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 ($1.{2}^{N}$) 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.
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 \ensuremathϵ(eN)N, where \ensuremathϵ is the estimation precision. In contrast, Gurvits' classical sampling algorithm has an estimation error of \ensuremathϵ(2N)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\ensuremathπN)1/4\ensuremathϵ(N/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.
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 present a randomized algorithm for estimating the permanent of an M×M real matrix A up to an additive error. We do this by viewing the permanent perm(A) of A as the expectation of a product of centered joint Gaussian random variables with a particular covariance matrix C. The algorithm outputs the empirical mean SN of this product after sampling N times. Our algorithm runs in total time O(M3+M2N+MN) with failure probability \begin{equation*} P(|S_{N}-\text{perm}(A)| > t) \leq \frac{3^{M}}{t^{2}N} \prod^{2M}_{i=1} C_{ii}. \end{equation*} In particular, we can estimate perm(A) to an additive error of ε(32M∏i=12MCii) in polynomial time. We compare to a previous procedure due to Gurvits. We discuss how to find a particular C using a semidefinite program and a relation to the Max-Cut problem and cut-norms.
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.
Computing permanents of matrices are known to be a classically hard problem that the computational cost grows exponentially with the size of the matrix increases. So far, there exist a few classical algorithms to compute the matrix permanents in deterministic and in randomized ways. By exploiting the series expansion of products of boson operators regarding collective boson operators, a generalized algorithm for computing permanents is developed that the algorithm can handle the arbitrary matrices with repeated columns and rows. In a particular case, the formula is reduced to Glynn's form. Not only the algorithm can be used for a deterministic direct calculation of the matrix permanent but also can be expressed as a sampling problem like Gurvits's randomized algorithm.
Computing the distribution of permanents of random matrices has been an outstanding open problem for several decades. In quantum computing, "anti-concentration" of this distribution is an unproven input for the proof of hardness of the task of boson-sampling. We study permanents of random i.i.d. complex Gaussian matrices, and more broadly, submatrices of random unitary matrices. Using a hybrid representation-theoretic and combinatorial approach, we prove strong lower bounds for all moments of the permanent distribution. We provide substantial evidence that our bounds are close to being tight and constitute accurate estimates for the moments. Let U(d)k×k be the distribution of k×k submatrices of d×d random unitary matrices, and Gk×k be the distribution of k×k complex Gaussian matrices. (1) Using the Schur-Weyl duality (or the Howe duality), we prove an expansion formula for the 2t-th moment of ∣Perm(M)∣ when M is drawn from U(d)k×k or Gk×k. (2) We prove a surprising size-moment duality: the 2t-th moment of the permanent of random k×k matrices is equal to the 2k-th moment of the permanent of t×t matrices. (3) We design an algorithm to exactly compute high moments of the permanent of small matrices. (4) We prove lower bounds for arbitrary moments of permanents of matrices drawn from Gk×k or U(k), and conjecture that our lower bounds are close to saturation up to a small multiplicative error. (5) Assuming our conjectures, we use the large deviation theory to compute the tail of the distribution of log-permanent of Gaussian matrices for the first time. (6) We argue that it is unlikely that the permanent distribution can be uniquely determined from the integer moments and one may need to supplement the moment calculations with extra assumptions to prove the anti-concentration conjecture.
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.