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arXiv:quant-ph/0406127

Permanents in linear optical networks

Stefan Scheel

Published Jun 18, 2004• 94 citations
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Abstract

We develop an abstract look at linear optical networks from the viewpoint of combinatorics and permanents. In particular we show that calculation of matrix elements of unitarily transformed photonic multi-mode states is intimately linked to the computation of permanents. An implication of this remarkable fact is that all calculations that are based on evaluating matrix elements are generically computationally hard. Moreover, quantum mechanics provides simpler derivations of certain matrix analysis results which we exemplify by showing that the permanent of any unitary matrix takes its values across the unit disk in the complex plane.

DOI
10.48550/arxiv.quant-ph/0406127

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We develop an abstract look at linear optical networks from the viewpoint of combinatorics and permanents. In particular we show that calculation of matrix elements of unitarily transformed photonic multi-mode states is intimately linked to the computation of permanents. An implication of this remarkable fact is that all calculations that are based on evaluating matrix elements are generically computationally hard. Moreover, quantum mechanics provides simpler derivations of certain matrix analysis results which we exemplify by showing that the permanent of any unitary matrix takes its values across the unit disk in the complex plane.
We give new evidence that quantum computers -- moreover, rudimentary quantum computers built entirely out of linear-optical elements -- cannot be efficiently simulated by classical computers. In particular, we define a model of computation in which identical photons are generated, sent through a linear-optical network, then nonadaptively measured to count the number of photons in each mode. This model is not known or believed to be universal for quantum computation, and indeed, we discuss the prospects for realizing the model using current technology. On the other hand, we prove that the model is able to solve sampling problems and search problems that are classically intractable under plausible assumptions. Our first result says that, if there exists a polynomial-time classical algorithm that samples from the same probability distribution as a linear-optical network, then P^#P=BPP^NP, and hence the polynomial hierarchy collapses to the third level. Unfortunately, this result assumes an extremely accurate simulation. Our main result suggests that even an approximate or noisy classical simulation would already imply a collapse of the polynomial hierarchy. For this, we need two unproven conjectures: the "Permanent-of-Gaussians Conjecture", which says that it is #P-hard to approximate the permanent of a matrix A of independent N(0,1) Gaussian entries, with high probability over A; and the "Permanent Anti-Concentration Conjecture", which says that |Per(A)|>=sqrt(n!)/poly(n) with high probability over A. We present evidence for these conjectures, both of which seem interesting even apart from our application. This paper does not assume knowledge of quantum optics. Indeed, part of its goal is to develop the beautiful theory of noninteracting bosons underlying our model, and its connection to the permanent function, in a self-contained way accessible to theoretical computer scientists.
The permanent is pivotal to both complexity theory and combinatorics. In quantum computing, the permanent appears in the expression of output amplitudes of linear optical computations, such as in the Boson Sampling model. Taking advantage of this connection, we give quantum-inspired proofs of many existing as well as new remarkable permanent identities. Most notably, we give a quantum-inspired proof of the MacMahon master theorem as well as proofs for new generalizations of this theorem. Previous proofs of this theorem used completely different ideas. Beyond their purely combinatorial applications, our results demonstrate the classical hardness of exact and approximate sampling of linear optical quantum computations with input cat states.
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.
Abstract The particular complexity of linear quantum optical networks has received certain deserved attention recently due to the possible implications for the theory of quantum computation. Two relevant boson models are discussed in the presented work. The symmetric product of the Hilbert spaces produces a rather abstract model; the second one is obtained by quantization of the harmonic oscillator. In contrast to the considered bosonic processes, the so-called ‘fermionic linear optics’ is effectively simulated on a classical computer. The comparison of the bosonic and fermionic case clarifies the controversy, and the more elaborate oscillator model provides a deeper analogy.
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.