Classical simulations of noisy variational quantum circuits

TL;DR

The paper introduces LOWESA, a classical algorithm to approximate the cost landscapes of noisy parameterised quantum circuits by decomposing the noisy circuit into low-weight process modes using a trigonometric basis. Leveraging Pauli transfer matrices and Pauli back-propagation, it achieves polynomial scaling in the number of qubits $n$ and circuit depth for circuits with independently parameterised non-Clifford gates under Pauli noise, with an error that decays exponentially with the cut-off $\ell$ and the noise rate $p$. It extends to general Pauli noise and to circuits with fixed non-Clifford gates, while showing that correlations between parameters can break the method’s guarantees, requiring much larger $\ell$ or exponential resources. The results help delineate the boundary between classical simulability and quantum advantage under realistic noise, offering a constructive tool for benchmarking and understanding fidelity thresholds in NISQ devices.

Abstract

Noise detrimentally affects quantum computations so that they not only become less accurate but also easier to simulate classically as systems scale up. We construct a classical simulation algorithm, LOWESA (low weight efficient simulation algorithm), for estimating expectation values of noisy parameterised quantum circuits. It combines previous results on spectral analysis of parameterised circuits with Pauli back-propagation and recent ideas for simulations of noisy random circuits. We show, under some conditions on the circuits and mild assumptions on the noise, that LOWESA gives an efficient, polynomial algorithm in the number of qubits (and depth), with approximation error that vanishes exponentially in the physical error rate and a controllable cut-off parameter. We also discuss the practical limitations of the method for circuit classes with correlated parameters and its scaling with decreasing error rates.

Figures (3)

• Figure 1: a) Schematic of the parameterised quantum circuits that can be simulated by lowesa. The light boxes are arbitrary (noisy) Clifford gates, the blue boxes are parameterised $z$-rotations and the red kites represent Pauli noise channels. b) Diagrammatic sketch of lowesa as described in Algorithm \ref{['algo:1']} applied to circuits given by Equation \ref{['eq:cliff_var_circ']}. The Pauli operator $P$ is propagated backwards through the circuit where every Clifford gate transforms it into another Pauli, and the decomposition of the parameterised $Rz$ rotations into process modes $D_0, D_1, D_{-1}$ splits the propagation up into paths that may annihilate. A cut-off of $\ell=2$ is chosen which artificially annihilates paths that branch into $D_1,D_{-1}$ more than $2$ times.
• Figure 2: Scaling of lowesa with the number of qubits $n$ and cut-off parameter $\ell$. The circuit structure consists of two parameterised layers of $H-Rz(\theta_i)-X-H$ on each qubit, where the Hadamard and X gates are chosen with 0.5 probability, followed by CNOTs placed on a 2D topology. a) Total number of paths for a given $\ell$, which equals $\sum_i^{\ell} {m \choose i} 2^i$. Note that the number of paths that lowesa needs to explore is dramatically lower. b) Wall time to run lowesa with truncation parameter $\ell$ on an average laptop without parallelisation. Each data point represents an average over 500 different randomized circuits with Pauli Z measurement operators that act on a random subset of qubits. The shading shows the 90% confidence interval. The simulation of the Clifford gates used a look-up table, meaning that the scaling in $n$ is entirely due the scaling of $m$ with $n$.
• Figure 3: Accuracy benchmark of lowesa compared to the error bounds as predicted in Theorem \ref{['thm:uncorr_cliff']}. We show the $L^2$ error of a single-qubit Pauli Y operator expectation with $\ell<m=60$ for two layers of a $n=10$ qubit circuit. The circuit consists of parametrized single-qubit gates $R_z(\boldsymbol{\theta}_i)\,R_x(\boldsymbol{\theta}_{i+1})\,R_z(\boldsymbol{\theta}_{i+2})$ on each qubit followed by CNOT gates in a 2D topology. For this particular circuit, each entangler in the 2D topology was placed with a 0.5 probability. The noise model is symmetric depolarising noise, where the parameters are set $p_X = p_Y = p_Z = p$. Each point is averaged over 1000 random parameterisations of the same circuit to compare to the integral definition of our error bounds. All paths below $\ell=3$ and above $\ell=21$ annihilate. Consequently, the simulation with $\ell=21$ is exact.

Theorems & Definitions (10)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• proof
• proof
• proof
• Lemma 5
• proof
• Corollary 6