Simulating Noisy Variational Quantum Algorithms: A Polynomial Approach

TL;DR

This paper addresses the challenge of classically simulating noisy variational quantum algorithms by introducing OBPPP, a polynomial-scale method based on a Pauli-path back-propagation path-integral. The approach rewrites the cost function as a sum over Pauli-path contributions, with noise scaling each path by factors $(1-2(p_y+p_z))^{|s|_X}(1-2(p_x+p_z))^{|s|_Y}(1-2(p_x+p_y))^{|s|_Z}$, enabling controlled truncation to bounded error. The authors prove that for fixed minimal non-zero noise γ, the algorithm runs in time and space Poly(n,L), and in Case 1 (at least two non-zero noise components) remains Poly(n,L) when γ ≥ 1/log L, while Case 2 can be exponential in L if γ ∼ 1/L. Numerical experiments on IBM's 127-qubit Eagle processor show OBPPP achieving higher accuracy and faster runtimes than the device in several circuits and capable of reproducing unmitigated results when appropriate, highlighting the nuanced role of noise in classical simulability and offering a versatile benchmark for quantum computer verification.

Abstract

Large-scale variational quantum algorithms are widely recognized as a potential pathway to achieve practical quantum advantages. However, the presence of quantum noise might suppress and undermine these advantages, which blurs the boundaries of classical simulability. To gain further clarity on this matter, we present a novel polynomial-scale method based on the path integral of observable's back-propagation on Pauli paths (OBPPP). This method efficiently approximates expectation values of operators in variational quantum algorithms with bounded truncation error in the presence of single-qubit Pauli noise. Theoretically, we rigorously prove: 1) For a constant minimal non-zero noise rate $γ$, OBPPP's time and space complexity exhibit a polynomial relationship with the number of qubits $n$, the circuit depth $L$. 2) For variable $γ$, in scenarios where more than two non-zero noise factors exist, the complexity remains $\mathrm{Poly}\left(n,L\right)$ if $γ$ exceeds $1/\log{L}$, but grows exponential with $L$ when $γ$ falls below $1/L$. Numerically, we conduct classical simulations of IBM's zero-noise extrapolated experimental results on the 127-qubit Eagle processor [Nature \textbf{618}, 500 (2023)]. Our method attains higher accuracy and faster runtime compared to the quantum device. Furthermore, our approach allows us to simulate noisy outcomes, enabling accurate reproduction of IBM's unmitigated results that directly correspond to raw experimental observations. Our research reveals the vital role of noise in classical simulations and the derived method is general in computing the expected value for a broad class of quantum circuits and can be applied in the verification of quantum computers.

Figures (17)

• Figure 1: (a) Noiseless parametrized quantum circuits consists of Pauli rotation gates (white boxes) and selected Clifford gates (pink boxes). (b) Single-qubit Pauli noise applied independently to each qubit (red points). (c) A diagrammatic illustration of our method. The Pauli operator back-propagates from the observable through the circuit and bifurcates during the propagation. (d) Under the presence of noise, the contributions of Pauli paths are suppressed exponentially with their Pauli weight. Translucent bars represent noiseless contributions, while opaque bars show the contributions after applying single-qubit Pauli noise. Contributions from Pauli paths with high Pauli weights are truncated and represented by dashed lines in (c) and the shaded area in (d).
• Figure 2: Classical simulation of different Pauli operators on IBM's 127-qubit Eagle processor. The green loops and blue circles in (a)-(e) correspond to direct experimental observations and error-mitigated results of different Pauli operators as reported in Ref. kim2023evidence. (a)-(c) simulate a circuit with $5$ Trotter steps, (d) simulates a circuit with $5$ Trotter steps and an additional layer of $R_X$ gates, and (e)-(f) simulate a circuit with $20$ Trotter steps. The rotation angle $\theta_J$ of the $R_{ZZ}$ gate in (a)-(e) is set to $\theta_J=-\pi/2$, while (f) is taken from Ref. anand2023classical with $\theta_J=-\pi/4$. The bottom subplots in (a)-(c) show the absolute difference between the simulated results of OBPPP and the exact results from Ref. kim2023evidence. In each figure, the red-dot line represents the output of OBPPP, which is an analytic expression of a trigonometric polynomial with respect to $\theta_h$. The orange-triangle line denotes the expectation from the suppressed path contributions under certain noise. The runtimes of (a)-(f) on two Xeon 6330 CPUs (28 cores per chip) are within 13 seconds, 146 seconds, 29 seconds, 137 seconds, 262 seconds, and 57 seconds, respectively.
• Figure 3: The tree representation of observable $O=1 X_0+1 Z_1+0.5 X_0X_1$.
• Figure 4: The tree representation of back-propagation process.
• Figure 5: The explanation of bifurcation points in Fig.\ref{['fig:bp_tree']}.

Theorems & Definitions (24)

• Proposition 1
• Remark
• lemma 1
• lemma 2
• Theorem 1
• Proposition 2
• proof
• Remark
• lemma 3
• proof : Proof of Lemma \ref{['lemma:MSE_l']}