Quantum estimation bound of Gaussian matrix permanent

TL;DR

The paper addresses the problem of estimating matrix permanents, which is #P-hard in general and central to boson sampling. It introduces a Glynn–Kan–Huh (GKH) reformulation that expresses Per(A) as an average of $N$-th powers of a binary function and then maps this to a quantum overlap involving a quantum Ising Hamiltonian, enabling an additive-error estimator. For real Gaussian matrices, the authors derive an average additive-error bound of the form $ ext{ε}_T le ext{ε} ( oot 2 ceil{eN})^{N}$, which improves upon Gurvits' classical sampler by an exponential factor and offers a quadratic speedup when using QPE-based amplitude estimation. While the approach cannot achieve the multiplicative-error bound without implying a collapse of the polynomial hierarchy, it provides a concrete quantum route to estimating matrix permanents and suggests extensions to related matrix functions and circuit-structure analyses. Overall, the work links classical permanence formulas to quantum estimation techniques, quantifies additive-error performance, and outlines directions for future quantum algorithms in linear-algebraic tasks.

Abstract

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 newly developed 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 $N$, the quantum estimator can approximate the matrix permanent with an additive error smaller than $ε(\sqrt{\mathrm{e}N})^{N}$, where $ε$ is the estimation precision. In contrast, Gurvits' classical sampling algorithm has an estimation error of $ε(2\sqrt{N})^{N}$, which is exponentially larger ($1.2^{N}$) than the quantum method. As expected, the quantum additive error bound fails to reach the multiplicative error bound of $(2πN)^{1/4}ε(\sqrt{N/\mathrm{e}})^{N}$. Additionally, the quantum permanent estimator can be up to quadratically faster than the classical estimator when using quantum phase estimation-based amplitude estimation.

Figures (1)

• Figure 1: Quantum circuits for evaluating a real part of the overlap integral using the Hadamard test. (a) The Hadamard test circuit for the additive error estimation of the real part of the quantum overlap integral $\mathrm{Re}\braket{\phi\vert U\vert\phi}$. The real part of the quantum overlap integral can be obtained via the difference between the probabilities of getting $0$ and $1$ in the ancilla qubit (denoted as $c$) measurement. The detailed quantum circuit implementation is presented: (b) the circuit synthesis of the controlled-$R_{ZZ}(\theta)=\exp(-\mathrm{i}\tfrac{\theta}{2} Z_{1}Z_{2})$ gate and (c) the circuit synthesis of controlled-$R_{Z}(\theta)$, where $R_{Z}(\theta)=\exp(-\mathrm{i}\tfrac{\theta}{2} Z)$.