[Paper Review] Generalizing and Derandomizing Gurvits's Approximation Algorithm for the Permanent

Scott Aaronson, Travis Hance|arXiv (Cornell University)|Nov 30, 2012

Sparse and Compressive Sensing Techniques4 citations

TL;DR

This paper generalizes and derandomizes Gurvits's randomized algorithm for approximating the permanent of an $n \times n$ matrix to within $\pm \varepsilon \|A\|^n$ additive error in $O(n^2 / \varepsilon^2)$ time. It improves the approximation for matrices with repeated rows or columns using bosonic creation operator analogies, and introduces complex $\varepsilon$-biased sets to derandomize the algorithm for nonnegative matrices, achieving polynomial-time deterministic approximation with $O(\log n + \log 1/\varepsilon)$ randomness.

ABSTRACT

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.

Motivation & Objective

• To improve Gurvits's randomized algorithm for approximating the permanent of arbitrary complex matrices, especially in cases with repeated rows or columns.
• To generalize the algorithm to handle matrices formed by repeating columns of a smaller matrix $B$, yielding tighter additive error bounds.
• To derandomize Gurvits's algorithm for nonnegative matrices using $\varepsilon$-biased sets, and extend this to complex $\varepsilon$-biased sets for the generalized case.
• To provide a classical simulation tool for quantum optics experiments involving photon bunching, particularly in the context of Aaronson-Arkhipov sampling problems.
• To resolve a conjecture on upper bounds for probabilities of bunched photon outcomes in linear optical networks.

Proposed method

• Generalizes Gurvits's algorithm by treating formal variables as bosonic creation operators, replacing $x_i$ with $\sqrt{s_i} x_i$ to model repeated columns.
• Derives a new upper bound on $|\operatorname{Per}(A)|$ as $\frac{s_1! \cdots s_k!}{\sqrt{s_1^{s_1} \cdots s_k^{s_k}}} \|B\|^n$ for matrices $A$ with $s_i$ copies of each column of $B$, improving error bounds for repeated structures.
• Introduces the concept of complex $\varepsilon$-biased sets, extending real $\varepsilon$-biased sets to the complex unit circle to enable derandomization.
• Uses deterministic amplification via complex $\varepsilon$-biased sets to ensure the expected value of a product of complex roots of unity has norm at most $\varepsilon/2$, enabling error control.
• Applies the generalized algorithm to estimate transition amplitudes in linear optical networks, particularly for outcomes with multiple photons in the same mode.
• Employs a hybrid randomization and deterministic construction: $\ell = O(\log n + \log 1/\varepsilon)$ complex roots of unity are selected deterministically, and $d_j \in \{0,1\}$ are chosen uniformly at random.

Experimental results

Research questions

• RQ1 Can Gurvits's algorithm be improved for matrices with repeated rows or columns, particularly in the context of quantum optics where such structures model photon bunching?
• RQ2 Can the randomized approximation of the permanent for nonnegative matrices be derandomized using $\varepsilon$-biased sets?
• RQ3 Can the notion of $\varepsilon$-biased sets be generalized to the complex unit circle to support derandomization of the generalized algorithm?
• RQ4 Is there a structural characterization of unitary or subunitary matrices for which the permanent is non-negligibly large, enabling deterministic estimation?
• RQ5 Can the additive error bound for estimating linear-optical amplitudes be improved to $\pm 1/\operatorname{poly}(n)$ in polynomial time?

Key findings

• The generalized algorithm approximates $\operatorname{Per}(A)$ to within $\pm \varepsilon \cdot \frac{s_1! \cdots s_k!}{\sqrt{s_1^{s_1} \cdots s_k^{s_k}}} \|B\|^n$ additive error in $O(nk / \varepsilon^2)$ time, improving on Gurvits’s original bound when $A$ has repeated columns.
• The paper proves a new upper bound $|\operatorname{Per}(A)| \leq \frac{s_1! \cdots s_k!}{\sqrt{s_1^{s_1} \cdots s_k^{s_k}}} \|B\|^n$, resolving a conjecture by Gurvits and providing a physical upper limit on bunched photon outcome probabilities.
• The authors construct complex $\varepsilon$-biased sets over the unit circle with $O(\log n + \log 1/\varepsilon)$ elements, enabling derandomization of the generalized algorithm.
• The derandomized algorithm for nonnegative matrices uses only $O(\log n + \log 1/\varepsilon)$ random bits, achieving $\varepsilon$-error control with high probability.
• The method enables classical estimation of any linear-optical amplitude, including those with bunched photons, to within the same additive error as the quantum experiment itself.
• The construction of complex $\varepsilon$-biased sets relies on a novel amplification technique using $\pi/8$-strong roots, differing from classical $\varepsilon$-biased set constructions that rely on single $-1$ values.