BosonSampling with Lost Photons

BosonSampling is an intermediate model of quantum computation where linear-optical networks are used to solve sampling problems expected to be hard for classical computers. Since these devices are not expected to be universal for quantum computation, it remains an open question of whether any error-correction techniques can be applied to them, and thus it is important to investigate how robust the model is under natural experimental imperfections, such as losses and imperfect control of parameters. Here we investigate the complexity of BosonSampling under photon losses---more specifically, the case where an unknown subset of the photons are randomly lost at the sources. We show that, if $k$ out of $n$ photons are lost, then we cannot sample classically from a distribution that is $1/n^{\Theta(k)}$-close (in total variation distance) to the ideal distribution, unless a $\text{BPP}^{\text{NP}}$ machine can estimate the permanents of Gaussian matrices in $n^{O(k)}$ time. In particular, if $k$ is constant, this implies that simulating lossy BosonSampling is hard for a classical computer, under exactly the same complexity assumption used for the original lossless case.Comment: 12 pages. v2: extended concluding sectio

Experimental Gaussian Boson Sampling

Gaussian Boson sampling (GBS) provides a highly efficient approach to make use of squeezed states from parametric down-conversion to solve a classically hard-to-solve sampling problem. The GBS protocol not only significantly enhances the photon generation probability, compared to standard boson sampling with single photon Fock states, but also links to potential applications such as dense subgraph problems and molecular vibronic spectra. Here, we report the first experimental demonstration of GBS using squeezed-state sources with simultaneously high photon indistinguishability and collection efficiency. We implement and validate 3-, 4- and 5-photon GBS with high sampling rates of 832 kHz, 163 kHz and 23 kHz, respectively, which is more than 4.4, 12.0, and 29.5 times faster than the previous experiments. Further, we observe a quantum speed-up on a NP-hard optimization problem when comparing with simulated thermal sampler and uniform sampler.Comment: 12 pages, 4 figures, published online on 2nd April 201

Quantum simulation of partially distinguishable boson sampling

Boson Sampling is the problem of sampling from the same output probability distribution as a collection of indistinguishable single photons input into a linear interferometer. It has been shown that, subject to certain computational complexity conjectures, in general the problem is difficult to solve classically, motivating optical experiments aimed at demonstrating quantum computational "supremacy". There are a number of challenges faced by such experiments, including the generation of indistinguishable single photons. We provide a quantum circuit that simulates bosonic sampling with arbitrarily distinguishable particles. This makes clear how distinguishabililty leads to decoherence in the standard quantum circuit model, allowing insight to be gained. At the heart of the circuit is the quantum Schur transform, which follows from a representation theoretic approach to the physics of distinguishable particles in first quantisation. The techniques are quite general and have application beyond boson sampling.Comment: 25 pages, 4 figures, 2 algorithms, comments welcom

Sufficient Conditions for Efficient Classical Simulation of Quantum Optics

We provide general sufficient conditions for the efficient classical simulation of quantum-optics experiments that involve inputting states to a quantum process and making measurements at the output. The first condition is based on the negativity of phase-space quasiprobability distributions (PQDs) of the output state of the process and the output measurements; the second one is based on the negativity of PQDs of the input states, the output measurements, and the transition function associated with the process. We show that these conditions provide useful practical tools for investigating the effects of imperfections in implementations of boson sampling. In particular, we apply our formalism to boson-sampling experiments that use single-photon or spontaneous-parametric-down-conversion sources and on-off photodetectors. Considering simple models for loss and noise, we show that above some threshold for the probability of random counts in the photodetectors, these boson-sampling experiments are classically simulatable. We identify mode mismatching as the major source of error contributing to random counts and suggest that this is the chief challenge for implementations of boson sampling of interesting size.Comment: 12 pages, 1 figur

Classical modelling of a bosonic sampler with photon collisions

When the problem of boson sampling was first proposed, it was assumed that little or no photon collisions occur. However, modern experimental realizations rely on setups where collisions are quite common, i.e. the number of photons $M$ injected into the circuit is close to the number of detectors $N$. Here we present a classical algorithm that simulates a bosonic sampler: it calculates the probability of a given photon distribution at the interferometer outputs for a given distribution at the inputs. This algorithm is most effective in cases with multiple photon collisions, and in those cases it outperforms known algorithms

Correlations for subsets of particles in symmetric states: what photons are doing within a beam of light when the rest are ignored

Given a state of light, how do its properties change when only some of the constituent photons are observed and the rest are neglected (traced out)? By developing formulae for mode-agnostic removal of photons from a beam, we show how the expectation value of any operator changes when only $q$ photons are inspected from a beam, ignoring the rest. We use this to reexpress expectation values of operators in terms of the state obtained by randomly selecting $q$ photons. Remarkably, this only equals the true expectation value for a unique value of $q$: expressing the operator as a monomial in normally ordered form, $q$ must be equal to the number of photons annihilated by the operator. A useful corollary is that the coefficients of any $q$-photon state chosen at random from an arbitrary state are exactly the $q$th order correlations of the original state; one can inspect the intensity moments to learn what any random photon will be doing and, conversely, one need only look at the $n$-photon subspace to discern what all of the $n$th order correlation functions are. The astute reader will be pleased to find no surprises here, only mathematical justification for intuition. Our results hold for any completely symmetric state of any type of particle with any combination of numbers of particles and can be used wherever bosonic correlations are found.Comment: 11+3 pages, 1 figure, comments always welcom

Boson Sampling with efficient scaling and efficient verification

A universal quantum computer of moderate scale is not available yet, however intermediate models of quantum computation would still permit demonstrations of a quantum computational advantage over classical computing and could challenge the Extended Church-Turing Thesis. One of these models based on single photons interacting via linear optics is called Boson Sampling. Proof-of-principle Boson Sampling has been demonstrated, but the number of photons used for these demonstrations is below the level required to claim quantum computational advantage. To make progress with this problem, here we conclude that the most practically achievable pathway to scale Boson Sampling experiments with current technologies is by combining continuous-variables quantum information and temporal encoding. We propose the use of switchable dual-homodyne and single-photon detections, the temporal loop technique and scattershot based Boson Sampling. This proposal gives details as to what the required assumptions are and a pathway for a quantum optical demonstration of quantum computational advantage. Furthermore, this particular combination of techniques permits a single efficient implementation of Boson Sampling and efficient verification in a single experimental setup