The Focked-up ZX Calculus: Picturing Continuous-Variable Quantum Computation

TL;DR

This paper introduces a new graphical calculus for continuous-variable quantum computation, extending existing tools to infinite-dimensional spaces and enabling visual reasoning about complex quantum protocols.

Contribution

It formulates a novel graphical language combining Z, X, and Fock spiders, and proves its completeness for Gaussian CVQC, advancing the understanding of infinite-dimensional quantum processes.

Findings

Graphical calculus for CVQC with Z, X, and Fock spiders.

Complete for Gaussian CVQC in infinite-dimensional Hilbert space.

Graphical proofs for GKP code and Gaussian boson sampling.

Abstract

While the ZX and ZW calculi have been effective as graphical reasoning tools for finite-dimensional quantum computation, the possibilities for continuous-variable quantum computation (CVQC) in infinite-dimensional Hilbert space are only beginning to be explored. In this work, we formulate a graphical language for CVQC. Each diagram is an undirected graph made of two types of spiders: the Z spider from the ZX calculus defined on the reals, and the newly introduced Fock spider defined on the natural numbers. The Z and X spiders represent functions in position and momentum space respectively, while the Fock spider represents functions in the discrete Fock basis. In addition to the Fourier transform between Z and X, and the Hermite transform between Z and Fock, we present exciting new graphical rules capturing heftier CVQC interactions. We ensure this calculus is complete for all of Gaussian CVQC interpreted in infinite-dimensional Hilbert space, by translating the completeness in affine Lagrangian relations by Booth, Carette, and Comfort. Applying our calculus for quantum error correction, we derive graphical representations of the Gottesman-Kitaev-Preskill (GKP) code encoder, syndrome measurement, and magic state distillation of Hadamard eigenstates. Finally, we elucidate Gaussian boson sampling by providing a fully graphical proof that its circuit samples submatrix hafnians.