eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2023-02-01
36:1
36:17
10.4230/LIPIcs.CSL.2023.36
article
Completeness of Sum-Over-Paths for Toffoli-Hadamard and the Dyadic Fragments of Quantum Computation
Vilmart, Renaud
1
https://orcid.org/0000-0002-8828-4671
Université Paris-Saclay, CNRS, ENS Paris-Saclay, Inria, LMF, 91190, Gif-sur-Yvette, France
The "Sum-Over-Paths" formalism is a way to symbolically manipulate linear maps that describe quantum systems, and is a tool that is used in formal verification of such systems.
We give here a new set of rewrite rules for the formalism, and show that it is complete for "Toffoli-Hadamard", the simplest approximately universal fragment of quantum mechanics. We show that the rewriting is terminating, but not confluent (which is expected from the universality of the fragment). We do so using the connection between Sum-over-Paths and graphical language ZH-Calculus, and also show how the axiomatisation translates into the latter.
Finally, we show how to enrich the rewrite system to reach completeness for the dyadic fragments of quantum computation - obtained by adding phase gates with dyadic multiples of π to the Toffoli-Hadamard gate-set - used in particular in the Quantum Fourier Transform.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol252-csl2023/LIPIcs.CSL.2023.36/LIPIcs.CSL.2023.36.pdf
Quantum Computation
Verification
Sum-Over-Paths
Rewrite Strategy
Toffoli-Hadamard
Completeness