Formal Modeling and Verification of Grover's Algorithm

TL;DR

This paper presents a formal approach to modeling and verifying Grover's quantum search algorithm using higher-order logic, ensuring correctness of key properties and demonstrating practical applications.

Contribution

It introduces a formal verification of Grover's algorithm in HOL Light, focusing on key properties and practical applications, which is novel in quantum algorithm verification.

Findings

Verified unitarity of oracle and diffusion operators

Proved success probability increases monotonically with iterations

Derived exact optimal iteration count for Grover's algorithm

Abstract

Grover's algorithm relies on the superposition and interference of quantum mechanics, which is more efficient than classical computing in specific tasks such as searching an unsorted database. Due to the high complexity of quantum mechanics, the correctness of quantum algorithms is difficult to guarantee through traditional simulation methods. By contrast, the fundamental concepts and mathematical structure of Grover's algorithm can be formalized into logical expressions and verified by higher-order logical reasoning. In this paper, we formally model and verify Grover's algorithm in the HOL Light theorem prover. We focus on proving key properties such as the unitarity of its oracle and diffusion operators, the monotonicity of the success probability with respect to the number of iterations, and an exact expression for the optimal iteration count. By analyzing a concrete application to integer factorization, we demonstrate the practicality and prospects of our work.