A polynomial-time classical algorithm for noisy random circuit sampling

TL;DR

The paper addresses the problem of sampling from noisy random circuit outputs and questions whether such sampling can provide a scalable quantum advantage under the extended Church-Turing thesis.It develops a Pauli-path Fourier framework that reformulates the problem as a low-weight Pauli-path sum, leveraging sparsity and a depolarizing-noise model to enable a polynomial-time classical sampler under anti-concentration.A sampling-to-computing reduction shows that the classical sampler can produce samples indistinguishable from the noisy quantum-circuit outputs, provided the truncation parameter scales as $\ell=O(\log(1/\varepsilon))$ and other parameters are favorable.The results hinge on gate-set orthogonality and anti-concentration, and they imply that, in the presence of a constant noise rate per gate, random circuit sampling does not achieve scalable quantum supremacy; however, the approach is not practical currently and does not address finite-size experiments or sublogarithmic-depth regimes.

Abstract

We give a polynomial time classical algorithm for sampling from the output distribution of a noisy random quantum circuit in the regime of anti-concentration to within inverse polynomial total variation distance. This gives strong evidence that, in the presence of a constant rate of noise per gate, random circuit sampling (RCS) cannot be the basis of a scalable experimental violation of the extended Church-Turing thesis. Our algorithm is not practical in its current form, and does not address finite-size RCS based quantum supremacy experiments.

Figures (3)

• Figure 1: Random circuit sampling, each white box is an independent Haar random 2-qubit gate. (a) Ideal RCS generates an output distribution $p(C)$ that satisfies anti-concentration when $d=\Omega(\log n)$. (b) Noisy RCS, where an arbitrarily small constant amount of depolarizing noise is applied to each qubit at each step, which generates a noisy output distribution $\tilde{p}(C)$. Here the 1D architecture is for illustration; the result applies to general architectures (Definition \ref{['def:architecture']}).
• Figure 2: A gate set related to Google and USTC's experiments, for which our main result holds. LHS: the gate set consists of a fixed $\mathrm{fSim}$ gate surrounded by random gates from $\{\sqrt{X},\sqrt{Y},\sqrt{W}\}$ as well as random $Z$ rotations. RHS: this is equivalent to LHS due to a special property of the $\mathrm{fSim}$ gates.
• Figure : Simulating noisy random circuits by low-degree Fourier approximation

Theorems & Definitions (43)

• Theorem 1: Main result
• Corollary 1
• Definition 1: Pauli path integral
• Definition 2: Pauli path integral for noisy quantum circuits
• Lemma 1: Properties of Haar random 2-qubit gates Harrow2009
• Lemma 2: Gate-set orthogonality
• proof
• Definition 3: Gate set and architecture of random circuits
• Lemma 3: Orthogonality of Fourier coefficients
• proof