Simulating Quantum Mean Values in Noisy Variational Quantum Algorithms: A Polynomial-Scale Approach

Abstract

Large-scale variational quantum algorithms possess an expressive capacity that is beyond the reach of classical computers and is widely regarded as a potential pathway to achieving 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 that efficiently approximates quantum mean values in variational quantum algorithms with bounded truncation error in the presence of independent single-qubit depolarizing noise. Our method is based on path integrals in the Pauli basis. We have rigorously proved that, for a fixed noise rate $\lambda$, our method's time and space complexity exhibits a polynomial relationship with the number of qubits $n$, the circuit depth $L$, the inverse truncation error $\frac{1}{\varepsilon}$, and the inverse success probability $\frac{1}{\delta}$. Furthermore, We also prove that computational complexity becomes $\mathrm{Poly}\left(n,L\right)$ when the noise rate $\lambda$ exceeds $\frac{1}{\log{L}}$ and it becomes exponential with $L$ when the noise rate $\lambda$ falls below $\frac{1}{L}$