We give new evidence that quantum computers-moreover, rudimentary quantum computers built entirely out of linear-optical elements-cannot be efficiently simulated by classical computers. In particular, we define a model of computation in which identical photons are generated, sent through a linear-optical network, then nonadaptively measured to count the number of photons in each mode. This model is not known or believed to be universal for quantum computation, and indeed, we discuss the prospects for realizing the model using current technology. On the other hand, we prove that the model is able to solve sampling problems and search problems that are classically intractable under plausible assumptions.Our first result says that, if there exists a polynomial-time classical algorithm that samples from the same probability distribution as a linear-optical network, then P #P = BPP NP , and hence the polynomial hierarchy collapses to the third level. Unfortunately, this result assumes an extremely accurate simulation.Our main result suggests that even an approximate or noisy classical simulation would already imply a collapse of the polynomial hierarchy. For this, we need two unproven conjectures: the Permanent-of-Gaussians Conjecture, which says that it is #P-hard to approximate the permanent of a matrix A of independent N (0, 1) Gaussian entries, with high probability over A; and the Permanent Anti-Concentration Conjecture, which says that |Per (A)| ≥ √ n!/ poly (n) with high probability over A. We present evidence for these conjectures, both of which seem interesting even apart from our application. This paper does not assume knowledge of quantum optics. Indeed, part of its goal is to develop the beautiful theory of noninteracting bosons underlying our model, and its connection to the permanent function, in a self-contained way accessible to theoretical computer scientists.

No abstract

We give new evidence that quantum computers-moreover, rudimentary quantum computers built entirely out of linear-optical elements-cannot be efficiently simulated by classical computers. In particular, we define a model of computation in which identical photons are generated, sent through a linear-optical network, then nonadaptively measured to count the number of photons in each mode. This model is not known or believed to be universal for quantum computation, and indeed, we discuss the prospects for realizing the model using current technology. On the other hand, we prove that the model is able to solve sampling problems and search problems that are classically intractable under plausible assumptions.Our first result says that, if there exists a polynomial-time classical algorithm that samples from the same probability distribution as a linear-optical network, then P #P = BPP NP , and hence the polynomial hierarchy collapses to the third level. Unfortunately, this result assumes an extremely accurate simulation.Our main result suggests that even an approximate or noisy classical simulation would already imply a collapse of the polynomial hierarchy. For this, we need two unproven conjectures: the Permanent-of-Gaussians Conjecture, which says that it is #P-hard to approximate the permanent of a matrix A of independent N (0, 1) Gaussian entries, with high probability over A; and the Permanent Anti-Concentration Conjecture, which says that |Per (A)| ≥ √ n!/ poly (n) with high probability over A. We present evidence for these conjectures, both of which seem interesting even apart from our application. This paper does not assume knowledge of quantum optics. Indeed, part of its goal is to develop the beautiful theory of noninteracting bosons underlying our model, and its connection to the permanent function, in a self-contained way accessible to theoretical computer scientists.

No abstract

BosonSampling, which we proposed three years ago, is a scheme for using linear-optical networks to solve sampling problems that appear to be intractable for a classical computer. \ In a recent manuscript, Gogolin et al.\ claimed that even an ideal BosonSampling device's output would be operationally indistinguishable\textquotedblright\ from a uniform random outcome, at least \textquotedblleft without detailed a priori knowledge; or at any rate, that telling the two apart might itself be a hard problem. We first answer these claims---explaining why the first is based on a definition of a priori knowledge such that, were it adopted, almost no quantum algorithm could be distinguished from a pure random-number source; while the second is neither new nor a practical obstacle to interesting BosonSampling experiments.However, we then go further, and address some interesting research questions inspired by Gogolin et al.'s arguments. We prove that, with high probability over a Haar-random matrix $A$, the BosonSampling distribution induced by $A$ is far from the uniform distribution in total variation distance. More surprisingly, and counter to Gogolin et al., we give an efficient algorithm that distinguishes these two distributions with constant bias. Finally, we offer three bonus results about BosonSampling. First, we report an observation of Fernando Brandao: that one can efficiently sample a distribution that has large entropy and that's indistinguishable from a BosonSampling distribution by any circuit of fixed polynomial size. Second, we show that BosonSampling distributions can be efficiently distinguished from uniform even with photon losses and for general initial states. Third, we offer the simplest known proof that Fermion Sampling is solvable in classical polynomial time, and we reuse techniques from our Boson Sampling analysis to characterize random FermionSampling distributions.