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 commutesilayers of losses to the input, wheresiis the length of the shortest path connecting theith input to any output. We then extend a recent classical simulation algorithm due to P. Clifford and R. Clifford to allow for arbitraryn-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) whenninput photons occupy a constant number of input modes; (B) when all butO(logn)photons are concentrated on a single input mode, while an additionalO(logn)modes contain one photon each.
Highlights
Introduction and relation to previous worksThe recent paradigm of quantum computational advantage, is regarded as a promising route towards demonstrating that quantum computers are more powerful than their classical counterparts [1]
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
We presented a comprehensive study of the effect of nonuniform losses on classical simulability of boson sampling
Summary
The recent paradigm of quantum computational advantage (or supremacy), is regarded as a promising route towards demonstrating that quantum computers are more powerful than their classical counterparts [1]. One candidate for demonstrating quantum advantage is nonadaptive linear optics, or boson sampling [2] In this model, an n-photon m-mode Fock state evolves according to a passive mmode linear-optical transformation U and is subsequently measured by particle-number resolving detectors [see Fig. 1(a)]. Several sources of imperfection affect linear-optical experiments, and it is essential to understand which can be mitigated and which degrade the computational power of the model This boundary between classical simulability and quantum advantage is an area of intense investigation, with losses in particular receiving most attention. The main result of [23] stated that, when less than n out of n photons are left, it is possible to approximate a lossy boson sampling state by a state of distinguishable photons, which is known to be classically simulable This approximation is done at the level of the input state, and holds for arbitrary linear-optical experiments. Boson sampling can be framed in the language of first quantization, which we briefly use (see e.g. [35] or the introductory section of [23] for the translation between the two descriptions)
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