Cutoff Theorems for the Equivalence of Parameterized Quantum Circuits (Extended)

TL;DR

This paper introduces a cutoff-based method for verifying the equivalence of parameterized quantum circuits, reducing the problem to finitely many simpler circuits, with a probabilistic approach for large parameter sets.

Contribution

It defines a generalized class of parameterized circuits, proves decidability for cyclotomic gates, and proposes both deterministic and probabilistic verification algorithms.

Findings

Decidability of circuit equivalence for cyclotomic gate sets.

A cutoff-based reduction to finitely many parameter-free circuits.

A probabilistic algorithm for large parameter spaces.

Abstract

Many promising quantum algorithms in economics, medical science, and material science rely on circuits that are parameterized by a large number of angles. To ensure that these algorithms are efficient, these parameterized circuits must be heavily optimized. However, most quantum circuit optimizers are not verified, so this procedure is known to be error-prone. For this reason, there is growing interest in the design of equivalence checking algorithms for parameterized quantum circuits. In this paper, we define a generalized class of parameterized circuits with arbitrary rotations and show that this problem is decidable for cyclotomic gate sets. We propose a cutoff-based procedure which reduces the problem of verifying the equivalence of parameterized quantum circuits to the problem of verifying the equivalence of finitely many parameter-free quantum circuits. Because the number of parameter-free circuits grows exponentially with the number of parameters, we also propose a probabilistic variant of the algorithm for cases when the number of parameters is intractably large. We show that our techniques extend to equivalence modulo global phase, and describe an efficient angle sampling procedure for cyclotomic gate sets.