Generalizing and Derandomizing Gurvitss Approximation Algorithm for the Permanent

Around 2002, Leonid Gurvits gave a striking randomized algorithm to approximate the permanent of an nn 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.

💡 Research Summary

The paper revisits Leonid Gurvits’s 2002 randomized algorithm for approximating the permanent of an arbitrary n × n matrix A. Gurvits’s method runs in O(n²/ε²) time and guarantees an additive error of ε·‖A‖ⁿ, where ‖A‖ is the maximum absolute entry of A. While powerful because it works for any complex matrix, the original algorithm has two notable limitations: it does not exploit structural repetitions that frequently appear in quantum‑optics experiments, and it relies on randomness, which limits its applicability in deterministic settings.

The authors address both issues. First, they observe that many Boson‑Sampling (Aaronson‑Arkhipov) experiments produce outcomes where several photons occupy the same mode, which translates mathematically into rows or columns of A being repeated. By modifying the sampling step to treat duplicated rows (or columns) as a single random variable, they derive a generalized version of Gurvits’s algorithm that achieves a tighter error bound: the additive error becomes ε·‖A‖^{n‑(k‑1)} when a row (or column) appears k times. Consequently, for matrices with multiple repeated rows and columns the bound improves dramatically, matching or surpassing the statistical precision that the physical experiment itself can provide. As a by‑product, the paper proves a conjecture of Gurvits concerning an upper bound on the permanent of “bunched” matrices, showing that the permanent of a matrix with repeated rows/columns cannot exceed ‖A‖^{n‑(k‑1)} in absolute value. This result has independent interest for quantum‑optics theory, where such bounds are used to argue about the hardness of sampling.

The second contribution concerns derandomization. The classic tool for derandomizing Gurvits’s algorithm in the non‑negative case is an ε‑biased set: a small deterministic collection of sign vectors that mimics the uniform distribution for all linear tests. The authors extend this concept to the complex domain. They define a “complex ε‑biased set” as a finite set of complex phases such that the expectation of any non‑trivial complex linear form over the set deviates from zero by at most ε. By exploiting characters of finite abelian groups and Fourier analysis, they construct such sets of size O(n/ε²). Substituting the random sign vectors in Gurvits’s procedure with the deterministic complex ε‑biased set yields a fully deterministic algorithm with the same O(n²/ε²) runtime and the same additive error guarantee, provided that A is entry‑wise non‑negative.

Importantly, the authors show that the same derandomization technique applies to their generalized multi‑row/multi‑column algorithm, again under the non‑negative assumption. Thus, for any non‑negative matrix that may have repeated rows or columns, one obtains a deterministic algorithm that approximates the permanent with the improved error bound derived earlier.

The paper leaves open the problem of derandomizing Gurvits’s algorithm for general complex matrices. The current complex ε‑biased construction relies on the non‑negativity of entries to control cancellations; extending it to arbitrary phase patterns would likely require new number‑theoretic or harmonic‑analysis tools.

In summary, the work makes two substantive advances: (1) a structural generalization that leverages row/column repetitions to obtain a substantially tighter additive error, and (2) a novel derandomization framework based on complex ε‑biased sets that yields deterministic algorithms for non‑negative matrices, including the new generalized version. These contributions deepen the theoretical understanding of permanent approximation, bridge a gap between classical simulation and quantum‑optics experiments, and open new avenues for deterministic algorithms in related combinatorial and quantum‑computational problems.