Circuit Relations for Real Stabilizers: Towards TOF+H

TL;DR

This paper develops a circuit-based framework for the real stabilizer fragment of quantum mechanics, connecting it with the ZX-calculus and exploring potential for a complete axiomatization of universal quantum computation.

Contribution

It extends the CNOT category with Hadamard and scalar generators, establishing inverse translations with the ZX-calculus fragment, and discusses prospects for axiomatizing universal quantum mechanics.

Findings

Constructed circuit relations for the real stabilizer fragment.

Established inverse translations between circuit relations and ZX-calculus.

Discussed potential for axiomatizing universal quantum computation.

Abstract

The real stabilizer fragment of quantum mechanics was shown to have a complete axiomatization in terms of the angle-free fragment of the ZX-calculus. This fragment of the ZX-calculus---although abstractly elegant---is stated in terms of identities, such as spider fusion which generally do not have interpretations as circuit transformations. We complete the category CNOT generated by the controlled not gate and the computational ancillary bits, presented by circuit relations, to the real stabilizer fragment of quantum mechanics. This is performed first, by adding the Hadamard gate and the scalar sqrt 2 as generators. We then construct translations to and from the angle-free fragment of the ZX-calculus, showing that they are inverses. We then discuss how this could potentially lead to a complete axiomatization, in terms of circuit relations, for the approximately universal fragment of quantum mechanics generated by the Toffoli gate, Hadamard gate and computational ancillary bits.