Classical simulation of linear optics subject to nonuniform losses

Abstract

We present a comprehensive study of the impact of non-uniform, i.e. path-dependent, photonic losses on the computational complexity of linear-optical processes. Our main result states that, if each beam splitter in a network induces some loss probability, non-uniform network designs cannot circumvent the efficient classical simulations based on losses.To achieve our result we obtain new intermediate results that can be of independent interest. First we show that, for any network of lossy beam-splitters, it is possible to extract a layer of non-uniform losses that depends on the network geometry. We prove that, for every input mode of the network it is possible to commute si layers of losses to the input, where si is the length of the shortest path connecting the ith input to any output. We then extend a recent classical simulation algorithm due to P. Clifford and R. Clifford to allow for arbitrary n-photon input Fock states (i.e. to include collision states). Consequently, we identify two types of input states where boson sampling becomes classically simulable: (A) when n input photons occupy a constant number of input modes; (B) when all but O(log⁡n) photons are concentrated on a single input mode, while an additional O(log⁡n) modes contain one photon each.

Classical simulation of lossy boson sampling using matrix product operators

“…In addition, it is natural to assume that the photon-loss rate is the same on each mode in practice. Note that our MPO algorithm is also applicable to nonuniform loss [52] although it requires more computational time (see Appendix B 2). Especially in the uniform loss case, one can easily verify that photon-loss channels commute with arbitrary beamsplitter circuits as shown in Fig.…”

“…We compare the MPS simulation of the standard boson sampling [5] with the one with a different type of an input state |ψ in = |N |0 M−1 (note that more general input states are analyzed in Ref. [52] and Appendix C.). It can be easily shown that the computation of the probability of an outcome in this case is not difficult because the corresponding permanent is constructed by repeating N times of the same column [5].…”

“…As a remark, in the case of nonuniform loss [52], we cannot simplify the problem by merging all loss channels as we did for uniform loss because nonuniform loss channels do not commute with beam splitters in general. Therefore, one needs to update an MPO by a completely positive trace-preserving map for a loss channel for each step, which requires more computational time [22].…”

Distinguishing noisy boson sampling from classical simulations

“…However, recent classical simulations [38,39] pushed the threshold to N ≈ 50 bosons. Inevitable experimental noise [35,46,40,41,42,43,44,45] additionally opens possibilities for efficient classical approximation algorithms [47,48,49,50,52,53,51]. Most importantly, if noise amplitudes do not vanish when the system size scales up, the recent algorithms of Refs.…”

Bosons vs. Fermions – A computational complexity perspective

“…Unfortunately, it is beyond the scope of this work to describe in further detail all the developments, both theoretical and experimental, in the field of quantum advantage (and BosonSampling in particular). A nonexhaustive list includes strengthened proofs more robust against experimental imperfections [67][68][69], the development of better competing classical algorithms [70][71][72][73][74][75][76][77], the raising of other important issues, such as how to efficiently verify the output data to guarantee that the device is working correctly [78][79][80][81], or proposal of alternative models, such as BosonSampling with Gaussian input states rather than Fock states [82,83]. We direct the interested reader to recent review articles for more thorough overviews of these developments [12,13,38,84].…”