Towards Large-scale Functional Verification of Universal Quantum Circuits

TL;DR

This paper presents a formal framework based on the Feynman path integral for the specification and automated verification of large-scale quantum circuits, including Clifford and Clifford+T circuits, with practical applications to quantum algorithms.

Contribution

It introduces a novel exponential sum formalism and rewrite system enabling polynomial-time equivalence checking of Clifford circuits and scalable verification of complex quantum algorithms.

Findings

Verified Clifford+T circuits with up to 100 qubits and thousands of T gates.

Automated verification of the Hidden Shift algorithm for large Boolean functions.

Achieved faster verification times compared to recent simulation methods.

Abstract

We introduce a framework for the formal specification and verification of quantum circuits based on the Feynman path integral. Our formalism, built around exponential sums of polynomial functions, provides a structured and natural way of specifying quantum operations, particularly for quantum implementations of classical functions. Verification of circuits over all levels of the Clifford hierarchy with respect to either a specification or reference circuit is enabled by a novel rewrite system for exponential sums with free variables. Our algorithm is further shown to give a polynomial-time decision procedure for checking the equivalence of Clifford group circuits. We evaluate our methods by performing automated verification of optimized Clifford+T circuits with up to 100 qubits and thousands of T gates, as well as the functional verification of quantum algorithms using hundreds of qubits. Our experiments culminate in the automated verification of the Hidden Shift algorithm for a class of Boolean functions in a fraction of the time it has taken recent algorithms to simulate.