Quantum Circuit Complexity

We propose a complexity model of quantum circuits analogous to the standard (acyclic) Boolean circuit model. It is shown that any function computable in polynomial time by a quantum Turing machine has a polynomial-size quantum circuit. This result also enables us to construct a universal quantum computer which can simulate, with a polynomial factor slowdown, a broader class of quantum machines than that considered by E. Bernstein and U. Vazirani (1993), thus answering an open question raised by them. We also develop a theory of quantum communication complexity, and use it as a tool to prove that the majority function does not have a linear-size quantum formula. >

[1]  R. Feynman.  Simulating physics with computers , 1999 .

[2]  Andrew Chi-Chih Yao,et al.  Some complexity questions related to distributive computing(Preliminary Report) , 1979, STOC.

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

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

[5]  A. Holevo.  Bounds for the quantity of information transmitted by a quantum communication channel , 1973 .

[6]  Richard J. Lipton,et al.  Multi-party protocols , 1983, STOC.

[7]  R. Jozsa.  Characterizing classes of functions computable by quantum parallelism , 1991, Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences.

[8]  John E. Savage,et al.  Computational Work and Time on Finite Machines , 1972, JACM.

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

[10]  E. Specker,et al.  Lengths of Formulas and Elimination of Quantifiers I , 1968 .

[11]  Michael T. Goodrich,et al.  Parallel algorithms column 1: models of computation , 1993, SIGA.

[12]  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.

[13]  Avi Wigderson,et al.  Monotone Circuits for Connectivity Require Super-Logarithmic Depth , 1990, SIAM J. Discret. Math..

[14]  Michael J. Fischer,et al.  Relations Among Complexity Measures , 1979, JACM.

[15]  Gilles Brassard,et al.  The quantum challenge to structural complexity theory , 1992, [1992] Proceedings of the Seventh Annual Structure in Complexity Theory Conference.