Quantum Circuit Completeness: Extensions and Simplifications
- URL: http://arxiv.org/abs/2303.03117v3
- Date: Fri, 1 Dec 2023 08:59:45 GMT
- Title: Quantum Circuit Completeness: Extensions and Simplifications
- Authors: Alexandre Cl\'ement, No\'e Delorme, Simon Perdrix, Renaud Vilmart
- Abstract summary: The first complete equational theory for quantum circuits has only recently been introduced.
We simplify the equational theory by proving that several rules can be derived from the remaining ones.
The complete equational theory can be extended to quantum circuits with ancillae or qubit discarding.
- Score: 44.99833362998488
- License: http://creativecommons.org/licenses/by/4.0/
- Abstract: Although quantum circuits have been ubiquitous for decades in quantum
computing, the first complete equational theory for quantum circuits has only
recently been introduced. Completeness guarantees that any true equation on
quantum circuits can be derived from the equational theory. We improve this
completeness result in two ways: (i) We simplify the equational theory by
proving that several rules can be derived from the remaining ones. In
particular, two out of the three most intricate rules are removed, the third
one being slightly simplified. (ii) The complete equational theory can be
extended to quantum circuits with ancillae or qubit discarding, to represent
respectively quantum computations using an additional workspace, and hybrid
quantum computations. We show that the remaining intricate rule can be greatly
simplified in these more expressive settings, leading to equational theories
where all equations act on a bounded number of qubits. The development of
simple and complete equational theories for expressive quantum circuit models
opens new avenues for reasoning about quantum circuits. It provides strong
formal foundations for various compiling tasks such as circuit optimisation,
hardware constraint satisfaction and verification.
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