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Abstract

Suppose we are given an oracle that claims to approximate the permanent for most matrices X, where X is chosen from the Gaussian ensemble (the matrix entries are i.i.d. univariate complex Gaussians). Can we test that the oracle satisfies this claim? This paper gives a polynomial-time algorithm for the task.

The oracle-testing problem is of interest because a recent paper of Aaronson and Arkhipov showed that if there is a polynomial-time algorithm for simulating boson-boson interactions in quantum mechanics, then an approximation oracle for the permanent (of the type described above) exists in BPP NP . Since computing the permanent of even 0/1 matrices is #P-complete, this seems to demonstrate more computational power in quantum mechanics than Shor’s factoring algorithm does. However, unlike factoring, which is in NP, it was unclear previously how to test the correctness of an approximation oracle for the permanent, and this is the contribution of the paper.

The technical difficulty overcome here is that univariate polynomial self-correction, which underlies similar oracle-testing algorithms for permanent over \(\textit{finite fields}\) —and whose discovery led to a revolution in complexity theory—does not seem to generalize to complex (or even, real) numbers. We believe that this tester will motivate further progress on understanding the permanent of Gaussian matrices.

Keywords

Additive Approximation Oracle Access Constant Factor Approximation Algorithm Gaussian Ensemble Interactive Proof System 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Sanjeev Arora
    • 1
  • Arnab Bhattacharyya
    • 1
  • Rajsekar Manokaran
    • 1
  • Sushant Sachdeva
    • 1
  1. 1.Department of Computer Science and Center for Computational IntractabilityPrinceton UniversityUSA

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