On Classical Simulation of Quantum Circuits Composed of Clifford Gates

TL;DR

This paper explains how quantum circuits made of Clifford gates can be efficiently simulated classically, providing a detailed, beginner-friendly breakdown of the Gottesman-Knill theorem and its underlying principles.

Contribution

It offers a step-by-step, intuitive explanation of the Gottesman-Knill theorem, making the classical simulation of Clifford circuits accessible to newcomers.

Findings

Clifford gates map Pauli strings to Pauli strings

Classical simulation of Clifford circuits is efficient

Provides detailed examples of superposition and entanglement

Abstract

The Gottesman-Knill theorem asserts that quantum circuits composed solely of Clifford gates can be efficiently simulated classically. This theorem hinges on the fact that Clifford gates map Pauli strings to other Pauli strings, thereby allowing for a structured simulation process using classical computations. In this work, we break down the step-by-step procedure of the Gottesman-Knill theorem in a beginner-friendly manner, leveraging concepts such as matrix products, tensor products, commutation, anti-commutation, eigenvalues, and eigenvectors of quantum mechanical operators. Through detailed examples illustrating superposition and entanglement phenomena, we aim to provide a clear understanding of the classical simulation of Clifford gate-based quantum circuits. While we do not provide a formal proof of the theorem, we offer intuitive physical insights at each stage where necessary, empowering readers to grasp the fundamental principles underpinning this intriguing aspect of quantum computation.