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6 On Google Cloud servers, we estimate that perform- ing the same task for m= 20 with 0:1% delity using the SFA algorithm would cost 50 trillion core-hours and consume 1 petawatt hour of energy. To put this in per- spective, it took 600 seconds to sample the circuit on the quantum processor 3 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 this largest circuit are archived online. One may wonder to what extent algorithmic innova- tion can enhance classical simulations. Our assumption, based on insights from complexity theory, is that the cost of this algorithmic task is exponential in nas well as m. Indeed, simulation methods have improved steadily over the past few years[42{50]. We expect that lower simula- tion costs than reported here will eventually be achieved, but we also expect 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 con- sidered digitized and localized [38, 51]. Under such a dig- ital model, all errors in the evolving quantum state may be characterized by a set of localized Pauli errors (bit- and/or phase- ips) interspersed into the circuit. Since continuous amplitudes are fundamental to quantum me- chanics, it needs to be tested whether errors in a quantum system could be treated as discrete and probabilistic. In- deed, our experimental observations support the validity of this model for our processor. Our system delity is well predicted by a simple model in which the individ- ually characterized delities 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 ran- domize and decorrelate errors, by optimizing control to minimize systematic errors and leakage, and by design- ing gates that operate much faster than correlated noise sources, such as 1=f ux noise [37]. Demonstrating a pre- dictive uncorrelated error model up to a Hilbert space of size 253 shows that we can build a system where quantum resources, such as entanglement, are not prohibitively fragile. WHAT DOES THE FUTURE HOLD? Quantum processors based on superconducting qubits can now perform computations in a Hilbert space of di- mension 253 ˇ9 1015, beyond the reach of the fastest classical supercomputers available today. To our knowl- edge, this experiment marks the rst computation that can only be performed on a quantum processor. Quan- tum processors have thus reached the regime of quantum supremacy. We expect their computational power will continue to grow at a double exponential rate: the clas- sical cost of simulating a quantum circuit increases expo- nentially with computational volume, and hardware im- provements will likely follow a quantum-processor equiv- alent of Moore’s law [52, 53], doubling this computational volume every few years. To sustain the double exponen- tial growth rate and to eventually o er the computational volume needed to run well-known quantum algorithms, such as the Shor or Grover algorithms [19, 54], the engi- neering of quantum error correction will have 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 eciently 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 with a physically realized quantum processor (with suciently low error rates), yet no ecient method is known to exist for classical comput- ing machinery. As a result of these developments, quan- tum computing is transitioning from a research topic to a technology that unlocks new computational capabilities. We are only one creative algorithm away from valuable near-term applications. Acknowledgments We are grateful to Eric Schmidt, Sergey Brin, Je Dean, and Jay Yagnik for their executive sponsorship of the Google AI Quantum team, and for their continued engagement and support. We thank Peter Norvig for reviewing a draft of the manuscript, and Sergey Knysh for useful discussions. We thank Kevin Kissel, Joey Raso, Davinci Yonge-Mallo, Orion Martin, and Niranjan Sridhar for their help with simulations. We thank Gina Bortoli and Lily Laws for keeping our team organized. This research used resources from the Oak Ridge Leadership Computing Facility, which is a DOE Oce of Science User Facility supported un- der Contract DE-AC05-00OR22725. A portion of this work was performed in the UCSB Nanofabrication Facility, an open access laboratory. Author contributions The Google AI Quantum team conceived of the experiment. The applications and algorithms team provided the theoretical foundation and the speci cs 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 supplement. Competing Interests The authors declare that they have no competing nancial interests. Correspondence and requests for materials should be addressed to John M. Martinis (jmartinis@google.com). |
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