In 2011, Aaronson gave a striking proof, based on quantum linear optics, that the problem of computing the permanent of a matrix is #P-hard. Aaronson's proof led naturally to hardness of approximation results for the permanent, and it was arguably simpler than Valiant's seminal proof of the same fact in 1979. Nevertheless, it did not show #P-hardness of the permanent for any class of matrices which was not previously known. In this paper, we present a collection of new results about matrix permanents that are derived primarily via these linear optical techniques. First, we show that the problem of computing the permanent of a real orthogonal matrix is #P-hard. Much like Aaronson's original proof, this implies that even a multiplicative approximation remains #P-hard to compute. The hardness result even translates to permanents of orthogonal matrices over the finite field F_{p^4} for p != 2, 3. Interestingly, this characterization is tight: in fields of characteristic 2, the permanent coincides with the determinant; in fields of characteristic 3, one can efficiently compute the permanent of an orthogonal matrix by a nontrivial result of Kogan. Finally, we use more elementary arguments to prove #P-hardness for the permanent of a positive semidefinite matrix. This result shows that certain probabilities of boson sampling experiments with thermal states are hard to compute exactly, despite the fact that they can be efficiently sampled by a classical computer.
Linear optical computing (LOC) with thermal light has recently gained attention because the problem is connected to the permanent of a Hermitian positive semidefinite matrix (HPSM), which is of importance in the computational complexity theory. Despite the several theoretical analyses on the computational structure of an HPSM in connection to LOC, the experimental demonstration and the computational complexity analysis via the linear optical system have not been performed yet. We present, herein, experimental LOC for estimating the permanent of an HPSM. From the linear optical experiments and theoretical analysis, we find that the LOC efficiency for a multiplicative error is dependent on the value of the permanent and that the lower bound of the computation time scales exponentially. Furthermore, our results are generalized and applied to LOC of permanents of unitary matrices, which can be implemented with a multiport quantum interferometer involving single photons at the input ports. We find that LOC with single photons, for the permanent estimation, is on average less efficient than the most efficient classical algorithm known to date, even in ideal conditions.
We study algorithms for approximating the permanent of a random matrix when the entries are slightly biased away from zero. This question is motivated by the goal of understanding the classical complexity of linear optics and \emph{boson sampling} (Aaronson and Arkhipov '11; Eldar and Mehraban '17). Barvinok's interpolation method enables efficient approximation of the permanent, provided one can establish a sufficiently large zero-free region for the polynomial
per(zJ+W), where
J is the all-ones matrix and
W is a random matrix with independent mean-zero entries.
We show that when the entries of
W are standard complex Gaussians, all zeros of the random polynomial
per(zJ+W) lie within a disk of radius
O~(n−1/3), which yields an approximation algorithm when the bias of the entries is
Ω~(n−1/3). Previously, there were no efficient algorithms at biases smaller than
1/polylog(n), and it was unknown whether there typically exist zeros
z with
∣z∣≥1. As a complementary result, we show that the bulk of the zeros, namely
(1−ε)n of them, have magnitude
Θ(n−1/2). This prevents our interpolation method from contradicting the conjectured average-case hardness of approximating the permanent. We also establish analogous zero-free regions for the hardcore model on general graphs with complex vertex fugacities. In addition, we prove universality results establishing zero-free regions for random matrices
W with i.i.d. subexponential entries.
Around 2002, Leonid Gurvits gave a striking randomized algorithm to approximate the permanent of an
n×n matrix A. The algorithm runs in
O(n2/ε2) time, and approximates
Per(A) to within
±ε∥A∥n additive error. A major advantage of Gurvits's algorithm is that it works for arbitrary matrices, not just for nonnegative matrices. \ This makes it highly relevant to quantum optics, where the permanents of bounded-norm complex matrices play a central role. (In particular,
n×n permanents arise as the transition amplitudes for
n identical photons.) Indeed, the existence of Gurvits's algorithm is why, in their recent work on the \textit{hardness} of quantum optics, Aaronson and Arkhipov (AA) had to talk about sampling problems rather than ordinary decision problems. In this paper, we improve Gurvits's algorithm in two ways. First, using an idea from quantum optics, we generalize the algorithm so that it yields a better approximation when the matrix A has either repeated rows or repeated columns. Translating back to quantum optics, this lets us classically estimate the probability of any outcome of an AA-type experiment -- even an outcome involving multiple photons "bunched" in the same mode -- at least as well as that probability can be estimated by the experiment itself. It also yields a general upper bound on the probabilities of "bunched" outcomes, which resolves a conjecture of Gurvits and might be of independent physical interest. Second, we use
ε-biased sets to derandomize Gurvits's algorithm, in the special case where the matrix A is nonnegative. More interestingly, we generalize the notion of
ε-biased sets to the complex numbers, construct "complex
ε-biased sets", then use those sets to derandomize even our generalization of Gurvits's algorithm to the case (again for nonnegative A) where some rows or columns are identical. Whether Gurvits's algorithm can be derandomized for general A remains an outstanding problem.
Recent demonstrations of quantum computational advantage have been driven largely by sampling problems. A prominent model, boson sampling, involves sampling from the output distribution of a linear optical network. However, its classical hardness hinges on two plausible yet less-studied conjectures: the average-case hardness of approximating Gaussian permanents, and the permanent anti-concentration conjecture (PACC). The PACC is a purely mathematical assertion regarding the distributional properties of random Gaussian matrices. While the typical magnitude of the permanent has been established for discrete random matrices, the complex Gaussian case, which governs transition amplitudes in linear optical networks, has remained open. Here, we establish a weak anti-concentration bound by upper-bounding the probability that a random Gaussian permanent is superexponentially smaller than its standard deviation. Tightening this bound to an inverse-polynomial fraction would prove the original PACC. As a corollary, we establish the typical magnitude of Gaussian permanents, on par with Tao and Vu's seminal result for Bernoulli matrices. Combined with the Aaronson-Arkhipov framework, our result implies that classically simulating boson sampling to within a superexponentially small total variation distance would collapse the polynomial hierarchy, assuming the remaining conjectures hold.