A Complete Equational Theory for Quantum Circuits

The work

AuthorsAlexandre Clément; Nicolas Heurtel; Shane Mansfield; Simon Perdrix; Benoît Valiron
Editors
Typepreprint
Year2022
Citekeyclement2022complete

Identifiers

arXiv2206.10577 from its doi
DOI10.48550/arxiv.2206.10577

Access

Landing pagehttps://arxiv.org/abs/2206.10577
Free full texthttps://arxiv.org/pdf/2206.10577

Settled

ContainerA preprint has no container. This is arXiv 2206.10577v2; the corpus's practice is that a repository is not a venue, as recorded on sewell2026escaping where 'arXiv' was removed from this field.

Abstract

We introduce the first complete equational theory for quantum circuits. More precisely, we introduce a set of circuit equations that we prove to be sound and complete: two circuits represent the same unitary map if and only if they can be transformed one into the other using the equations. The proof is based on the properties of multi-controlled gates -- that are defined using elementary gates -- together with an encoding of quantum circuits into linear optical circuits, which have been proved to have a complete axiomatisation.

A copy is held

pdf, 1.4 MB. Not published — it may be under copyright. The facts and links here are.

How it got here

How it got hereagent via arxiv
Added2026-09-01 23:05 UTC
Approved bya person 2026-09-02 08:55 UTC

Filed under

quantum-between-lab-and-threat what-a-program-means

Cite it as

@unpublished{clement2022complete,
  title        = {A Complete Equational Theory for Quantum Circuits},
  author       = {Alexandre Clément and Nicolas Heurtel and Shane Mansfield and Simon Perdrix and Benoît Valiron},
  year         = {2022},
  doi          = {10.48550/arxiv.2206.10577},
}

This record lives at https://refs.drheap.org/clement2022complete/ and will keep doing so.