Complete Graphical Language for Linear Optical Circuits with Finite-Photon-Number Sources&Detectors

Speaker: Nicolas Heurtel

arXiv: https://arxiv.org/pdf/2402.17693

Abstract: Linear optical circuits can be used to manipulate the quantum states of photons as they pass through components including beam splitters and phase shifters. Those photonic states possess a particularly high level of expressiveness, as they reside within the bosonic Fock space, an infinite-dimensional Hilbert space. However, in the domain of linear optical quantum computation, these basic components may not be sufficient to efficiently perform all computations of interest. To address this limitation it is common to add auxiliary sources and detectors, which enable projections onto auxiliary photonic states and thus increase the versatility of the processes. In this paper, we introduce the LO_{fi}-calculus, a graphical language to reason on the infinite-dimensional bosonic Fock space with circuits composed of four core elements of linear optics: the phase shifter, the beam splitter, and auxiliary sources and detectors with bounded photon number. We present an equational theory that we prove to be complete: two LO_{fi}-circuits represent the same quantum process if and only if one can be transformed into the other with the rules of the LO_{fi}-calculus. We give a unique and compact universal form for such circuits.

Presented at the ZX-calculus seminar on the 11th of June 2024.