New Hardness Results for the Permanent Using Linear Optics

Abstract

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.

Computational advantage of quantum random sampling

“…This is the case if the function f (z) is holomorphic on the unit disc-for matrices with entries a i,j satisfying |a i,j −1| ≤ 0.19 (Barvinok, 2016b), satisfying δ < a i,j ≤ 1 (Barvinok, 2017), and diagonally dominant matrices (Barvinok, 2019). An interesting case is that of positive semi-definite matrices, as it has been shown that exactly computing the permanent of such matrices remains #P-hard (Grier and Schaeffer, 2018), but multiplicative-error approximation algorithms in BPP NP (Rahimi-Keshari et al, 2015) and with quasi-polynomial runtime (Anari et al, 2017;Barvinok, 2020) exist in some circumstances. Building on the approach of Barvinok, Eldar and Mehraban (2018) show that for random Gaussian matrices with non-zero but vanishing mean there is a quasi-polynomial time algorithm that approximates the permanent to within a multiplicative error.…”

Partial distinguishability and photon counting probabilities in linear multiport devices

“…( 6). Numerical computations of the generating function can turn out to be hard with increase of N and M, at least in some cases, as the example considered below, due to hardness of the matrix permanent of positive definite Hermitian matrices [29]. The same applies to the formulae for the photon counting probabilities, computational hardness of even approximate calculation of which is at the core of the computational advantage of the Boson Sampling [1,3].…”

Quantum-inspired permanent identities

“…Interestingly, the permanent appears in the expression of the output amplitudes of linear optical quantum computations with noninteracting bosons [4,5], as in the Boson Sampling model of quantum computation [6]. This connection has lead to several linear optical proofs of existing and new classical complexity results: computation of the permanent is #P-hard [7], (inverse polynomial) multiplicative estimation of the permanent of positive semidefinite matrices is in BPP NP [8], multiplicative estimation of the permanent of orthogonal matrices is #P-hard [9], and computation of a class of multidimensional integrals is #P-hard [10]. It has also lead to the introduction of a quantum-inspired classical algorithm for additive estimation of the permanent of positive semidefinite matrices [11].…”