Approximating Permanent of Random Matrices with Vanishing Mean: Made Better and Simpler

Bibliographic Details
Title: Approximating Permanent of Random Matrices with Vanishing Mean: Made Better and Simpler
Authors: Ji, Zhengfeng, Jin, Zhihan, Lu, Pinyan
Publication Year: 2019
Collection: Computer Science
Quantum Physics
Subject Terms: Computer Science - Data Structures and Algorithms, Quantum Physics
More Details: The algorithm and complexity of approximating the permanent of a matrix is an extensively studied topic. Recently, its connection with quantum supremacy and more specifically BosonSampling draws special attention to the average-case approximation problem of the permanent of random matrices with zero or small mean value for each entry. Eldar and Mehraban (FOCS 2018) gave a quasi-polynomial time algorithm for random matrices with mean at least $1/\mathbf{\mathrm{polyloglog}} (n)$. In this paper, we improve the result by designing a deterministic quasi-polynomial time algorithm and a PTAS for random matrices with mean at least $1/\mathbf{\mathrm{polylog}}(n)$. We note that if it can be further improved to $1/\mathbf{\mathrm{poly}}(n)$, it will disprove a central conjecture for quantum supremacy. Our algorithm is also much simpler and has a better and flexible trade-off for running time. The running time can be quasi-polynomial in both $n$ and $1/\epsilon$, or PTAS (polynomial in $n$ but exponential in $1/\epsilon$), where $\epsilon$ is the approximation parameter.
Comment: 30 pages
Document Type: Working Paper
Access URL: http://arxiv.org/abs/1911.11962
Accession Number: edsarx.1911.11962
Database: arXiv
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