DeepAI
Log In Sign Up

Completeness of Sum-Over-Paths for Toffoli-Hadamard and the Clifford Hierarchy

05/05/2022
by   Renaud Vilmart, et al.
0

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. 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 whole Clifford hierarchy.

READ FULL TEXT

page 1

page 9

page 13

03/13/2019

Completeness of the ZX-Calculus

The ZX-Calculus is a graphical language for diagrammatic reasoning in qu...
02/22/2021

Kindergarden quantum mechanics graduates (...or how I learned to stop gluing LEGO together and love the ZX-calculus)

This paper is a `spiritual child' of the 2005 lecture notes Kindergarten...
11/02/2022

Exact Completeness of LP Hierarchies for Linear Codes

Determining the maximum size A_2(n,d) of a binary code of blocklength n ...
10/02/2022

Beyond the Existential Theory of the Reals

We show that completeness at higher levels of the theory of the reals is...
09/26/2017

Towards a Minimal Stabilizer ZX-calculus

The stabilizer ZX-calculus is a rigorous graphical language for reasonin...
10/09/2021

Nonlocal Games, Compression Theorems, and the Arithmetical Hierarchy

We investigate the connection between the complexity of nonlocal games a...