A Complete Equational Theory for Real-Clifford+CH Quantum Circuits

TL;DR

This paper presents a complete set of equations for a specific quantum circuit fragment involving real Clifford and controlled-Hadamard gates, enabling derivation of all true circuit equalities within this fragment.

Contribution

It provides the first complete equational theory for a finitely-generated, universal quantum circuit fragment without parameters or ancillas.

Findings

Established a simple set of circuit equalities.

Proved all true equations in the fragment can be derived.

First completeness result for this class of quantum circuits.

Abstract

We introduce a complete equational theory for the fragment of quantum circuits generated by the real Clifford gates plus the two-qubit controlled-Hadamard gate. That is, we give a simple set of equalities between circuits of this fragment, and prove that any other true equation can be derived from these. This is the first such completeness result for a finitely-generated, universal fragment of quantum circuits, with no parameterized gates and no need for ancillas.