Efficient classical simulation of noisy quantum computation

TL;DR

This paper introduces a tensor network method to efficiently simulate noisy quantum circuits classically, revealing that most such circuits with constant noise levels are computationally manageable, even with some perfect gates.

Contribution

The authors develop a tensor network formalism for simulating noisy quantum circuits and prove that most circuits with constant noise are classically simulatable with polynomial complexity.

Findings

Most noisy quantum circuits at constant noise levels can be efficiently simulated classically.

The simulation remains efficient even with some perfect noiseless gates, such as Clifford gates.

The results connect classical simulatability, quantum supremacy, and fault-tolerance in quantum computing.

Abstract

Understanding the boundary between classical simulatability and the power of quantum computation is a fascinating topic. Direct simulation of noisy quantum computation requires solving an open quantum many-body system, which is very costly. Here, we develop a tensor network formalism to simulate the time-dynamics and the Fourier spectrum of noisy quantum circuits. We prove that under general conditions most of the quantum circuits at any constant level of noise per gate can be efficiently simulated classically with the cost increasing only polynomially with the size of the circuits. The result holds even if we have perfect noiseless quantum gates for some subsets of operations, such as all the gates in the Clifford group. This surprising result reveals the subtle relations between classical simulatability, quantum supremacy, and fault-tolerant quantum computation. The developed simulation tools may also be useful for solving other open quantum many-body systems.