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, such as universal quantum computation. 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 $\textbf{LO}_{fi}$-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 $\textbf{LO}_{fi}$-circuits represent the same quantum process if and only if one can be transformed into the other with the rules of the $\textbf{LO}_{fi}$-calculus. We give a unique and compact universal form for such circuits.

翻译：线性光学电路可用于操纵光子穿过分束器和移相器等组件时的量子态。这些光子态具有特别高的表达能力，因为它们存在于玻色子福克空间（一种无限维希尔伯特空间）中。然而，在线性光学量子计算领域，这些基本组件可能不足以高效执行所有感兴趣的计算，例如通用量子计算。为克服此局限，通常添加辅助源和探测器，它们能够投影到辅助光子态上，从而增强过程的多样性。本文引入$\textbf{LO}_{fi}$-演算，一种用于在无限维玻色子福克空间上进行推理的图形语言，其电路由线性光学的四个核心要素组成：移相器、分束器以及具有有界光子数的辅助源和探测器。我们提出了一种等式理论，并证明其是完备的：两个$\textbf{LO}_{fi}$-电路表示相同的量子过程当且仅当其中一个可以通过$\textbf{LO}_{fi}$-演算的规则变换为另一个。我们给出了此类电路的一种唯一且紧凑的通用形式。