Quantum measurements and the Abelian Stabilizer Problem

We present a polynomial quantum algorithm for the Abelian stabilizer problem which includes both factoring and the discrete logarithm. Thus we extend famous Shor’s results [7]. Our method is based on a procedure for measuring an eigenvalue of a unitary operator. Another application of this procedure is a polynomial quantum Fourier transform algorithm for an arbitrary finite Abelian group. The paper also contains a rather detailed introduction to the theory of quantum computation.

[1]  Gary L. Miller,et al.  Riemann's Hypothesis and tests for primality , 1975, STOC.

[2]  Charles H. Bennett,et al.  Logical reversibility of computation , 1973 .

[3]  Pérès,et al.  Reversible logic and quantum computers. , 1985, Physical review. A, General physics.

[4]  Peter W. Shor,et al.  Algorithms for Quantum Computation: Discrete Log and Factoring (Extended Abstract) , 1994, FOCS 1994.

[5]  D. Coppersmith.  An approximate Fourier transform useful in quantum factoring , 2002, quant-ph/0201067.

[6]  P. Benioff.  Quantum mechanical hamiltonian models of turing machines , 1982 .

[7]  D. Deutsch,et al.  Rapid solution of problems by quantum computation , 1992, Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences.

[8]  Andrew Chi-Chih Yao,et al.  Quantum Circuit Complexity , 1993, FOCS.

[9]  Barenco,et al.  Elementary gates for quantum computation. , 1995, Physical review. A, Atomic, molecular, and optical physics.

[10]  I PaulBenioff.  Quantum Mechanical Hamiltonian Models of Turing Machines , 1982 .

[11]  Umesh V. Vazirani,et al.  Quantum complexity theory , 1993, STOC.

[12]  D. Deutsch.  Quantum computational networks , 1989, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[13]  D. Deutsch.  Quantum theory, the Church–Turing principle and the universal quantum computer , 1985, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[14]  Dima Grigoriev,et al.  Testing shift-equivalence of polynomials using quantum machines , 1996, ISSAC '96.

[15]  Daniel R. Simon,et al.  On the power of quantum computation , 1994, Proceedings 35th Annual Symposium on Foundations of Computer Science.

[16]  S. Lloyd.  Quantum-Mechanical Computers , 1995 .