We generalize an approach for description of multi-photon experiments with multi-port unitary linear optical devices, started in Phys. Rev. A 89, 022333 (2014) with single photons in mixed spectral states, to arbitrary (multi-photon) input and arbitrary photon detectors. We show that output probabilities are always given in terms of the matrix permanents of the Hadamard product of a matrix built from the network matrix and matrices built from spectral state of photons and spectral sensitivities of detectors. Moreover, in case of input with up to one photon per mode, the output probabilities are given by a sum (or integral) with each term being the absolute value squared of such a matrix permanent. We conjecture that, for an arbitrary multi-photon input, zero output probability of an output configuration is always the result of an exact cancellation of quantum transition amplitudes of completely indistinguishable photons (a subset of all input photons) and, moreover, is independent of coherence between only partially indistinguishable photons. The conjecture is supported by examples. Furthermore, we propose a measure of partial indistinguishability of photons which generalizes Mandel's observation, and find the law of degradation of quantum coherence in a realistic Boson-Sampling device with increase of the total number of photons and/or their "classicality parameter".

The boson sampler proposed by Aaronson and Arkhipov is a non-universal quantum computer, which can serve as evidence against the extended Church-Turing thesis. It samples the probability distribution at the output of linear unitary optical network, with indistinguishable single photons at the input. Four experimental groups have already tested their small-scale prototypes with up to four photons. The boson sampler with few dozens of single photons is believed to be hard to simulate on a classical computer. For scalability of a realistic boson sampler with current technology it is necessary to know the effect of the photon mode mismatch on its operation. Here a nondeterministic model of the boson sampler is analyzed, which employs partially indistinguishable single photons emitted by identical sources. A sufficient condition on the average mutual fidelity F of the single photons is found, which guarantees that the realistic boson sampler outperforms the classical computer. Moreover, the boson sampler computer with partially indistinguishable single photons is scalable while being beyond the power of classical computers when the single photon mode mismatch 1 − F scales as O(N −3/2 ) with the total number of photons N .

Photon losses are among the strongest imperfections affecting multi-photon interference. Despite their importance, little is known about their effect on boson sampling experiments. In this work we show that using classical computers, one can efficiently simulate multi-photon interference in all architectures that suffer from an exponential decay of the transmission with the depth of the circuit, such as integrated photonic circuits or optical fibers. We prove that either the depth of the circuit is large enough that it can be simulated by thermal noise with an algorithm running in polynomial time, or it is shallow enough that a tensor network simulation runs in quasi-polynomial time. This result suggests that in order to implement a quantum advantage experiment with single-photons and linear optics new experimental platforms may be needed.

It is found that identical bosons (fermions) show generalized bunching (antibunching) property in linear networks: The absolute maximum (minimum) of probability that all N input particles are detected in a subset of K output modes of any nontrivial linear M -mode network is attained only by completely indistinguishable bosons (fermions). For fermions K is arbitrary, for bosons it is either (i) arbitrary for only classically correlated bosons or (ii) satisfies K ≥ N (or K = 1) for arbitrary input states of N particles. The generalized bunching allows to certify in a polynomial in N number of runs that a physical device realizing Boson Sampling with an arbitrary network operates in the regime of full quantum coherence compatible only with completely indistinguishable bosons. The protocol needs only polynomial classical computations for the standard BosonSampling, whereas an analytic formula is available for the Scattershot version.PACS numbers: 42.50. St, 03.67.Ac, 42.50.Ar Introduction.-Optical networks with photons have become a growing research field with potential application in quantum computing [1][2][3]. Boson Sampling (BS) idea [2], a non-universal but near-future feasible device aiming at the Extended Church-Turing thesis (ECT), followed by spectacular experiments with optical networks of growing size and number of photons [4][5][6][7][8][9][10], can be a way for benchmark demonstration of quantum supremacy. A BS device with a random but known Mmode network, N ∼ 30 single photons at known input modes for M ≫ N 2 would achieve this ultimate goal [2]. However, the very computational complexity of BS [11, 12] requires an exponential in N number of runs of such a device and computation of an exponential number (at least O(N N )) of classically hard permanents to prove BS by comparing an output distribution with theoretical probabilities. Quantum supremacy demonstration thus faces a big challenge of maintaining the N th order quantum coherence in a BS device with N ∼ 30 for an exponential number of runs. Though a distribution claimed to simulate BS, as the uniform one [13], can be exposed with only polynomial number of runs of a device [14] (see also Ref. [9]), finding such a protocol for a given sampler, e.g., the one with distinguishable particles, is a hard open problem.The proof of BS being exponential both in the number of runs and computations does not prevent efficient verification of the very source of quantum supremacy of BS, i.e., the full N th-order quantum coherence in a device with N bosons, where unwanted distinguishability, photon losses, and higher photon numbers are the leading adversary factors in optical setups [4][5][6][7][8][9][10]. Such an assessment of BS may require only a polynomial number of runs and polynomial classical computations. Step (I): the source is checked for output with only one particle per mode.Step (II), the statistics of all N input particles to land in K output modes of a M -mode network U is gathered by on-off type detectors in complementary L = M −K modes: The...

We show that controlled dissipation can be used as a tool for exploring fundamental phenomena and managing mesoscopic systems of cold atoms and Bose-Einstein condensates. Even the simplest boson-Josephson junction, that is, a Bose-Einstein condensate in a double-well trap, subjected to removal of atoms from one of the two potential minima allows one to observe such phenomena as the suppression of losses and the nonlinear Zeno effect. In such a system the controlled dissipation can be used to create desired macroscopic states and implement controlled switching among different quantum regimes.