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arXiv · 2003.13564

Hypergraph Simplification: Linking the Path-sum Approach to the ZH-calculus

Louis Lemonnier·John van de Wetering·Aleks Kissinger
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Abstract

The ZH-calculus is a complete graphical calculus for linear maps between qubits that admits a straightforward encoding of hypergraph states and circuits arising from the Toffoli+Hadamard gate set. In this paper, we establish a correspondence between the ZH-calculus and the path-sum formalism, a technique recently introduced by Amy to verify quantum circuits. In particular, we find a bijection between certain canonical forms of ZH-diagrams and path-sum expressions. We then introduce and prove several new simplification rules for the ZH-calculus, which are in direct correspondence to the simplification rules of the path-sum formalism. The relatively opaque path-sum rules are shown to arise naturally from two powerful families of rewrite rules in the ZH-calculus. The first is the extension of the familiar graph-theoretic simplifications based on local complementation and pivoting to their hypergraph-theoretic analogues: hyper-local complementation and hyper-pivoting. The second is the graphical Fourier transform introduced by Kuijpers et al., which enables effective simplification of ZH-diagrams encoding multi-linear phase polynomials with arbitrary real coefficients.

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Louis Lemonnier, John van de Wetering, Aleks Kissinger. 2021-09-06. Hypergraph Simplification: Linking the Path-sum Approach to the ZH-calculus. https://doi.org/10.4204/eptcs.340.10

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