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arXiv:2206.05642

Average-case hardness of estimating probabilities of random quantum circuits with a linear scaling in the error exponent

Hari Krovi

Published Jun 12, 2022• 2 citations
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Abstract

We consider the hardness of computing additive approximations to output probabilities of random quantum circuits. We consider three random circuit families, namely, Haar random, $p=1$ QAOA, and random IQP circuits. Our results are as follows. For Haar random circuits with $m$ gates, we improve on prior results by showing $\mathsf{coC_=P}$ hardness of average-case additive approximations to an imprecision of $2^{-O(m)}$. Efficient classical simulation of such problems would imply the collapse of the polynomial hierarchy. For constant depth circuits i.e., when $m=O(n)$, this linear scaling in the exponent is within a constant of the scaling required to show hardness of sampling. Prior to our work, such a result was shown only for Boson Sampling in Bouland et al (2021). We also use recent results in polynomial interpolation to show $\mathsf{coC_=P}$ hardness under $\mathsf{BPP}$ reductions rather than $\mathsf{BPP}^{\mathsf{NP}}$ reductions. This improves the results of prior work for Haar random circuits both in terms of the error scaling and the power of reductions. Next, we consider random $p=1$ QAOA and IQP circuits and show that in the average-case, it is $\mathsf{coC_=P}$ hard to approximate the output probability to within an additive error of $2^{-O(n)}$. For $p=1$ QAOA circuits, this work constitutes the first average-case hardness result for the problem of approximating output probabilities for random QAOA circuits, which include Sherrington-Kirkpatrick and Erdös-Renyi graphs. For IQP circuits, a consequence of our results is that approximating the Ising partition function with imaginary couplings to an additive error of $2^{-O(n)}$ is hard even in the average-case, which extends prior work on worst-case hardness of multiplicative approximation to Ising partition functions.

DOI
10.48550/arxiv.2206.05642

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BosonSampling and Random Circuit Sampling are important both as a theoretical tool for separating quantum and classical computation, and as an experimental means of demonstrating quantum speedups. Prior works have shown that average-case hardness of sampling follows from certain unproven conjectures about the hardness of computing output probabilities, such as the Permanent-of-Gaussians Conjecture (PGC), which states that e−nlog⁡n−n−O(log⁡n)e^{-n \log n-n-O(\log n)}e−nlogn−n−O(logn) additive-error estimates to the output probability of most random BosonSampling experiments are #P-hard. Prior works have only shown weaker average-case hardness results that do not imply sampling hardness. Proving these conjectures has become a central question in quantum complexity. In this work, we show that e−nlog⁡n−n−O(nδ)e^{-n \log n-n-O\left(n^{\delta}\right)}e−nlogn−n−O(nδ) additive-error estimates to output probabilities of most random BosonSampling experiments are #P-hard for any δ>0\delta\gt 0δ>0, exponentially improving on prior work. In the process, we circumvent all known barrier results for proving PGC. The remaining hurdle to prove PGC is now “merely” to show that the O(nδ)O\left(n^{\delta}\right)O(nδ) in the exponent can be improved to O(log⁡n)O(\log n)O(logn). We also obtain an analogous result for Random Circuit Sampling. We then show, for the first time, a hardness of average-case classical sampling result for BosonSampling, under an anticoncentration conjecture. Specifically, we prove the impossibility of multiplicative-error sampling from random BosonSampling experiments with probability 1−2−O~(N1/3)1-2^{-\tilde{O}\left(N^{1 / 3}\right)}1−2−O~(N1/3) for input size N, unless the Polynomial Hierarchy collapses. This exponentially improves upon the state-of-the-art. To do this, we introduce new proof techniques which tolerate exponential loss in the worst-to-average-case reduction. This opens the possibility to show the hardness of average-case sampling without ever proving PGC.
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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 γγγ. 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(log⁡n/(nd))γ=O(\log n/(nd))γ=O(logn/(nd)) for nnn-qubit circuits of depth ddd, 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(log⁡n/(nd))γ=ω(\log n/(nd))γ=ω(logn/(nd)) on layered, regularly connected architectures. Thus, wherever the two architectural settings overlap, this identifies γ=Θ(log⁡n/(nd))γ=Θ(\log n/(nd))γ=Θ(logn/(nd)) as the asymptotic complexity-transition scale.
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