Minimal Equational Theories for Quantum Circuits (
arXiv:2311.07476), with Alexandre Clément and Simon Perdrix at
LICS’24 in Tallinn (Estonia) on 9th July 2024 (
slide of the talk). Also presented at
QPL’24 in Buenos Aires (Argentina) on 16th July 2024 (
slide of the talk), and at the
Workshop Defi EQIP 2023 in Lyon (France) on 14th Nov. 2023 (
slide of the talk).
We introduce the first minimal and complete equational theory for quantum circuits. Hence, we show that any true equation on quantum circuits can be derived from simple rules, all of them being standard except a novel but intuitive one which states that a multi-control 2π rotation is nothing but the identity. Our work improves on the recent complete equational theories for quantum circuits, by getting rid of several rules including a fairly impractical one. One of our main contributions is to prove the minimality of the equational theory, i.e. none of the rules can be derived from the other ones. More generally, we demonstrate that any complete equational theory on quantum circuits (when all gates are unitary) requires rules acting on an unbounded number of qubits. Finally, we also simplify the complete equational theories for quantum circuits with ancillary qubits and/or qubit discarding.
Quantum Circuit Completeness: Extensions and Simplifications (
arXiv:2303.03117), with Alexandre Clément, Simon Perdrix, and Renaud Vilmart at
CSL’24 in Naples (Italy) on 20th Feb. 2024 (
slide of the talk).
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.