Classical simulation of peaked shallow quantum circuits

TL;DR

This paper investigates the classical simulability of peaked shallow quantum circuits, introducing peakedness as $\max_x |\langle x|U|0^n\rangle|^2 \ge n^{-a}$ and proving a sampling algorithm with runtime $n^{O(\log n)}$ that approximates the output distribution within inverse-polynomial error. For geometrically local circuits in fixed dimensions, the authors obtain faster runtimes: polynomial in $n$ for $D=2$ and quasi-polynomial $n^{O(\log\log n)}$ for $D>2$, by a side-dissection technique and a heavy-slice/pseudomixture framework that reduces global sampling to marginals of small blocks. They further show how to compute a high-fidelity description of the output state $U|0^n\rangle$ when the circuit is peaked, enabling estimation of quantities such as the magnitude of Pauli observables and certain distance measures, and provide a software implementation demonstrating practical performance up to tens of qubits. The work advances understanding of when shallow quantum circuits resist classical simulation and offers concrete tools for probability estimation, mean values, and state similarity tasks in the peaked regime.

Abstract

An $n$-qubit quantum circuit is said to be peaked if it has an output probability that is at least inverse-polynomially large as a function of $n$. We describe a classical algorithm with quasipolynomial runtime $n^{O(\log{n})}$ that approximately samples from the output distribution of a peaked constant-depth circuit. We give even faster algorithms for circuits composed of nearest-neighbor gates on a $D$-dimensional grid of qubits, with polynomial runtime $n^{O(1)}$ if $D=2$ and almost-polynomial runtime $n^{O(\log{\log{n}})}$ for $D>2$. Our sampling algorithms can be used to estimate output probabilities of shallow circuits to within a given inverse-polynomial additive error, improving previously known methods. As a simple application, we obtain a quasipolynomial algorithm to estimate the magnitude of the expected value of any Pauli observable in the output state of a shallow circuit (which may or may not be peaked). This is a dramatic improvement over the prior state-of-the-art algorithm which had an exponential scaling in $\sqrt{n}$.

Figures (4)

• Figure 1: Runtime of classical algorithms for approximating output probabilities of $n$-qubit shallow quantum circuits. In contrast with prior work, the algorithms presented here solve the more general problem of sampling from the output distribution of peaked shallow circuits to within a given error inverse polynomial in $n$.
• Figure 2: Circuit $W$ from Claim \ref{['corol:trace']}. Here $n=3$.
• Figure 3: Numerical simulation of 2D random Clifford circuits interspersed with $R(\theta)$ rotations; the number of qubits $n$ ranges from $30$ to $56$, and all simulated $U(\theta)$ circuits have depth $d=3$ (so the $V(\theta)$'s all have depth $5$); $\lambda_1(G)$ is the largest eigenvalue of the $G$ matrix defined in Lemma \ref{['lemma:truncation']}, and $W$ is the radius of Hamming ball centered at $z$; the algorithm is self-certifying since the simulation error is inversely proportional to $\lambda_1(G)$.
• Figure 4: A partition of the grid of qubits into equal-width rectangular regions, depicted here in the two-dimensional case. Lemma \ref{['lem:lotsofslices']} describes how we can find lightcone-separated heavy slices within each region $H^{j}$ for $1\leq j\leq T$.

Theorems & Definitions (28)

• Theorem 1: Output distribution of peaked shallow circuits
• Theorem 1a: Output state of peaked shallow circuits
• Theorem 2: Output distribution with geometric locality
• Claim 1
• proof
• Lemma 1: Peaked shallow purification
• proof
• Lemma 2
• proof
• Lemma 3