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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Aaronson, S., Arkhipov, A.: The computational complexity of linear optics. In: Fortnow, L., Vadhan, S.P. (eds.) Proc. 43rd Annual ACM Symposium on the Theory of Computing, pp. 333–342. ACM (2011)
Valiant, L.G.: The complexity of computing the permanent. Theor. Comp. Sci. 8(2), 189–201 (1979)
Shor, P.W.: Algorithms for quantum computation: Discrete logarithms and factoring. In: Proc. 35th Annual IEEE Symposium on Foundations of Computer Science, pp. 124–134. IEEE Computer Society (1994)
Broder, A.Z.: How hard is to marry at random (on the approximation of the permanent). In: Hartmanis, J. (ed.) Proc. 18th Annual ACM Symposium on the Theory of Computing, pp. 50–58. ACM (1986)
Jerrum, M., Sinclair, A.: Approximating the permanent. SIAM J. on Comput. 18(6), 1149–1178 (1989)
Jerrum, M., Sinclair, A., Vigoda, E.: A polynomial-time approximation algorithm for the permanent of a matrix with nonnegative entries. J. ACM 51(4), 671–697 (2004)
Gemmell, P., Lipton, R.J., Rubinfeld, R., Sudan, M., Wigderson, A.: Self-testing/correcting for polynomials and for approximate functions. In: Koutsougeras, C., Vitter, J.S. (eds.) Proc. 23rd Annual ACM Symposium on the Theory of Computing, pp. 32–42. ACM (1991)
Gemmell, P., Sudan, M.: Highly resilient correctors for polynomials. Inform. Process. Lett. 43(4), 169–174 (1992)
Cai, J.-Y., Pavan, A., Sivakumar, D.: On the Hardness of Permanent. In: Meinel, C., Tison, S. (eds.) STACS 1999. LNCS, vol. 1563, pp. 90–99. Springer, Heidelberg (1999)
Lund, C., Fortnow, L., Karloff, H., Nisan, N.: Algebraic methods for interactive proof systems. J. ACM 39(4), 859–868 (1992)
Arora, S., Khot, S.: Fitting algebraic curves to noisy data. J. Comp. Sys. Sci. 67(2), 325–340 (2003)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Arora, S., Bhattacharyya, A., Manokaran, R., Sachdeva, S. (2012). Testing Permanent Oracles – Revisited. In: Gupta, A., Jansen, K., Rolim, J., Servedio, R. (eds) Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques. APPROX RANDOM 2012 2012. Lecture Notes in Computer Science, vol 7408. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32512-0_31
Download citation
DOI: https://doi.org/10.1007/978-3-642-32512-0_31
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-32511-3
Online ISBN: 978-3-642-32512-0
eBook Packages: Computer ScienceComputer Science (R0)