Code and manuscript for The energy-advantage frontier of random circuit sampling across the simulability transition (Wakaura & Tanimae, QIRI). We study the energy-advantage frontier of random circuit sampling (RCS) — the locus in the noise-rate/size plane where the classical and quantum energy-to-solution are equal — and its relation to the noise-induced classical-simulability transition. The package provides: exact and tensor-network (Liouville-MPS) operator-space entanglement, Landauer-floor energy accounting, the energy/computational advantage frontiers, an energy-charged cross-entropy witness, an IBM hardware interface, and the exs energy-to-solution benchmark CLI. The included PDF is the accompanying manuscript. Validated against exact dense evolution (<1e-2 in OSEE at n=9); full test suite passes. The Liouville-MPS sampler extends the operator-entanglement transition beyond exact density matrices.
Code and manuscript for The energy-advantage frontier of random circuit sampling across the simulability transition (Wakaura & Tanimae, QIRI). We study the energy-advantage frontier of random circuit sampling (RCS) — the locus in the noise-rate/size plane where the classical and quantum energy-to-solution are equal — and its relation to the noise-induced classical-simulability transition. The package provides: exact and tensor-network (Liouville-MPS) operator-space entanglement, Landauer-floor energy accounting, the energy/computational advantage frontiers, an energy-charged cross-entropy witness, an IBM hardware interface, and the exs energy-to-solution benchmark CLI. The included PDF is the accompanying manuscript. Validated against exact dense evolution (<1e-2 in OSEE at n=9); full test suite passes. The Liouville-MPS sampler extends the operator-entanglement transition beyond exact density matrices.
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 ∼1022 complex operations. At FXEB(36)=2.3×10−3 under favorable memory assumptions this implies 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 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.
We introduce the energy-advantage frontier of random circuit sampling (RCS)—the locus in the noise-rate/size plane where the classical and quantum energy-to-solution are equal—and locate it relative to the noise-induced classical-simulability transition. Our firm, model-free result is that the classical sampling-energy floor collapses sharply across the transition, exhibiting a finite-size scaling collapse over exact systems n≤12 (with a collapse exponent of order unity). Built on it, the energy-advantage frontier is distinct from the computational (wall-clock) frontier: in the exascale-classical, few-femtojoule regime, the energy frontier is the more noise-robust, so an energy advantage can persist where a time advantage does not—a projection, not a deployment claim, whose direction reverses outside that regime. We give a conditional Landauer lower bound for sampling, RE≡Ecl/Eq≳2(1.0±0.2)n/n under standard RCS hardness, package RE as a single figure of merit computed by a released open tool (exs), and map an energy-normalized cross-entropy witness within a cost model. A run on a 156-qubit IBM Heron processor tests the fidelity model (no energy is measured on hardware) and shows real noise exceeds the depolarizing idealization, so quoted frontier locations are upper bounds.
Random circuit sampling (RCS) is a leading candidate for demonstrating quantum advantage, supported by strong complexity-theoretic evidence of hardness in the ideal setting and by rapid experimental progress to date. In practice, however, noise is unavoidable, and a central problem is to identify the noise-strength boundary between classically simulable and classically hard regimes. In this work, we establish an architecture-general hardness bound for this boundary for the standard local depolarizing noise of strength γ. Assuming the standard average-case #P-hardness conjecture for ideal RCS, we show that, for any circuit architecture satisfying this conjecture, noisy RCS on the same architecture remains hard to simulate classically within any inverse-polynomial total variation distance whenever γ=O(logn/(nd)) for n-qubit circuits of depth d, unless the polynomial hierarchy collapses. Crucially, noisy-RCS hardness follows without any additional conjectural or architecture-specific assumption beyond those already entering the ideal-RCS hardness framework. Our proof combines a low-degree polynomial extrapolation with a monotonicity reduction showing that efficient classical simulation at one depolarizing noise strength implies efficient simulation at every larger strength. Together, these ingredients transfer the standard ideal-RCS hardness conjecture to sampling hardness at a prespecified noise strength. Finally, combining the convergence-to-uniformity result of Dalzell et al. [Commun. Math. Phys. 405, 78 (2024)] with our monotonicity reduction yields efficient classical simulation for γ=ω(logn/(nd)) on layered, regularly connected architectures. Thus, wherever the two architectural settings overlap, this identifies γ=Θ(logn/(nd)) as the asymptotic complexity-transition scale.
Random circuit sampling, the task to sample bit strings from a random unitary operator, has been performed to demonstrate quantum advantage on the Sycamore quantum processor with 53 qubits and on the Zuchongzhi quantum processor with 56 and 61 qubits. Recently, it has been claimed that classical computers using tensor network simulation could catch on current noisy quantum processors for random circuit sampling. While the linear cross entropy benchmark fidelity is used to certify all these claims, it may not capture in detail statistical properties of outputs. Here, we compare the bit strings sampled from classical computers using tensor network simulation by Pan et al. [Phys. Rev. Lett. 129, 090502 (2022)] and by Kalachev et al. [arXiv:2112.15083 (2021)] and from the Sycamore and Zuchongzhi quantum processors. It is shown that all Kalachev et al.'s samples pass the NIST random number tests. The heat maps of bit strings show that Pan et al.'s and Kalachev et al.'s samples are quite different from the Sycamore or Zuzhongzhi samples. The analysis with the Marchenko-Pastur distribution and the Wasssertein distances demonstrates that Kalachev et al.'s samples are statistically close to the Sycamore samples than Pan et al.'s while the three datasets have similar values of the linear cross entropy fidelity. Our finding implies that further study is needed to certify or beat the claims of quantum advantage for random circuit sampling.
We classically simulate the IBM doped Clifford random circuit sampling experiment, comprising 70 qubits, 70 entangling layers, and 468 inserted T gates, as reported in the first version of arXiv:2607.25941. A deterministic temporal-boundary tensor network contraction approach is specifically designed to tackle such open-boundary one-dimensional brickwork circuits with operator-Schmidt-rank-2 entangling gates. For an n-qubit circuit of depth d, the resulting unsliced path evaluates an exact amplitude with contraction width ⌈d/2⌉; Ratcatcher calculations certify that no smaller width is possible for the tested instances. Because one-qubit gates are absorbed without changing the network topology, the width and dense scheduled contraction cost are independent of their values and of the number and placement of T gates. For this initial instance, its largest intermediate tensor contains 235 complex64 entries (256 times smaller than IBM's estimation), corresponding to a tensor payload of 256 GiB, and is distributed across eight GPUs within a node. Using 32 nodes, with eight NVIDIA H100 GPUs per node, we completed all 2051 amplitude batches corresponding to IBM's published output bitstrings in 37.3 minutes. The resulting probabilities yield a log-XEB estimate of 0.35034 with a 95% interval of [0.29763,0.40305]. Under the Porter--Thomas and scrambled-noise assumptions, this is numerically compatible with IBM's fidelity lower bound; separately, fidelity-weighted resource accounting projects a 583-contraction workload with a 10.6-minute makespan on the same 32 nodes. More broadly, the approach provides a practical diagnostic for experimental outputs and a quantitative tool for designing future doped Clifford sampling experiments.
Random circuit sampling, the task of sampling bit strings from a random unitary operator, has been implemented to demonstrate quantum advantage on the Sycamore quantum processor with 53 qubits and on the Zuchongzhi quantum processor with 56 and 61 qubits. Recently, it was claimed that classical computers using tensor network simulation could catch on to current noisy quantum processors for random circuit sampling. While the linear cross-entropy benchmark fidelity was used to certify all these claims, it may not capture statistical properties of outputs in detail. Here, we compare the bit strings sampled from classical computers using tensor network simulation by Pan et al. [F. Pan, K. Chen, and P. Zhang, Phys. Rev. Lett. 129, 090502 (2022)] and by Kalachev et al. [G. Kalachev, P. Panteleev, P. Zhou, and M.-H. Yung, arXiv:2112.15083] with the bit strings from the Sycamore quantum processor. It is shown that all of Kalachev et al.'s samples passed the NIST random number tests. The heat maps of bit strings show that Pan et al.'s and Kalachev et al.'s samples are quite different from the Sycamore or Zuchongzhi samples. The analysis with the Marchenko-Pastur distribution and the Wasssertein distances demonstrates that Kalachev et al.'s samples are statistically closer to the Sycamore samples than Pan et al.'s while the three datasets have similar values for the linear cross-entropy fidelity. Our finding implies that further study is needed to certify or beat the claims of quantum advantage using random circuit sampling.