Feynman path sum approach for simulation of linear optics

TL;DR

The paper develops a Linear-Optical Feynman Path (LOFP) framework to compute exact probability amplitudes in boson sampling by summing over valid Feynman paths through linear optical networks. It combines a path-sum formulation with tensor contraction to achieve near-linear runtime in the number of modes for shallow circuits, at the cost of exponential memory with depth. Through rigorous benchmarking against Ryser and the Cifuentes-Parrilo algorithm, LOFP demonstrates competitive accuracy (TVD near 10^-13 to 10^-16) and favorable scaling in regimes with larger mode counts or photon densities. The work provides an open-source C implementation, highlights the trade-offs between memory and speed, and discusses extensions to nonlinear gates and partially indistinguishable photons, with potential for parallelization and design-general interferometers.

Abstract

The Feynman path integral formalism has inspired the development of memory-efficient and parallelizable classical algorithms for simulating quantum computers. We adapt this approach for the calculation of probability amplitudes of linear-optical boson sampling experiments, which involve Fock-state inputs, linear optical circuits, and photo-detection at the output. We describe this simulation method and compare it with alternative approaches. Additionally, we implement a Linear-Optical Feynman Path simulator in open-source C code, enhancing its performance using tensor contraction techniques. Our method is benchmarked for low-depth linear optical circuits, where it offers advantages in runtime and memory efficiency.

Figures (13)

• Figure 1: Feynman path simulation of Fock-state linear optics. (a) The interferometers are built from a mesh of locally-connected BS (squares). Blue circles represent target choices of photon number occupations at input and output modes. (b) The Feynman path simulation algorithm adds the probability amplitudes of all possible paths, each determined by allowed photon numbers at internal green and red waveguides. The path enumeration only needs to iterate over possible occupations of green-colored waveguides, as those of the red-colored ones are then uniquely determined due to photon number conservation. (c) We further simplify path enumeration by using light-cone reasoning to quickly identify invalid, zero-amplitude paths. As shown in this example, invalid paths can result from an incompatibility between the target input and output configurations and the interferometer connectivity.
• Figure 2: Boson sampling setup using a planar interferometer design consisting of locally connected BS, as in the universal design proposed by Clements et al.Clements16. Input photon occupations are denoted by $x_i$, and output occupations by $y_i$.
• Figure 3: Planar interferometer design comprising (a) one, (b) two, and (c) three layers of locally-connected BS. The modes colored in black represent the target configurations of input and output. Modes colored green need to be iterated over for the Feynman path sum, while modes shown in red have their occupation numbers determined by those colored green or black, and the fixed input/output occupations, due to photon number conservation at each BS.
• Figure 4: Planar interferometer design consisting of $D$ layers of locally-connected BS. A Feynman path amplitude calculation requires iterating over all occupation numbers for green-colored modes, which then uniquely determine those of the red-colored ones.
• Figure 5: Past and future light cones for a green waveguide in the second BS layer of a 5-layer interferometer.