A polynomial-time classical algorithm for noisy quantum circuits

TL;DR

The paper investigates how noise constrains quantum advantage by presenting a classical algorithm that can compute expectation values for noisy quantum circuits on most inputs in polynomial time (uniform noise) or quasi-polynomial time (gate-based noise). The core method evolves observables in the Pauli basis and truncates high-weight Pauli operators, exploiting noise damping to bound complexity. The authors derive rigorous error bounds, extend results to sampling under anti-concentration and certain non-unital noise models, and show that many error-mitigation strategies would be classically simulable on most inputs, thereby providing a practical test for quantum advantage. The findings stress that without quantum error correction, robust quantum advantage may require noise rates to scale down with system size, and they outline clear directions for future work in continuous-time dynamics and broader noise models.

Abstract

We provide a polynomial-time classical algorithm for noisy quantum circuits. The algorithm computes the expectation value of any observable for any circuit, with a small average error over input states drawn from an ensemble (e.g. the computational basis). Our approach is based upon the intuition that noise exponentially damps non-local correlations relative to local correlations. This enables one to classically simulate a noisy quantum circuit by only keeping track of the dynamics of local quantum information. Our algorithm also enables sampling from the output distribution of a circuit in quasi-polynomial time, so long as the distribution anti-concentrates. A number of practical implications are discussed, including a fundamental limit on the efficacy of noise mitigation strategies: for constant noise rates, any quantum circuit for which error mitigation is efficient on most input states, is also classically simulable on most input states.

Figures (4)

• Figure 1: (a) Schematic of a noisy quantum circuit. The input state $\rho$ is acted on by a circuit of arbitrary two-qubit gates (pink) and local noise channels (dots), and concludes with measurement of an observable $O$. In the gate-based noise model, the noise only acts on a qubit when a gate is performed (black dots). In the uniform noise model, even when the qubit is idle, noise can occur (dashed dots). (b) For circuits with uniform noise, our classical algorithm decomposes the expectation value of $O$ as a sum of Pauli paths. We depict each path by a space-time grid, where each square denotes whether the path is the identity (white) or $X$, $Y$, or $Z$ (red, yellow, or purple) at that qubit and circuit layer. Our algorithm computes the sum of all low-weight paths (left), and truncates high-weight paths (right) since they are strongly damped by noise (fading). (c) For circuits with gate-based noise, our algorithm instead simulates the Heisenberg time-evolution of $O$ within the subspace of low-weight Pauli operators. That is, at each layer, we truncate all Pauli operators with weight above a threshold $\ell$ (dashed red line). The plot depicts the weight of various components of the time-evolved operator (blue lines), as they are acted on by noise (fading) and, potentially, truncated by our algorithm (red scissors).
• Figure 2: (a) The Pauli tree framework used to prove Theorem \ref{['thm 1']}, which groups Pauli paths according to their weight $w_t$ at each circuit layer $t$. Our algorithm truncates paths with a summed weight above $\ell$. We group these truncations into ${\ell \choose d}$ individual truncations (red scissors), which enables a tight bound on the algorithm's error. In the example shown, the branchings at each node correspond to weights $w = 1,\ldots,5$ and we set $\ell = 6, d = 2$. (b) To prove Theorem \ref{['thm 2']}, we analyze the flow of Pauli operators from one weight to another under circuit gates. Here, each bubble depicts the set of Pauli operators of a given weight, and arrows indicate flow from one weight to another. We show that this flow is lossy in the presence of gate-based noise, in the sense that a decrease $-J$ in support at weight $w$ can increase the support at weight $w+1$ by at most $e^{-2\gamma} J$. This leads to our bound on the cumulative operator weight distribution in Lemma \ref{['lemma: operator norm noise']}.
• Figure 3: Schematic of our results' implications for quantum error mitigation. For concreteness, we depict a specific error mitigation strategy, zero noise extrapolation, in which one measures an expectation value $\langle O \rangle$ for several different noise rates (orange lines), and performs an extrapolation (green arrow and dashed line) to estimate the ideal expectation value (black line). (a) If error mitigation succeeds in recovering the ideal expectation value, then the expectation value must be dominated by low-weight Pauli operators (green). Thus, our classical algorithm can also compute the ideal expectation value. (b) On the other hand, if the ideal expectation value is hard to compute classically, then it must contain contributions from high-weight Pauli operators (red). Error mitigation cannot capture these contributions, since they are exponentially suppressed by noise. Thus, the extrapolated expectation value necessarily differs from the ideal value (red arrows).
• Figure : Classical algorithm for uniform noise

Theorems & Definitions (19)

• Theorem 1
• Theorem 2
• Lemma 1
• Lemma 2
• Corollary 1: Lower bound on error mitigation
• Corollary 2
• Definition 1
• Lemma 3
• Lemma 4
• Lemma 5