Approximating the Permanent of a Random Matrix with Vanishing Mean

We show an algorithm for computing the permanent of a random matrix with vanishing mean in quasi-polynomial time. Among special cases are the Gaussian, and biased-Bernoulli random matrices with mean 1/lnln(n)^{1/8}. In addition, we can compute the permanent of a random matrix with mean 1/poly(ln(n)) in time 2^{O(n^{\eps})} for any small constant \eps>0. Our algorithm counters the intuition that the permanent is hard because of the "sign problem" - namely the interference between entries of a matrix with different signs. A major open question then remains whether one can provide an efficient algorithm for random matrices of mean 1/poly(n), whose conjectured #P-hardness is one of the baseline assumptions of the BosonSampling paradigm

The Computational Complexity of Linear Optics

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)|>=sqrt(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.Comment: 94 pages, 4 figure

Quantum Sampling Problems, BosonSampling and Quantum Supremacy

There is a large body of evidence for the potential of greater computational power using information carriers that are quantum mechanical over those governed by the laws of classical mechanics. But the question of the exact nature of the power contributed by quantum mechanics remains only partially answered. Furthermore, there exists doubt over the practicality of achieving a large enough quantum computation that definitively demonstrates quantum supremacy. Recently the study of computational problems that produce samples from probability distributions has added to both our understanding of the power of quantum algorithms and lowered the requirements for demonstration of fast quantum algorithms. The proposed quantum sampling problems do not require a quantum computer capable of universal operations and also permit physically realistic errors in their operation. This is an encouraging step towards an experimental demonstration of quantum algorithmic supremacy. In this paper, we will review sampling problems and the arguments that have been used to deduce when sampling problems are hard for classical computers to simulate. Two classes of quantum sampling problems that demonstrate the supremacy of quantum algorithms are BosonSampling and IQP Sampling. We will present the details of these classes and recent experimental progress towards demonstrating quantum supremacy in BosonSampling.Comment: Survey paper first submitted for publication in October 2016. 10 pages, 4 figures, 1 tabl

Computing the partition function of the Sherrington-Kirkpatrick model is hard on average

We establish the average-case hardness of the algorithmic problem of exact computation of the partition function associated with the Sherrington-Kirkpatrick model of spin glasses with Gaussian couplings and random external field. In particular, we establish that unless $P= \#P$, there does not exist a polynomial-time algorithm to exactly compute the partition function on average. This is done by showing that if there exists a polynomial time algorithm, which exactly computes the partition function for inverse polynomial fraction ($1/n^{O(1)}$) of all inputs, then there is a polynomial time algorithm, which exactly computes the partition function for all inputs, with high probability, yielding $P=\#P$. The computational model that we adopt is {\em finite-precision arithmetic}, where the algorithmic inputs are truncated first to a certain level $N$ of digital precision. The ingredients of our proof include the random and downward self-reducibility of the partition function with random external field; an argument of Cai et al. \cite{cai1999hardness} for establishing the average-case hardness of computing the permanent of a matrix; a list-decoding algorithm of Sudan \cite{sudan1996maximum}, for reconstructing polynomials intersecting a given list of numbers at sufficiently many points; and near-uniformity of the log-normal distribution, modulo a large prime $p$. To the best of our knowledge, our result is the first one establishing a provable hardness of a model arising in the field of spin glasses. Furthermore, we extend our result to the same problem under a different {\em real-valued} computational model, e.g. using a Blum-Shub-Smale machine \cite{blum1988theory} operating over real-valued inputs.Comment: 31 page

Simply Exponential Approximation of the Permanent of Positive Semidefinite Matrices

We design a deterministic polynomial time $c^n$ approximation algorithm for the permanent of positive semidefinite matrices where $c=e^{\gamma+1}\simeq 4.84$. We write a natural convex relaxation and show that its optimum solution gives a $c^n$ approximation of the permanent. We further show that this factor is asymptotically tight by constructing a family of positive semidefinite matrices