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

• URL: http://arxiv.org/abs/2406.02905v1
• Date: Wed, 5 Jun 2024 03:56:18 GMT
• Title: The Focked-up ZX Calculus: Picturing Continuous-Variable Quantum Computation
• Authors: Razin A. Shaikh, Lia Yeh, Stefano Gogioso,
• Abstract summary: We formulate a graphical language for continuous-variable quantum computation. We present exciting new graphical rules capturing heftier CVQC interactions. Applying our calculus for quantum error correction, we derive graphical representations of the Gottesman-Kitaev-Preskill code encoder, syndrome measurement, and magic state distillation of Hadamard eigenstates.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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