2018 2 of 4

Fig. 1. Device andexperimental protocol.(A) Optical micrographof the nine-qubit array.Gray regions are alumi-num; dark regions arewhere the aluminum hasbeen etched away todefine features. Colorshave been added to dis-tinguish readout circuitry,qubits, couplers, andcontrol wiring. (B) Five-qubit example of thepulse sequences used inthese experiments. First,the qubits are initializedusing microwave pulses(red).Three of the qubitsstart in the ground statej0i and the rest start inthe excited state j1i. Next,the qubit frequenciesare set using rectangularpulses (orange). Duringthis time, all couplingsare simultaneouslypulsed (green); eachpulse has a randomlyselected duration. Last,we measure the state ofevery qubit. The mea-surement is repeatedmany times to estimatethe probability of eachoutput state. (C) Werepeat this pulsesequence for randomlyselected control parameters. Each instance corresponds to a different set of qubit frequencies, couplingpulse heights and lengths. Here we plot the measured probabilities for two instances after 10 couplerpulses (cycles). Error bars (±3 SD) represent the statistical uncertainty from 50,000 samples.Predictions from a control model are overlaid as red circles.

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for constructing this map is outlined in (16). Sec-ond, wemodel theHamiltonian using only single-qubit calibrations, which we find to be accurateeven when all of the couplers are used simulta-neously. This is a scalable approach to calibration.

Third, when truncating the Hamiltonian to twolevels, we find poor agreementwith both an exacttheoretical model and experimental results. Wefind that a three-level description must be usedto account for virtual transitions to the second

excited state during the evolution. When includ-ing these states, truncating to a fixed number ofexcitations lowers the size of the computationalHilbert space from 3N to approximately 0.15 ×2.42N (table S1): Thus, a nine-qubit experimentrequires accurately modeling a 414-dimensionalunitary operation. Determining how many ofthese states are needed for sufficient accuracydepends on the magnitude of the coupling andis an open question, but the number should scalesomewhere between 2.0N and 2.5N. The predic-tions are overlaid onto the data inFig. 1C and showexcellent agreement.In Fig. 3B, we show how techniques from

machine learning were used to achieve low errorrates. To set thematrix elements of theHamiltonian,we built a physical model for our gmon qubits.This model is parameterized in terms of capaci-tances, inductances, and control currents. Theparameters in this model were calibrated usingsimple single-qubit experiments (16). We useda search algorithm to find offsets in the controlmodel thatminimize the error (1 - Fidelity). Figure3B shows the error, averaged over cycles, versusthe number of optimization steps. Before train-ing the model, the data were split into twohalves: a training set (red) and a verification set(black). The optimization algorithm was usedonly to access the training set, whereas the ver-ification set was used only to verify the optimalparameters.We find that the error in both the training set

and the verification set fall considerably by theend of the optimization procedure. The highdegree of correlation between the training andverification data suggests that we are genuinelylearning a better physicalmodel. Optimizing overmore parameters does not further reduce theerror. This suggests that the remaining error isnot the product of an inaccurate control modelbut rather results from decoherence. Using thecross-entropy as a cost function for optimizingthe parameters of a physical model was the key toachieving high-fidelity control in this experiment.It is important to note how these experiments

might change at the level of a few tens of qubits.At this level, it becomes exponentially unlikely thatany state will appear twice, making it impracticalto measure probabilities in an experiment. How-ever, even for these large systems, sampling fromthe output states is sufficient to determine thefidelity (14). Therefore, the distribution of prob-abilities can be inferred from the classical com-putations, and a high-fidelity experimental resultindicates that we are likely solving a difficult com-putational problem.Ideally, in addition to exponential complexity

and high fidelity, a quantum platform shouldoffer valuable applications. In Fig. 4, we illustrateapplications of our algorithms to many-bodyphysicswhere the exponential growth in complexityis a substantial barrier to ongoing research (19–24).By varying the amount of disorder in the system,weare able to study disorder-induced localization. Thisis done using two-body correlations

jhninji � hniihnjij ð4Þ

Neill et al., Science 360, 195–199 (2018) 13 April 2018 3 of 4

Fig. 2. Complexity:uniform sampling of anexponentially growingstate-space.(A) Histogram of the rawprobabilities (see Fig. 1C)for five- to nine-qubitexperiments after five cyclesof evolution. Before makingthe histogram, probabilitieswere weighted by the numberof states in the Hilbert space,with all curves placed on auniversal axis.The widthof the bars representsthe size of the bins used toconstruct the histogram.Thedata are taken frommorethan 29.7million experiments.For dynamics that uniformlyexplore all states, this histo-gram decays exponentially;an exponential decay isshown as a solid line forcomparison. A histogram ofthe probabilities for sevenqubits after 100 cycles isshown for contrast. In thisplot, decoherence dominates and we observe a tall narrow peak around 1. (B) To measure convergenceof the measured histograms to an exponential distribution, we compute their distance as a function of thenumber of cycles. Distance is measured using the Kullback-Leibler divergence (Eq. 2).We find that amaximum overlap occurs after just two cycles, and decoherence subsequently increases the distancebetween the distributions.

Fig. 3. Fidelity: learning abetter control model.(A) Average fidelity decayversus number of cycles forfive- to nine-qubit experiments(circles). The fidelity iscomputed from Eq. 3. Theerror per cycle, presented inthe inset, is the slope of thedashed line that best fits thedata. (B) Using the fidelity as acost function, we learn optimalparameters for our controlmodel. We take half of theexperimental data to train ourmodel. The other half of thedata is used to verify thisnew model; the optimizerdoes not have access to thesedata. The correspondingimprovement in fidelity of theverification set providesevidence that we are indeedlearning a better control model.

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whichwe average over qubit pairs, cycles (numberof coupler pulses), and instances (choice of ran-domly selected pulse parameters). In Fig. 4A, weplot the average two-body correlations againstthe separation between qubits. This experimentis performed for both low and high disorder inthe qubit frequencies (shown in gold and blue,respectively). Figure 4B depicts the results of ourexperiment as we continuously vary the amountof disorder.At low disorder, we find that the correlations

are independent of separation: qubits at oppositeends of the chain are as correlated as nearestneighbors. At high disorder, the correlations falloff exponentiallywith separation. The rate atwhichthis exponential decays allows us to determinethe correlation length. A fit to the data is shown

in Fig. 4A as a solid blue line where we find acorrelation length of roughly four qubits. Thestudy of localization and delocalization in inter-acting systems provides a promising applicationof our algorithms.

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ACKNOWLEDGMENTS

We thank E. Kapit and J. Fitzsimons for discussions. Funding:This work was supported by Google. C.Q. and Z.C. acknowledgesupport from the NSF Graduate Research Fellowship undergrant DGE-1144085. Devices were made at the UCSBNanofabrication Facility, a part of the NSF-funded NationalNanotechnology Infrastructure Network. K.K. acknowledgessupport from NASA Academic Mission Services, under contractnumber NNA16BD14C. The views and conclusions containedherein are those of the authors and should not be interpreted asnecessarily representing the official policies or endorsements,either expressed or implied, of the U.S. government. The U.S.government is authorized to reproduce and distribute reprintsfor governmental purposes, notwithstanding any copyrightannotation thereon. Author contributions: C.N. designedand fabricated the device. C.N. and P.R. designed the experiment.C.N. performed the experiment and analyzed the data. C.N., K.K.,and V.S. developed the physical control model. S.B. and S.V.I.numerically validated the protocol for large qubit arrays. All membersof the UCSB and Google teams contributed to the experimentalsetup and manuscript preparation. Competing interests: Nonedeclared. Data and materials availability: The data that support theplots presented in this paper and other findings of this studyare available in the supplementary materials.

SUPPLEMENTARY MATERIALS

www.sciencemag.org/content/360/6385/195/suppl/DC1Supplementary TextFigs. S1 to S27Tables S1 and S2References (25–39)Data S1

19 July 2017; accepted 14 February 201810.1126/science.aao4309

Neill et al., Science 360, 195–199 (2018) 13 April 2018 4 of 4

Fig. 4. Applications:localization anddelocalization.(A) Average two-bodycorrelations (Eq. 4) asa function of theseparation betweenqubits. Data are shownfor two values ofdisorder strength. Atlow disorder, the qubitfrequencies are setover a range of±5 MHz, and thetwo-body correlationsare independent ofseparation (i.e., qubitsat the ends of thechain are just ascorrelated as nearestneighbors). At highdisorder, the qubitfrequencies are setover a range of±30 MHz, and we findan exponential decayin correlations as a function of separation. (B) Map of correlations as a continuous function offrequency disorder. Arrows indicate the location of line cuts used in (A).We observe a clear transitionfrom long-range to short-range correlations.

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A blueprint for demonstrating quantum supremacy with superconducting qubits

Lucero, J. Mutus, M. Neeley, C. Quintana, D. Sank, A. Vainsencher, J. Wenner, T. C. White, H. Neven and J. M. MartinisBarends, B. Burkett, Y. Chen, Z. Chen, A. Fowler, B. Foxen, M. Giustina, R. Graff, E. Jeffrey, T. Huang, J. Kelly, P. Klimov, E. C. Neill, P. Roushan, K. Kechedzhi, S. Boixo, S. V. Isakov, V. Smelyanskiy, A. Megrant, B. Chiaro, A. Dunsworth, K. Arya, R.

DOI: 10.1126/science.aao4309 (6385), 195-199.360Science 

, this issue p. 195Scienceachievable with current technologies.continues to increase at the same rate, a quantum computer with about 60 qubits and reasonable fidelity might beaffects the quality of the output of their superconducting qubit device. If, as the number of qubits grows further, the error

explore how increasing the number of qubits from five to nineet al.up existing architectures to this number is tricky. Neill that a classical computer cannot. It has been estimated that such a computer would need around 50 qubits, but scaling

Quantum information scientists are getting closer to building a quantum computer that can perform calculationsScaling up to supremacy

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