(MATH) We study approximation algorithms for the permanent of an n x n (0,1) matrix A based on the following simple idea: obtain a random matrix B by replacing each 1-entry of A independently by ± e, where e is a random basis element of a suitable algebra; then output |det(B)|2. This estimator is always unbiased, but it may have exponentially large variance. In our first main result we show that, if we take the algebra to be a Clifford algebra of dimension polynomial in n, then we get an estimator with small variance. Hence only a constant number of trials suffices to estimate the permanent to good accuracy. The idea of using Clifford algebras is a natural extension of earlier work by Godsil and Gutman, Karmarkar et al., and Barvinok, who used the real numbers, complex numbers and quaternions respectively.(MATH) The above result implies that, in principle, this approach gives a fully-polynomial randomized approximation scheme for the permanent, provided |det(B)|2 can be efficiently computed in the Clifford algebras. Since these algebras are non-commutative it is not clear how to do this. However, our second main result shows how to compute in polynomial time an estimator with the same mean and variance over the 4-dimensional algebra (which is the quaternions, and is non-commutative); in addition to providing some hope that the computations can be performed in higher dimensions, this quaternion algorithm provides an exponential improvement in the variance over that of the 2-dimensional complex version studied by Karmarkar et al.
(MATH) We study approximation algorithms for the permanent of an n x n (0,1) matrix A based on the following simple idea: obtain a random matrix B by replacing each 1-entry of A independently by ± e, where e is a random basis element of a suitable algebra; then output |det(B)|2. This estimator is always unbiased, but it may have exponentially large variance. In our first main result we show that, if we take the algebra to be a Clifford algebra of dimension polynomial in n, then we get an estimator with small variance. Hence only a constant number of trials suffices to estimate the permanent to good accuracy. The idea of using Clifford algebras is a natural extension of earlier work by Godsil and Gutman, Karmarkar et al., and Barvinok, who used the real numbers, complex numbers and quaternions respectively.(MATH) The above result implies that, in principle, this approach gives a fully-polynomial randomized approximation scheme for the permanent, provided |det(B)|2 can be efficiently computed in the Clifford algebras. Since these algebras are non-commutative it is not clear how to do this. However, our second main result shows how to compute in polynomial time an estimator with the same mean and variance over the 4-dimensional algebra (which is the quaternions, and is non-commutative); in addition to providing some hope that the computations can be performed in higher dimensions, this quaternion algorithm provides an exponential improvement in the variance over that of the 2-dimensional complex version studied by Karmarkar et al.
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.
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.
The permanent is #P-hard to compute exactly on average for natural random matrices including matrices over finite fields or Gaussian ensembles. Should we expect that it remains #P-hard to compute on average if we only care about approximation instead of exact computation? In this work we take a first step towards resolving this question: We present a quasi-polynomial time deterministic algorithm for approximating the permanent of a typical n × n random matrix with unit variance and vanishing mean μ = O(ln ln n)-1/8to within inverse polynomial multiplicative error. (alternatively, one can achieve permanent approximation for matrices with mean μ = 1/polylog(n) in time 2n(ε), for arbitrarily small ε>0). The proposed algorithm significantly extends the regime of matrices for which efficient approximation of the permanent is known. This is because unlike previous algorithms which require a stringent correlation between the signs of the entries of the matrix [1], [2] it can tolerate random ensembles in which this correlation is negligible (albeit non-zero). Among important special cases we note: 1) Biased Gaussian: each entry is a complex Gaussian with unit variance 1 and mean μ. 2) Biased Bernoulli: each entry is -1 + μ with probability 1/2, and 1 with probability 1/2. These results counter the common intuition that the difficulty of computing the permanent, even approximately, stems merely from our inability to treat matrices with many opposing signs. The Gaussian ensemble approaches the threshold of a conjectured hardness [3] of computing the permanent of a zero mean Gaussian matrix. This conjecture is one of the baseline assumptions of the BosonSampling paradigm that has received vast attention in recent years in the context of quantum supremacy experiments. We furthermore show that the permanent of the biased Gaussian ensemble is #P-hard to compute exactly on average. To our knowledge, this is the first natural example of a counting problem that becomes easy only when average case analysis and approximation are combined. On a technical level, our approach stems from a recent approach taken by Barvinok [1], [4], [5], [6] who used Taylor series approximation of the logarithm of a certain univariate polynomial related to the permanent. Our main contribution is to introduce an average-case analysis of such related polynomials. We complement our approach with a new technique for iteratively computing a Taylor series approximation of a function that is analytical in the vicinity of a curve in the complex plane. This method can be viewed as a computational version of analytic continuation in complex analysis.
The algorithm and complexity of approximating the permanent of a matrix is an extensively studied topic. Recently, its connection with quantum supremacy and more specifically BosonSampling draws a special attention to the average-case approximation problem of the permanent of random matrices with zero or small mean value for each entry. Eldar and Mehraban (FOCS 2018) gave a quasi-polynomial time algorithm for random matrices with mean at least 1/polyloglog(n). In this paper, we improve the result by designing a deterministic quasi-polynomial time algorithm and a PTAS for random matrices whose module of mean is at least 1/ polylog(n). We note that if the algorithm can be further improved to work with a mean value that is a sufficiently small 1/poly(n), it will disprove a central conjecture for quantum supremacy. Our algorithm is also much simpler and has a better and flexible trade-off for running time. The running time can be quasi-polynomial in both n and 1/∊, or PTAS (polynomial in n but exponential in 1/∊), where ∊ is the approximation parameter.
In this work, we study the computational complexity of quantum determinants, a q-deformation of matrix permanents: Given a complex number q on the unit circle in the complex plane and an n×n matrix X, the q-permanent of X is defined as Perq(X)=∑σ∈Snqℓ(σ)X1,σ(1)…Xn,σ(n), where ℓ(σ) is the inversion number of permutation σ in the symmetric group Sn on n elements. The function family generalizes determinant and permanent, which correspond to the cases q=−1 and q=1 respectively. For worst-case hardness, by Liouville's approximation theorem and facts from algebraic number theory, we show that for primitive m-th root of unity q for odd prime power m=pk, exactly computing q-permanent is ModpP-hard. This implies that an efficient algorithm for computing q-permanent results in a collapse of the polynomial hierarchy. Next, we show that computing q-permanent can be achieved using an oracle that approximates to within a polynomial multiplicative error and a membership oracle for a finite set of algebraic integers. From this, an efficient approximation algorithm would also imply a collapse of the polynomial hierarchy. By random self-reducibility, computing q-permanent remains to be hard for a wide range of distributions satisfying a property called the strong autocorrelation property. Specifically, this is proved via a reduction from 1-permanent to q-permanent for O(1/n2) points z on the unit circle. Since the family of permanent functions shares common algebraic structure, various techniques developed for the hardness of permanent can be generalized to q-permanents.
We present an algorithm for efficiently approximating of qubit unitaries over gate sets derived from totally definite quaternion algebras. It achieves ε-approximations using circuits of length O(log(1/ε)), which is asymptotically optimal. The algorithm achieves the same quality of approximation as previously-known algorithms for Clifford+T [arXiv:1212.6253], V-basis [arXiv:1303.1411] and Clifford+π/12 [arXiv:1409.3552], running on average in time polynomial in O(log(1/ε)) (conditional on a number-theoretic conjecture). Ours is the first such algorithm that works for a wide range of gate sets and provides insight into what should constitute a "good" gate set for a fault-tolerant quantum computer. Joint work with Alex Bocharov, Martin Roetteler, Jon Yard: http://arxiv.org/abs/1510.03888