Simulating quantum circuits with arbitrary local noise using Pauli Propagation

TL;DR

This work proves that typical quantum circuits subjected to arbitrary local incoherent noise, including non-unital and dephasing, can be efficiently simulated classically for estimating observable expectation values using Pauli-path propagation with path-weight truncation. It develops a general framework tying normal-form noise parameters and contraction coefficients to average-case tractability across broad circuit geometries, and shows that any local noise induces an effective logarithmic-depth reduction for the estimation task. The authors establish polynomial-time guarantees for non-unital noise with one-design layer ensembles and for unital noise with approximately scrambling layers, along with depth truncation results, and they validate the approach with substantial numerical experiments on lattice models and real-time dynamics. The work broadens the regime where classical simulation is feasible, offering practical tools and insights for assessing quantum advantage under realistic noisy conditions.

Abstract

We present a polynomial-time classical algorithm for estimating expectation values of arbitrary observables on typical quantum circuits under any incoherent local noise, including non-unital or dephasing. Although previous research demonstrated that some carefully designed quantum circuits affected by non-unital noise cannot be efficiently simulated, we show that this does not apply to average-case circuits, as these can be efficiently simulated using Pauli-path methods. Specifically, we prove that, with high probability over the circuit gates choice, Pauli propagation algorithms with tailored truncation strategies achieve an inversely polynomially small simulation error. This result holds for arbitrary circuit topologies and for any local noise, under the assumption that the distribution of each circuit layer is invariant under single-qubit random gates. Under the same minimal assumptions, we also prove that most noisy circuits can be truncated to an effective logarithmic depth for the task of {estimating} expectation values of observables, thus generalizing prior results to a significantly broader class of circuit ensembles. We further numerically validate our algorithm with simulations on a $6\times6$ lattice of qubits under the effects of amplitude damping and dephasing noise, as well as real-time dynamics on an $11\times11$ lattice of qubits affected by amplitude damping.

Figures (6)

• Figure 1: Schematic summary of our main results. a) Prior work has provided guarantees for simulating a quantum circuit either for highly random quantum circuits or assuming a special noise model. b) Here we show that the combination of a very general noise model (arbitrary local incoherent noise) with some randomness allows quantum circuits to be efficiently simulated. c) Our algorithm combines Pauli propagation with a truncation scheme based on the so-called path weight, which is the total cumulative weight of Paulis operators on a given branch aharonov2022polynomialschuster2024polynomialgonzalez2024pauli in contrast to current Pauli weight rudolph2023classicalschuster2024polynomialangrisani2024classically.
• Figure 2: Overview of noise models. Here we provide a geometric sketch of three commonly considered families of local noise: a) Depolarizing noise, which drives any input state towards the maximally mixed state, b) Dephasing-like noise, which drives states to $z$-axis connecting poles of the Bloch sphere, and c) Non-unital noise, which drives states to a given fixed point on the Bloch sphere. While prior work considered special cases of these families our analysis applies to any local noise channel with a constant noise rate.
• Figure 3: Error of simulating circuits with amplitude damping noise. a) Frobenius norm error and b) mean square error (MSE) for different path-weight truncation values and noise strengths in a periodic $6\times6$ lattice of qubits. The observable is a Pauli-Z operator in the middle of the lattice with a circuit consisting of RX, RZ single-qubit rotations and RZZ entangling gates and in b) the initial state is the all zero state. The theoretical bound, established in Thm. \ref{['thm:mse-core']}, is calculated via $(1-\gamma + \gamma^2)^k$, where $k$ is the path-weight truncation order and $1-\gamma + \gamma^2$ is the squared contraction coefficient computed in Example \ref{['obs:coeff-noise']}.
• Figure 4: Error of simulating circuits with dephasing noise. a) Frobenius norm error and b) mean square error (MSE) for the same 36-qubit simulation as in Fig. \ref{['fig:ampdamp_error']}. The theoretical bound, established in Thm. \ref{['thm:mse-core']}, is calculated via $\left(\frac{1+(1-2p)^2}{2}\right)^k$, where $k$ is the path-weight truncation order and $\frac{1+(1-2p)^2}{2}$ is the mean squared contraction coefficient computed in Example \ref{['obs:coeff-noise']}.
• Figure 5: Simulation of noisy real-time quantum dynamics. We simulate the expectation value of a Pauli-Z observable in the middle of an $11\times11$ arrangement of 121 qubits evolving under a rotated transverse-field Ising model at a quantum critical point. In addition to the path-weight truncations 20, 25 and 30, we truncate small coefficients below $2^{-23}$ and propagating Pauli operators with more than 5 X or Y Paulis. This setup was used in Ref. beguvsic2024real to accurately simulate the noise-free case with significantly more computational resources.

Theorems & Definitions (62)

• Theorem 1: Non-unital noise, informal
• Theorem 2: Unital noise, informal
• Theorem 3: Noise-induced shallow depth, informal
• Definition 1: Haar measure and $t$-designs
• Example 1: Unitary 1 and 2-designs
• Definition 2: Locally unbiased distribution
• Definition 3: Pauli-invariant distribution
• Lemma 1: Orthogonality
• proof
• Example 2: A class of unitary 1-designs which are not Pauli-invariant