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arXiv:2304.11119 · Nature

Phase transitions in random circuit sampling

Alexis Morvan, Benjamin Villalonga, Xiao Mi, Salvatore Mandrà, Andreas Bengtsson, Paul V. Klimov, Z. Chen, Sabrina Hong, Catherine Erickson, Ilya Drozdov, J. Chau, G. Laun, Ramis Movassagh, Abraham T. Asfaw, Luís T. A. N. Brandão, René Peralta, Dmitry A. Abanin, Rajeev Acharya, R. M. Allen, Trond I. Andersen, K. Anderson, M. Ansmann, Frank Arute, Kunal Arya, Juan Atalaya, Joseph C. Bardin, Alexander Bilmes, Gina Bortoli, Alexandre Bourassa, Jenna Bovaird, L. Brill, Michael Broughton, Bob B. Buckley, David A. Buell, Tim Burger, Brian Burkett, Nicholas Bushnell, Juan Campero, Hung-Shen Chang, B. Chiaro, Desmond Chik, Chihong Chou, J. G. Cogan, Roberto Collins, P. Conner, William Courtney, A. L. Crook, Ben Curtin, Dripto M. Debroy, Alexander Del Toro Barba, Sean Demura, Agustín Di Paolo, A. Dunsworth, Lara Faoro, Edward Farhi, Reza Fatemi, V. S. Ferreira, Leslie Flores Burgos, Ebrahim Forati, Austin G. Fowler, Brooks Foxen, G. Garcia, Élie Genois, William Giang, Craig Gidney, Dar Gilboa, Marissa Giustina, Raja Gosula, Alejandro Grajales Dau, Jonathan A. Gross, Steve Habegger, M. C. Hamilton, M. Hansen, Matthew P. Harrigan, Sean David Harrington, Paula Heu, M. R. Hoffmann, Trent Huang, Ashley Huff, William J. Huggins, L. B. Ioffe, Sergei V. Isakov, Justin Iveland, Evan R. Jeffrey, Z. Jiang, Cody Jones, Pavol Juhás, Dvir Kafri, Tanuj Khattar, Mostafa Khezri, Mária Kieferová, S. Kim, Alexei Kitaev, A. R. Klots, A. N. Korotkov, Fedor Kostritsa, John Mark Kreikebaum, David Landhuis, Pavel Laptev, K.-M. Lau

Published Oct 9, 2024• 138 citations
arXivHTMLPDF

Abstract

Undesired coupling to the surrounding environment destroys long-range correlations in quantum processors and hinders coherent evolution in the nominally available computational space. This noise is an outstanding challenge when leveraging the computation power of near-term quantum processors1. It has been shown that benchmarking random circuit sampling with cross-entropy benchmarking can provide an estimate of the effective size of the Hilbert space coherently available2–8. Nevertheless, quantum algorithms’ outputs can be trivialized by noise, making them susceptible to classical computation spoofing. Here, by implementing an algorithm for random circuit sampling, we demonstrate experimentally that two phase transitions are observable with cross-entropy benchmarking, which we explain theoretically with a statistical model. The first is a dynamical transition as a function of the number of cycles and is the continuation of the anti-concentration point in the noiseless case. The second is a quantum phase transition controlled by the error per cycle; to identify it analytically and experimentally, we create a weak-link model, which allows us to vary the strength of the noise versus coherent evolution. Furthermore, by presenting a random circuit sampling experiment in the weak-noise phase with 67 qubits at 32 cycles, we demonstrate that the computational cost of our experiment is beyond the capabilities of existing classical supercomputers. Our experimental and theoretical work establishes the existence of transitions to a stable, computationally complex phase that is reachable with current quantum processors. By implementing random circuit sampling, experimental and theoretical results establish the existence of transitions to a stable, computationally complex phase that is reachable with current quantum processors.

DOI
10.1038/s41586-024-07998-6

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Undesired coupling to the surrounding environment destroys long-range correlations in quantum processors and hinders coherent evolution in the nominally available computational space. This noise is an outstanding challenge when leveraging the computation power of near-term quantum processors1. It has been shown that benchmarking random circuit sampling with cross-entropy benchmarking can provide an estimate of the effective size of the Hilbert space coherently available2–8. Nevertheless, quantum algorithms’ outputs can be trivialized by noise, making them susceptible to classical computation spoofing. Here, by implementing an algorithm for random circuit sampling, we demonstrate experimentally that two phase transitions are observable with cross-entropy benchmarking, which we explain theoretically with a statistical model. The first is a dynamical transition as a function of the number of cycles and is the continuation of the anti-concentration point in the noiseless case. The second is a quantum phase transition controlled by the error per cycle; to identify it analytically and experimentally, we create a weak-link model, which allows us to vary the strength of the noise versus coherent evolution. Furthermore, by presenting a random circuit sampling experiment in the weak-noise phase with 67 qubits at 32 cycles, we demonstrate that the computational cost of our experiment is beyond the capabilities of existing classical supercomputers. Our experimental and theoretical work establishes the existence of transitions to a stable, computationally complex phase that is reachable with current quantum processors. By implementing random circuit sampling, experimental and theoretical results establish the existence of transitions to a stable, computationally complex phase that is reachable with current quantum processors.
Demonstrations of quantum computational advantage and benchmarks of quantum processors via quantum random circuit sampling are based on evaluating the linear cross-entropy benchmark (XEB). A key question in the theory of XEB is whether it approximates the fidelity of the quantum state preparation. Previous works have shown that the XEB generically approximates the fidelity in a regime where the noise rate per qudit ε\varepsilonε satisfies εN≪1\varepsilon N \ll 1εN≪1 for a system of NNN qudits and that this approximation breaks down at large noise rates. Here, we show that the breakdown of XEB as a fidelity proxy occurs as a sharp phase transition at a critical value of εN\varepsilon NεN that depends on the circuit architecture and properties of the two-qubit gates, including in particular their entangling power. We study the phase transition using a mapping of average two-copy quantities to statistical mechanics models in random quantum circuit architectures with full or one-dimensional connectivity. We explain the phase transition behavior in terms of spectral properties of the transfer matrix of the statistical mechanics model and identify two-qubit gate sets that exhibit the largest noise robustness.
Random circuit sampling (RCS) remains one of the most competitive frameworks for demonstrating quantum advantage in near-term noisy intermediate-scale quantum (NISQ) hardware. Unfortunately, absent error-correction, existing benchmarks to characterize these experiments, like linear cross-entropy, have been classically spoofed due to noise. Because of this, there are interesting regimes, like shallow-depth random quantum circuits, where sampling is plausibly classically intractable, but no existing benchmark can distinguish between a noisy quantum computer and an adversarial classical spoofer. In this paper, we demonstrate that the nonlinear cross-entropy provides a sample-efficient benchmark for shallow-depth all-to-all random quantum circuits whose score cleanly separates noisy quantum computers from state-of-the-art classical spoofers, even in the presence of depolarizing noise. Further, we develop a binary classifier based on the notion of heavy output generation that features logarithmic sample complexity at short depth. Our evidence comes from exact analytic expressions for all-to-all Brownian circuit ensembles derived using replica tricks, and numerical simulations that corroborate these results for discrete Haar-random unitary circuits.
The increasing scale of near-term quantum hardware motivates the need for efficient noise characterization methods, since qubit and gate level techniques cannot capture crosstalk and correlated noise in many qubit systems. While scalable approaches, such as cycle benchmarking, are known for special classes of quantum circuits, the characterization of noise in general circuits with non-Clifford gates has been an unreachable task. We develop an algorithm that can sample-efficiently estimate the total amount of noise induced by a layer of arbitrary non-Clifford gates, including all crosstalks, and experimentally demonstrate the method on IBM Quantum hardware. Our algorithm is inspired by Google's quantum supremacy experiment and is based on random circuit sampling. In their paper, Google observed that their experimental linear cross entropy was consistent with a simple uncorrelated noise model, and claimed this coincidence indicated that the noise in their device was uncorrelated -- a key step in hardware development towards fault tolerance. As an application, we show that our result provides formal evidence to support such a conclusion.
Random unitaries scramble quantum information into highly nonlocal degrees of freedom, thereby providing protection from local noise. However, as demonstrated by Turkeshi and Sierant (Phys. Rev. Lett. 132, 140401, 2024 and arXiv:2603.00235), this protection may fail once local noise grows sufficiently strong. These works describe a phase transition in a noisy random encoding-decoding circuit model between an error-protecting phase, in which encoded information is recoverable, and an error-vulnerable phase, in which encoded information is lost, with the transition occurring at a critical coherent noise angle αc≈1.14α_c \approx 1.14αc​≈1.14. Here, we present the first experimental observation of this transition, conducted on IBM's ibm_pittsburgh Heron r3 device for system sizes of up to N=26N = 26N=26 qubits. We detect this transition via three order parameters: fidelity, Rényi-2 entanglement entropy, and stabilizer Rényi-2 entropy. For the latter two, we employ randomized measurement protocols to construct estimators, and for all three, we rely on zero-noise extrapolation and M3 SPAM error mitigation. We then jointly model the finite-size scaling collapse across all three order parameters with a Gaussian process regression, yielding a critical angle αc≈1.01α_c \approx 1.01αc​≈1.01, well below the theoretical value. Inputting noiseless simulated data into the same model yields αc≈1.05α_c \approx 1.05αc​≈1.05, implying that the discrepancy between experiment and theory is due to our encoding circuits, which are not deep enough to be truly Haar-random. Finally, we observe that the same model cannot adequately identify critical behavior in unmitigated, raw experimental data. This indicates that error mitigation techniques are crucial for detecting the transition on hardware.
We report forward random-circuit sampling (RCS) on the 120-qubit Nighthawk r2 superconducting processor (ibm_phoenix) with square-lattice connectivity, using 61 qubits, native CZ gates, and the standard cloud execution stack with no benchmark-specific calibration. Two independent fidelity estimators---mirror benchmarking and three- and four-patch cross-entropy benchmarking (XEB)---agree with each other at every measured depth, the mirror from 4 to 40 cycles and the patched estimators from 20 to 40 cycles, across more than two orders of magnitude of fidelity decay, and exceed the first-generation Nighthawk r1 device by more than an order of magnitude at fixed depth. The 36-cycle circuits sit at the depth where tensor-network contraction cost saturates at system size: a contraction-cost estimator validated against the published Sycamore and Zuchongzhi networks places the single-amplitude cost at ∼\sim∼102210^{22}1022 complex operations. At FXEB(36)=2.3×10−3F_{\mathrm{XEB}}(36)=2.3\times10^{-3}FXEB​(36)=2.3×10−3 under favorable memory assumptions this implies 1.2×10271.2\times10^{27}1.2×1027 machine operations within the bounded-fidelity rejection-sampling model --- more than a century of runtime on the Frontier supercomputer --- to collect a 10610^{6}106-sample ensemble, which takes only 19\,s on Nighthawk r2. To our knowledge, this is the first demonstration of quantum advantage for a vanilla random-circuit sampling on a commercially and broadly accessible quantum processor that most non-expert quantum computer users can easily replicate.
Cross-entropy benchmarking is a central technique adopted to certify a quantum chip in recent investigations. To better understand its mathematical foundation and develop new benchmarking schemes, the concept of ergodicity was introduced to random circuit sampling and it was found that the Haar random quantum circuit could satisfy an ergodicity condition—the average of certain types of postprocessing function over the output bit strings is close to the average over the unitary ensemble. For noiseless random circuits, it was proven that the ergodicity holds for polynomials of degree t with positive coefficients when the random circuits form a unitary 2 t -design. For strong enough noise, the ergodicity condition is violated, which suggests that ergodicity is a property that can be exploited to certify a quantum chip. The deviation of ergodicity was formulated as a measure for quantum chip benchmarking, and it was demonstrated that it can be used to estimate the circuit fidelity for global depolarizing noise and weakly correlated noise. For a quadratic postprocessing function, our framework recovered Google's result on estimating the circuit fidelity via linear cross-entropy benchmarking (XEB), and we gave a sufficient condition on the noise model characterizing when such estimation is valid. The results establish an interesting connection between ergodicity and noise in random circuits and provide new insights into designing quantum benchmarking schemes.