Abstract
We present matching upper and lower bounds for the “weak” polynomial degree of the recursive Fourier sampling problem from quantum complexity theory. The degree bound is h + 1, where h is the order of recursion in the problem’s definition, and this bound is exponentially lower than the bound implied by the existence of a BQP algorithm for the problem. For the upper bound we exhibit a degree-h + 1 real polynomial that represents the problem on its entire domain. For the lower bound, we show that any non-zero polynomial agreeing with the problem, even on just its zero-inputs, must have degree at least h + 1. The lower bound applies to representing polynomials over any Field.
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References
Bernstein, E., Vazirani, U.: Quantum complexity theory. In: STOC 1993: Proceedings of the Twenty-Fifth Annual ACM symposium on Theory of computing, pp. 11–20. ACM Press, New York (1993)
Bernstein, E., Vazirani, U.: Quantum complexity theory. SIAM J. Comput. 26, 1411–1473 (1997)
Shor, P.W.: Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM Journal on Computing 26, 1484–1509 (1997)
Beals, R., Buhrman, H., Cleve, R., Mosca, M., de Wolf, R.: Quantum lower bounds by polynomials. In: IEEE Symposium on Foundations of Computer Science, pp. 352–361 (1998)
Aaronson, S.: Quantum lower bound for the collision problem. In: STOC 2002: Proceedings of the Thiry-Fourth Annual ACM symposium on Theory of Computing, pp. 635–642. ACM, New York (2002)
Aaronson, S.: Quantum lower bound for recursive fourier sampling. Quantum Information and Computation 3, 2–72 (2003)
Aaronson, S.: Bqp and the polynomial hierarchy. Technical Report ECCC TR09-104 (2009)
Furst, M.L., Saxe, J.B., Sipser, M.: Parity, circuits, and the polynomial-time hierarchy. Mathematical Systems Theory 17, 13–27 (1984)
Johnson, B.E.: Upper and Lower Bounds for Recursive Fourier Sampling. PhD thesis, University of California at Berkeley, Berkeley, CA, USA, chair - Leo Harrington (2008)
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Johnson, B. (2011). The Polynomial Degree of Recursive Fourier Sampling. In: van Dam, W., Kendon, V.M., Severini, S. (eds) Theory of Quantum Computation, Communication, and Cryptography. TQC 2010. Lecture Notes in Computer Science, vol 6519. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-18073-6_9
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DOI: https://doi.org/10.1007/978-3-642-18073-6_9
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