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arXiv:1609.02416 · Physical Review A

Quantum-inspired algorithm for estimating the permanent of positive semidefinite matrices

Levon Chakhmakhchyan, Nicolas J. Cerf, Raúl García−Patrón

Published Aug 31, 2017• 22 citations
arXivHTMLPDF

Abstract

We construct a quantum-inspired classical algorithm for computing the permanent of Hermitian positive semidefinite matrices by exploiting a connection between these mathematical structures and the boson sampling model. Specifically, the permanent of a Hermitian positive semidefinite matrix can be expressed in terms of the expected value of a random variable, which stands for a specific photon-counting probability when measuring a linear-optically evolved random multimode coherent state. Our algorithm then approximates the matrix permanent from the corresponding sample mean and is shown to run in polynomial time for various sets of Hermitian positive semidefinite matrices, achieving a precision that improves over known techniques. This work illustrates how quantum optics may benefit algorithm development.

DOI
10.1103/physreva.96.022329

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We construct a quantum-inspired classical algorithm for computing the permanent of Hermitian positive semidefinite matrices by exploiting a connection between these mathematical structures and the boson sampling model. Specifically, the permanent of a Hermitian positive semidefinite matrix can be expressed in terms of the expected value of a random variable, which stands for a specific photon-counting probability when measuring a linear-optically evolved random multimode coherent state. Our algorithm then approximates the matrix permanent from the corresponding sample mean and is shown to run in polynomial time for various sets of Hermitian positive semidefinite matrices, achieving a precision that improves over known techniques. This work illustrates how quantum optics may benefit algorithm development.
We consider the problem of calculating the approximate value (estimate) of the permanent of a square matrix over the field of real numbers using a photonic quantum computer that sampling bosons from a given distribution. It is pointed out that it is possible to effectively solve this problem for the following three classes of matrices: symmetric matrices which eigenvalues do not exceed 1 in absolute value; orthogonal matrices; doubly stochastic matrices. For the last two classes, we are talking about evaluating the square of the permanent. The mathematical basis for solving the problem is the formula, which represents the permanent of the Hermitian matrix as the mathematical expectation of some random variable which values are equal to the products of the powers of the eigenvalues of the Hermitian matrix. And the corresponding probabilities with which these products are accepted as values of a random variable are also completely determined by the same Hermitian matrix. This circumstance allows using the methods of probability theory in order to solve the problem of permanent estimation.
Considering the problem of sampling from the output photon-counting probability distribution of a linear-optical network for input Gaussian states, we obtain results that are of interest from both quantum theory and the computational complexity theory point of view. We derive a general formula for calculating the output probabilities, and by considering input thermal states, we show that the output probabilities are proportional to permanents of positive-semidefinite Hermitian matrices. It is believed that approximating permanents of complex matrices in general is a #P-hard problem. However, we show that these permanents can be approximated with an algorithm in the BPP^{NP} complexity class, as there exists an efficient classical algorithm for sampling from the output probability distribution. We further consider input squeezed-vacuum states and discuss the complexity of sampling from the probability distribution at the output.
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 NNN, 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 (1.2N1.{2}^{N}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.
In order to find the outcome probabilities of quantum mechanical systems like the optical networks underlying Boson sampling, it is necessary to be able to compute the permanents of unitary matrices, a computationally hard task. Here we first discuss how to compute the permanent efficiently on a parallel computer, followed by algorithms which provide an exponential speed-up for sparse matrices and linear run times for matrices of limited bandwidth. The parallel algorithm has been implemented in a freely available software package, also available in an efficient serial version. As part of the timing runs for this package we set a new world record for the matrix order on which a permanent has been computed. Next we perform a simulation study of several conjectures regarding the distribution of the permanent for random matrices. Here we focus on permanent anti-concentration conjecture, which has been used to find the classical computational complexity of Boson sampling. We find a good agreement with the basic versions of these conjectures, and based on our data we propose refined versions of some of them. For small systems we also find noticeable deviations from a proposed strengthening of a bound for the number of photons in a Boson sampling system.
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.