Abstract
The promise of quantum computers is that certain computational tasks might be executed exponentially faster on a quantum processor than on a classical processor^{1}. A fundamental challenge is to build a highfidelity processor capable of running quantum algorithms in an exponentially large computational space. Here we report the use of a processor with programmable superconducting qubits^{2,3,4,5,6,7} to create quantum states on 53 qubits, corresponding to a computational statespace of dimension 2^{53} (about 10^{16}). Measurements from repeated experiments sample the resulting probability distribution, which we verify using classical simulations. Our Sycamore processor takes about 200 seconds to sample one instance of a quantum circuit a million times—our benchmarks currently indicate that the equivalent task for a stateoftheart classical supercomputer would take approximately 10,000 years. This dramatic increase in speed compared to all known classical algorithms is an experimental realization of quantum supremacy^{8,9,10,11,12,13,14} for this specific computational task, heralding a muchanticipated computing paradigm.
Main
In the early 1980s, Richard Feynman proposed that a quantum computer would be an effective tool with which to solve problems in physics and chemistry, given that it is exponentially costly to simulate large quantum systems with classical computers^{1}. Realizing Feynman’s vision poses substantial experimental and theoretical challenges. First, can a quantum system be engineered to perform a computation in a large enough computational (Hilbert) space and with a low enough error rate to provide a quantum speedup? Second, can we formulate a problem that is hard for a classical computer but easy for a quantum computer? By computing such a benchmark task on our superconducting qubit processor, we tackle both questions. Our experiment achieves quantum supremacy, a milestone on the path to fullscale quantum computing^{8,9,10,11,12,13,14}.
In reaching this milestone, we show that quantum speedup is achievable in a realworld system and is not precluded by any hidden physical laws. Quantum supremacy also heralds the era of noisy intermediatescale quantum (NISQ) technologies^{15}. The benchmark task we demonstrate has an immediate application in generating certifiable random numbers (S. Aaronson, manuscript in preparation); other initial uses for this new computational capability may include optimization^{16,17}, machine learning^{18,19,20,21}, materials science and chemistry^{22,23,24}. However, realizing the full promise of quantum computing (using Shor’s algorithm for factoring, for example) still requires technical leaps to engineer faulttolerant logical qubits^{25,26,27,28,29}.
To achieve quantum supremacy, we made a number of technical advances which also pave the way towards error correction. We developed fast, highfidelity gates that can be executed simultaneously across a twodimensional qubit array. We calibrated and benchmarked the processor at both the component and system level using a powerful new tool: crossentropy benchmarking^{11}. Finally, we used componentlevel fidelities to accurately predict the performance of the whole system, further showing that quantum information behaves as expected when scaling to large systems.
A suitable computational task
To demonstrate quantum supremacy, we compare our quantum processor against stateoftheart classical computers in the task of sampling the output of a pseudorandom quantum circuit^{11,13,14}. Random circuits are a suitable choice for benchmarking because they do not possess structure and therefore allow for limited guarantees of computational hardness^{10,11,12}. We design the circuits to entangle a set of quantum bits (qubits) by repeated application of singlequbit and twoqubit logical operations. Sampling the quantum circuit’s output produces a set of bitstrings, for example {0000101, 1011100, …}. Owing to quantum interference, the probability distribution of the bitstrings resembles a speckled intensity pattern produced by light interference in laser scatter, such that some bitstrings are much more likely to occur than others. Classically computing this probability distribution becomes exponentially more difficult as the number of qubits (width) and number of gate cycles (depth) grow.
We verify that the quantum processor is working properly using a method called crossentropy benchmarking^{11,12,14}, which compares how often each bitstring is observed experimentally with its corresponding ideal probability computed via simulation on a classical computer. For a given circuit, we collect the measured bitstrings {x_{i}} and compute the linear crossentropy benchmarking fidelity^{11,13,14} (see also Supplementary Information), which is the mean of the simulated probabilities of the bitstrings we measured:
where n is the number of qubits, P(x_{i}) is the probability of bitstring x_{i} computed for the ideal quantum circuit, and the average is over the observed bitstrings. Intuitively, \({ {\mathcal F} }_{{\rm{XEB}}}\) is correlated with how often we sample highprobability bitstrings. When there are no errors in the quantum circuit, the distribution of probabilities is exponential (see Supplementary Information), and sampling from this distribution will produce \({{\mathscr{F}}}_{{\rm{X}}{\rm{E}}{\rm{B}}}=1\). On the other hand, sampling from the uniform distribution will give ⟨P(x_{i})⟩_{i} = 1/2^{n} and produce \({{\mathscr{F}}}_{{\rm{X}}{\rm{E}}{\rm{B}}}=0\). Values of \({ {\mathcal F} }_{{\rm{XEB}}}\) between 0 and 1 correspond to the probability that no error has occurred while running the circuit. The probabilities P(x_{i}) must be obtained from classically simulating the quantum circuit, and thus computing \({ {\mathcal F} }_{{\rm{XEB}}}\) is intractable in the regime of quantum supremacy. However, with certain circuit simplifications, we can obtain quantitative fidelity estimates of a fully operating processor running wide and deep quantum circuits.
Our goal is to achieve a high enough \({ {\mathcal F} }_{{\rm{XEB}}}\) for a circuit with sufficient width and depth such that the classical computing cost is prohibitively large. This is a difficult task because our logic gates are imperfect and the quantum states we intend to create are sensitive to errors. A single bit or phase flip over the course of the algorithm will completely shuffle the speckle pattern and result in close to zero fidelity^{11} (see also Supplementary Information). Therefore, in order to claim quantum supremacy we need a quantum processor that executes the program with sufficiently low error rates.
Building a highfidelity processor
We designed a quantum processor named ‘Sycamore’ which consists of a twodimensional array of 54 transmon qubits, where each qubit is tunably coupled to four nearest neighbours, in a rectangular lattice. The connectivity was chosen to be forwardcompatible with error correction using the surface code^{26}. A key systems engineering advance of this device is achieving highfidelity single and twoqubit operations, not just in isolation but also while performing a realistic computation with simultaneous gate operations on many qubits. We discuss the highlights below; see also the Supplementary Information.
In a superconducting circuit, conduction electrons condense into a macroscopic quantum state, such that currents and voltages behave quantum mechanically^{2,30}. Our processor uses transmon qubits^{6}, which can be thought of as nonlinear superconducting resonators at 5–7 GHz. The qubit is encoded as the two lowest quantum eigenstates of the resonant circuit. Each transmon has two controls: a microwave drive to excite the qubit, and a magnetic flux control to tune the frequency. Each qubit is connected to a linear resonator used to read out the qubit state^{5}. As shown in Fig. 1, each qubit is also connected to its neighbouring qubits using a new adjustable coupler^{31,32}. Our coupler design allows us to quickly tune the qubit–qubit coupling from completely off to 40 MHz. One qubit did not function properly, so the device uses 53 qubits and 86 couplers.
The processor is fabricated using aluminium for metallization and Josephson junctions, and indium for bumpbonds between two silicon wafers. The chip is wirebonded to a superconducting circuit board and cooled to below 20 mK in a dilution refrigerator to reduce ambient thermal energy to well below the qubit energy. The processor is connected through filters and attenuators to roomtemperature electronics, which synthesize the control signals. The state of all qubits can be read simultaneously by using a frequencymultiplexing technique^{33,34}. We use two stages of cryogenic amplifiers to boost the signal, which is digitized (8 bits at 1 GHz) and demultiplexed digitally at room temperature. In total, we orchestrate 277 digitaltoanalog converters (14 bits at 1 GHz) for complete control of the quantum processor.
We execute singlequbit gates by driving 25ns microwave pulses resonant with the qubit frequency while the qubit–qubit coupling is turned off. The pulses are shaped to minimize transitions to higher transmon states^{35}. Gate performance varies strongly with frequency owing to twolevelsystem defects^{36,37}, stray microwave modes, coupling to control lines and the readout resonator, residual stray coupling between qubits, flux noise and pulse distortions. We therefore optimize the singlequbit operation frequencies to mitigate these error mechanisms.
We benchmark singlequbit gate performance by using the crossentropy benchmarking protocol described above, reduced to the singlequbit level (n = 1), to measure the probability of an error occurring during a singlequbit gate. On each qubit, we apply a variable number m of randomly selected gates and measure \({ {\mathcal F} }_{{\rm{XEB}}}\) averaged over many sequences; as m increases, errors accumulate and average \({ {\mathcal F} }_{{\rm{XEB}}}\) decays. We model this decay by [1 − e_{1}/(1 − 1/D^{2})]^{m} where e_{1} is the Pauli error probability. The state (Hilbert) space dimension term, D = 2^{n}, which equals 2 for this case, corrects for the depolarizing model where states with errors partially overlap with the ideal state. This procedure is similar to the more typical technique of randomized benchmarking^{27,38,39}, but supports nonCliffordgate sets^{40} and can separate out decoherence error from coherent control error. We then repeat the experiment with all qubits executing singlequbit gates simultaneously (Fig. 2), which shows only a small increase in the error probabilities, demonstrating that our device has low microwave crosstalk.
We perform twoqubit iSWAPlike entangling gates by bringing neighbouring qubits onresonance and turning on a 20MHz coupling for 12 ns, which allows the qubits to swap excitations. During this time, the qubits also experience a controlledphase (CZ) interaction, which originates from the higher levels of the transmon. The twoqubit gate frequency trajectories of each pair of qubits are optimized to mitigate the same error mechanisms considered in optimizing singlequbit operation frequencies.
To characterize and benchmark the twoqubit gates, we run twoqubit circuits with m cycles, where each cycle contains a randomly chosen singlequbit gate on each of the two qubits followed by a fixed twoqubit gate. We learn the parameters of the twoqubit unitary (such as the amount of iSWAP and CZ interaction) by using \({ {\mathcal F} }_{{\rm{XEB}}}\) as a cost function. After this optimization, we extract the percycle error e_{2c} from the decay of \({ {\mathcal F} }_{{\rm{XEB}}}\) with m, and isolate the twoqubit error e_{2} by subtracting the two singlequbit errors e_{1}. We find an average e_{2} of 0.36%. Additionally, we repeat the same procedure while simultaneously running twoqubit circuits for the entire array. After updating the unitary parameters to account for effects such as dispersive shifts and crosstalk, we find an average e_{2} of 0.62%.
For the full experiment, we generate quantum circuits using the twoqubit unitaries measured for each pair during simultaneous operation, rather than a standard gate for all pairs. The typical twoqubit gate is a full iSWAP with 1/6th of a full CZ. Using individually calibrated gates in no way limits the universality of the demonstration. One can compose, for example, controlledNOT (CNOT) gates from 1qubit gates and two of the unique 2qubit gates of any given pair. The implementation of highfidelity ‘textbook gates’ natively, such as CZ or \(\sqrt{{\rm{iSWAP}}}\), is work in progress.
Finally, we benchmark qubit readout using standard dispersive measurement^{41}. Measurement errors averaged over the 0 and 1 states are shown in Fig. 2a. We have also measured the error when operating all qubits simultaneously, by randomly preparing each qubit in the 0 or 1 state and then measuring all qubits for the probability of the correct result. We find that simultaneous readout incurs only a modest increase in perqubit measurement errors.
Having found the error rates of the individual gates and readout, we can model the fidelity of a quantum circuit as the product of the probabilities of errorfree operation of all gates and measurements. Our largest random quantum circuits have 53 qubits, 1,113 singlequbit gates, 430 twoqubit gates, and a measurement on each qubit, for which we predict a total fidelity of 0.2%. This fidelity should be resolvable with a few million measurements, since the uncertainty on \({ {\mathcal F} }_{{\rm{XEB}}}\) is \(1/\sqrt{{N}_{{\rm{s}}}}\), where N_{s} is the number of samples. Our model assumes that entangling larger and larger systems does not introduce additional error sources beyond the errors we measure at the single and twoqubit level. In the next section we will see how well this hypothesis holds up.
Fidelity estimation in the supremacy regime
The gate sequence for our pseudorandom quantum circuit generation is shown in Fig. 3. One cycle of the algorithm consists of applying singlequbit gates chosen randomly from \(\{\sqrt{X},\sqrt{Y},\sqrt{W}\}\) on all qubits, followed by twoqubit gates on pairs of qubits. The sequences of gates which form the ‘supremacy circuits’ are designed to minimize the circuit depth required to create a highly entangled state, which is needed for computational complexity and classical hardness.
Although we cannot compute \({ {\mathcal F} }_{{\rm{XEB}}}\) in the supremacy regime, we can estimate it using three variations to reduce the complexity of the circuits. In ‘patch circuits’, we remove a slice of twoqubit gates (a small fraction of the total number of twoqubit gates), splitting the circuit into two spatially isolated, noninteracting patches of qubits. We then compute the total fidelity as the product of the patch fidelities, each of which can be easily calculated. In ‘elided circuits’, we remove only a fraction of the initial twoqubit gates along the slice, allowing for entanglement between patches, which more closely mimics the full experiment while still maintaining simulation feasibility. Finally, we can also run full ‘verification circuits’, with the same gate counts as our supremacy circuits, but with a different pattern for the sequence of twoqubit gates, which is much easier to simulate classically (see also Supplementary Information). Comparison between these three variations allows us to track the system fidelity as we approach the supremacy regime.
We first check that the patch and elided versions of the verification circuits produce the same fidelity as the full verification circuits up to 53 qubits, as shown in Fig. 4a. For each data point, we typically collect N_{s} = 5 × 10^{6} total samples over ten circuit instances, where instances differ only in the choices of singlequbit gates in each cycle. We also show predicted \({ {\mathcal F} }_{{\rm{XEB}}}\) values, computed by multiplying the noerror probabilities of single and twoqubit gates and measurement (see also Supplementary Information). The predicted, patch and elided fidelities all show good agreement with the fidelities of the corresponding full circuits, despite the vast differences in computational complexity and entanglement. This gives us confidence that elided circuits can be used to accurately estimate the fidelity of morecomplex circuits.
The largest circuits for which the fidelity can still be directly verified have 53 qubits and a simplified gate arrangement. Performing random circuit sampling on these at 0.8% fidelity takes one million cores 130 seconds, corresponding to a millionfold speedup of the quantum processor relative to a single core.
We proceed now to benchmark our computationally most difficult circuits, which are simply a rearrangement of the twoqubit gates. In Fig. 4b, we show the measured \({ {\mathcal F} }_{{\rm{XEB}}}\) for 53qubit patch and elided versions of the full supremacy circuits with increasing depth. For the largest circuit with 53 qubits and 20 cycles, we collected N_{s} = 30 × 10^{6} samples over ten circuit instances, obtaining \({ {\mathcal F} }_{{\rm{XEB}}}=(2.24\pm 0.21)\times {10}^{3}\) for the elided circuits. With 5σ confidence, we assert that the average fidelity of running these circuits on the quantum processor is greater than at least 0.1%. We expect that the full data for Fig. 4b should have similar fidelities, but since the simulation times (red numbers) take too long to check, we have archived the data (see ‘Data availability’ section). The data is thus in the quantum supremacy regime.
The classical computational cost
We simulate the quantum circuits used in the experiment on classical computers for two purposes: (1) verifying our quantum processor and benchmarking methods by computing \({ {\mathcal F} }_{{\rm{XEB}}}\) where possible using simplifiable circuits (Fig. 4a), and (2) estimating \({ {\mathcal F} }_{{\rm{XEB}}}\) as well as the classical cost of sampling our hardest circuits (Fig. 4b). Up to 43 qubits, we use a Schrödinger algorithm, which simulates the evolution of the full quantum state; the Jülich supercomputer (with 100,000 cores, 250 terabytes) runs the largest cases. Above this size, there is not enough random access memory (RAM) to store the quantum state^{42}. For larger qubit numbers, we use a hybrid Schrödinger–Feynman algorithm^{43} running on Google data centres to compute the amplitudes of individual bitstrings. This algorithm breaks the circuit up into two patches of qubits and efficiently simulates each patch using a Schrödinger method, before connecting them using an approach reminiscent of the Feynman pathintegral. Although it is more memoryefficient, the Schrödinger–Feynman algorithm becomes exponentially more computationally expensive with increasing circuit depth owing to the exponential growth of paths with the number of gates connecting the patches.
To estimate the classical computational cost of the supremacy circuits (grey numbers in Fig. 4b), we ran portions of the quantum circuit simulation on both the Summit supercomputer as well as on Google clusters and extrapolated to the full cost. In this extrapolation, we account for the computation cost of sampling by scaling the verification cost with \({ {\mathcal F} }_{{\rm{XEB}}}\), for example^{43,44}, a 0.1% fidelity decreases the cost by about 1,000. On the Summit supercomputer, which is currently the most powerful in the world, we used a method inspired by Feynman pathintegrals that is most efficient at low depth^{44,45,46,47}. At m = 20 the tensors do not reasonably fit into node memory, so we can only measure runtimes up to m = 14, for which we estimate that sampling three million bitstrings with 1% fidelity would require a year.
On Google Cloud servers, we estimate that performing the same task for m = 20 with 0.1% fidelity using the Schrödinger–Feynman algorithm would cost 50 trillion corehours and consume one petawatt hour of energy. To put this in perspective, it took 600 seconds to sample the circuit on the quantum processor three million times, where sampling time is limited by control hardware communications; in fact, the net quantum processor time is only about 30 seconds. The bitstring samples from all circuits have been archived online (see ‘Data availability’ section) to encourage development and testing of more advanced verification algorithms.
One may wonder to what extent algorithmic innovation can enhance classical simulations. Our assumption, based on insights from complexity theory^{11,12,13}, is that the cost of this algorithmic task is exponential in circuit size. Indeed, simulation methods have improved steadily over the past few years^{42,43,44,45,46,47,48,49,50}. We expect that lower simulation costs than reported here will eventually be achieved, but we also expect that they will be consistently outpaced by hardware improvements on larger quantum processors.
Verifying the digital error model
A key assumption underlying the theory of quantum error correction is that quantum state errors may be considered digitized and localized^{38,51}. Under such a digital model, all errors in the evolving quantum state may be characterized by a set of localized Pauli errors (bitflips or phaseflips) interspersed into the circuit. Since continuous amplitudes are fundamental to quantum mechanics, it needs to be tested whether errors in a quantum system could be treated as discrete and probabilistic. Indeed, our experimental observations support the validity of this model for our processor. Our system fidelity is well predicted by a simple model in which the individually characterized fidelities of each gate are multiplied together (Fig. 4).
To be successfully described by a digitized error model, a system should be low in correlated errors. We achieve this in our experiment by choosing circuits that randomize and decorrelate errors, by optimizing control to minimize systematic errors and leakage, and by designing gates that operate much faster than correlated noise sources, such as 1/f flux noise^{37}. Demonstrating a predictive uncorrelated error model up to a Hilbert space of size 2^{53} shows that we can build a system where quantum resources, such as entanglement, are not prohibitively fragile.
The future
Quantum processors based on superconducting qubits can now perform computations in a Hilbert space of dimension 2^{53} ≈ 9 × 10^{15}, beyond the reach of the fastest classical supercomputers available today. To our knowledge, this experiment marks the first computation that can be performed only on a quantum processor. Quantum processors have thus reached the regime of quantum supremacy. We expect that their computational power will continue to grow at a doubleexponential rate: the classical cost of simulating a quantum circuit increases exponentially with computational volume, and hardware improvements will probably follow a quantumprocessor equivalent of Moore’s law^{52,53}, doubling this computational volume every few years. To sustain the doubleexponential growth rate and to eventually offer the computational volume needed to run well known quantum algorithms, such as the Shor or Grover algorithms^{25,54}, the engineering of quantum error correction will need to become a focus of attention.
The extended Church–Turing thesis formulated by Bernstein and Vazirani^{55} asserts that any ‘reasonable’ model of computation can be efficiently simulated by a Turing machine. Our experiment suggests that a model of computation may now be available that violates this assertion. We have performed random quantum circuit sampling in polynomial time using a physically realizable quantum processor (with sufficiently low error rates), yet no efficient method is known to exist for classical computing machinery. As a result of these developments, quantum computing is transitioning from a research topic to a technology that unlocks new computational capabilities. We are only one creative algorithm away from valuable nearterm applications.
Data availability
The datasets generated and analysed for this study are available at our public Dryad repository (https://doi.org/10.5061/dryad.k6t1rj8).
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Acknowledgements
We are grateful to E. Schmidt, S. Brin, S. Pichai, J. Dean, J. Yagnik and J. Giannandrea for their executive sponsorship of the Google AI Quantum team, and for their continued engagement and support. We thank P. Norvig, J. Yagnik, U. Hölzle and S. Pichai for advice on the manuscript. We acknowledge K. Kissel, J. Raso, D. L. YongeMallo, O. Martin and N. Sridhar for their help with simulations. We thank G. Bortoli and L. Laws for keeping our team organized. This research used resources from the Oak Ridge Leadership Computing Facility, which is a DOE Office of Science User Facility (supported by contract DEAC0500OR22725). A portion of this work was performed in the UCSB Nanofabrication Facility, an open access laboratory. R.B., S.M., and E.G.R. appreciate support from the NASA Ames Research Center and from the Air Force Research (AFRL) Information Directorate (grant F4HBKC4162G001). T.S.H. is supported by the DOE Early Career Research Program. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of AFRL or the US government.
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The Google AI Quantum team conceived the experiment. The applications and algorithms team provided the theoretical foundation and the specifics of the algorithm. The hardware team carried out the experiment and collected the data. The data analysis was done jointly with outside collaborators. All authors wrote and revised the manuscript and the Supplementary Information.
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Correspondence to John M. Martinis.
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This file contains Supplementary Information I–XI, which contains supplementary figures S1–S44 and Supplementary Tables I–X
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Arute, F., Arya, K., Babbush, R. et al. Quantum supremacy using a programmable superconducting processor. Nature 574, 505–510 (2019) doi:10.1038/s4158601916665
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