2020/03/31 by Louis Lemonnier, John van de Wetering, Aleks Kissinger
Physics and Astronomy · #quant-ph
paper · pdf · doi:10.4204/eptcs.340.10
published as EPTCS 340, 2021, pp. 188-212 · In Proceedings QPL 2020, arXiv:2109.01534
arxiv created 2021/09/06 · arxiv updated 2021/09/07
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.