ZH: A Complete Graphical Calculus for Quantum Computations Involving Classical Non-linearity

TL;DR

This paper introduces ZH, a new graphical calculus for quantum circuits that efficiently encodes classical non-linear functions and simplifies reasoning about quantum-classical interactions.

Contribution

The paper presents ZH, a complete graphical calculus extending existing formalisms to include compact encodings of non-linear classical functions in quantum circuits.

Findings

ZH calculus can encode AND with only 2 generators.

It simplifies the representation of N-controlled gates and hypergraph states.

The calculus is sound and complete for universal quantum computation.

Abstract

We present a new graphical calculus that is sound and complete for a universal family of quantum circuits, which can be seen as the natural string-diagrammatic extension of the approximately (real-valued) universal family of Hadamard+CCZ circuits. The diagrammatic language is generated by two kinds of nodes: the so-called 'spider' associated with the computational basis, as well as a new arity-N generalisation of the Hadamard gate, which satisfies a variation of the spider fusion law. Unlike previous graphical calculi, this admits compact encodings of non-linear classical functions. For example, the AND gate can be depicted as a diagram of just 2 generators, compared to ~25 in the ZX-calculus. Consequently, N-controlled gates, hypergraph states, Hadamard+Toffoli circuits, and diagonal circuits at arbitrary levels of the Clifford hierarchy also enjoy encodings with low constant overhead. This suggests that this calculus will be significantly more convenient for reasoning about the interplay between classical non-linear behaviour (e.g. in an oracle) and purely quantum operations. After presenting the calculus, we will prove it is sound and complete for universal quantum computation by demonstrating the reduction of any diagram to an easily describable normal form.