The permanent of a matrix has numerous applications but is notoriously hard to compute. While nonnegative matrices admit polynomial approximation schemes based on rapidly mixing Markov chains, the known practical estimators of the permanent rely on importance or rejection sampling. We advance the rejection sampling approach, which provides probabilistic accuracy guarantees, unlike importance sampling. Specifically, we give a novel class of nesting upper bounds and a simple preprocessing method that, in comparison to previous works, enable faster sampling with better acceptance rate; we demonstrate order-of-magnitude improvements with both theoretical and empirical analyses. In addition, we display instances on which our approximation scheme is competitive against state-of-the-art importance sampling based estimators.
The permanent of a matrix has numerous applications but is notoriously hard to compute. While nonnegative matrices admit polynomial approximation schemes based on rapidly mixing Markov chains, the known practical estimators of the permanent rely on importance or rejection sampling. We advance the rejection sampling approach, which provides probabilistic accuracy guarantees, unlike importance sampling. Specifically, we give a novel class of nesting upper bounds and a simple preprocessing method that, in comparison to previous works, enable faster sampling with better acceptance rate; we demonstrate order-of-magnitude improvements with both theoretical and empirical analyses. In addition, we display instances on which our approximation scheme is competitive against state-of-the-art importance sampling based estimators.
Exact calculation and even multiplicative error estimation of matrix permanent are challenging for both classical and quantum computers. Regarding the permanents of random Gaussian matrices, the additive error estimation is closely linked to boson sampling and achieving multiplicative error estimation requires exponentially many samplings. Our formula for matrix permanents and its corresponding quantum expression have enabled better estimation of the average additive error for random Gaussian matrices compared to Gurvits' classical sampling algorithm. The well-known Ryser formula has been converted into a quantum permanent estimator. When dealing with real random Gaussian square matrices of size N, the quantum estimator can approximate the matrix permanent with an additive error smaller than $\ensuremath{\epsilon}{(\sqrt{\mathrm{e}N})}^{N}$, where $\ensuremath{\epsilon}$ is the estimation precision. In contrast, Gurvits' classical sampling algorithm has an estimation error of $\ensuremath{\epsilon}{(2\sqrt{N})}^{N}$, which is exponentially larger (1.2N) than the quantum method. As expected, the quantum additive error bound fails to reach the multiplicative error bound of ${(2\ensuremath{\pi}N)}^{1/4}\ensuremath{\epsilon}{(\sqrt{N/\mathrm{e}})}^{N}$. Additionally, the quantum permanent estimator can be up to quadratically faster than the classical estimator when using quantum phase estimation-based amplitude estimation.
The matrix permanent belongs to the complexity class #P-Complete. It is generally believed to be computationally infeasible for large problem sizes, and significant research has been done on approximation algorithms for the matrix permanent. We present an implementation and detailed runtime analysis of one such Markov Chain Monte Carlo (MCMC) based Fully Polynomial Randomized Approximation Scheme (FPRAS) for the matrix permanent, which has previously only been described theoretically and with big-Oh runtime analysis. We demonstrate by analysis and experiment that the constant factors hidden by previous big-Oh analyses result in computational infeasibility.
Since the celebrated work of Jerrum, Sinclair, and Vigoda [J. ACM, 51 (2004), pp. 671–697], we have known that the permanent of a matrix with entries in {0,1} can be approximated in randomized polynomial time by using a rapidly mixing Markov chain to sample perfect matchings of a bipartite graph. A separate strand of the literature has pursued the possibility of an alternate, algebraic polynomial-time approximation scheme. These schemes work by replacing each 1 with a random element of an algebra A and considering the determinant of the resulting matrix. In the case where A is noncommutative, this determinant can be defined in several ways. We show that for some estimators based on the conventional determinant, the critical ratio of the second moment to the square of the first—and therefore the number of trials we need to obtain a good estimate of the permanent—is (1+O(1/d))n when A is the algebra of d×d matrices. These results can be extended to group algebras and semisimple algebras in general. We also study the symmetrized determinant of Barvinok, showing that the resulting estimator has small variance when d is large enough. However, if d is constant—the only case in which an efficient algorithm is known—we show that the critical ratio exceeds 2n/nO(d). Thus our results do not provide a new polynomial-time approximation scheme for the permanent. Indeed, they suggest that the algebraic approach to approximating the permanent faces significant obstacles. We obtain these results using diagrammatic techniques in which we express matrix products as contractions of tensor products. When these matrices are chosen randomly according to the Gaussian distribution, we can evaluate the trace of these products in terms of the cycle structure of a suitably random permutation. In the symmetrized case, our estimates are then derived by a connection with the character theory of the symmetric group.
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.
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(n^2/eps^2) time, and approximates Per(A) to within eps*||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. Indeed, the existence of Gurvits's algorithm is why, in their recent work on the hardness of quantum optics, Aaronson and Arkhipov (AA) had to talk about sampling problems rather than estimation 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. (This does not, of course, let us solve the AA sampling problem.) 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 eps-biased sets to derandomize Gurvits's algorithm, in the special case where the matrix A is nonnegative. More interestingly, we generalize the notion of eps-biased sets to the complex numbers, construct "complex eps-biased sets," then use those sets to derandomize even our generalization of Gurvits's algorithm to the multirow/multicolumn case (again for nonnegative A). Whether Gurvits's algorithm can be derandomized for general A remains an outstanding problem.