Space-Bounded Quantum Complexity

This paper investigates the computational power of space-bounded quantum Turing machines. The following facts are proved for space-constructible space bounds s satisfying s(n)=?(logn): 1.Any quantum Turing machine (QTM) running in space s can be simulated by an unbounded error probabilistic Turing machine (PTM) run- ning in space O(s). No assumptions on the probability of error or running time for the QTM are required, although it is assumed that all transition amplitudes of the QTM are rational. 2.Any PTM that runs in space s and halts absolutely (i.e., has finite worst-case running time) can be simulated by a QTM running in space O(s). If the PTM operates with bounded error, then the QTM may be taken to operate with bounded error as well, although the QTM may not halt absolutely in this case. In the case of unbounded error, the QTM may be taken to halt absolutely. We therefore have that unbounded error, space O(s) bounded quantum Turing machines and probabilistic Turing machines are equivalent in power and, furthermore, that any QTM running in space s can be simulated deterministically in NC2(2s)?DSPACE(s2)?DTIME(2O(s)). We also consider quantum analogues of nondeterministic and one-sided error probabilistic space-bounded classes and prove some simple facts regarding these classes.

[1]  Janos Simon,et al.  Space-bounded hierarchies and probabilistic computations , 1982, STOC '82.

[2]  Alston S. Householder,et al.  The Theory of Matrices in Numerical Analysis , 1964 .

[3]  Stephen A. Fenner,et al.  Determining acceptance possibility for a quantum computation is hard for the polynomial hierarchy , 1998, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[4]  Stephen A. Fenner,et al.  Determining Acceptance Possibility for a Quantum Computation is Hard for PH , 1998 .

[5]  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.

[6]  Hermann Jung.  On Probabilistic Time and Space , 1985, ICALP.

[7]  John T. Gill,et al.  Computational complexity of probabilistic Turing machines , 1974, STOC '74.

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

[9]  Allan Borodin,et al.  Parallel Computation for Well-Endowed Rings and Space-Bounded Probabilistic Machines , 1984, Inf. Control..

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

[11]  G. Brassard,et al.  Oracle Quantum Computing , 1992, Workshop on Physics and Computation.

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

[13]  Michael Saks,et al.  RSPACE(S) DSPACE(S^3/2) , 1995, FOCS 1995.

[14]  Christos H. Papadimitriou,et al.  Reversible Simulation of Space-Bounded Computations , 1995, Theor. Comput. Sci..

[15]  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.

[16]  H. S. Allen.  The Quantum Theory , 1928, Nature.

[17]  John Watrous,et al.  On the power of quantum finite state automata , 1997, Proceedings 38th Annual Symposium on Foundations of Computer Science.

[18]  Pierre McKenzie,et al.  Reversible Space Equals Deterministic Space (Extended Astract) , 1997 .

[19]  Michael E. Saks,et al.  Randomization and derandomization in space-bounded computation , 1996, Proceedings of Computational Complexity (Formerly Structure in Complexity Theory).

[20]  John Watrous,et al.  Relationships between quantum and classical space-bounded complexity classes , 1998, Proceedings. Thirteenth Annual IEEE Conference on Computational Complexity (Formerly: Structure in Complexity Theory Conference) (Cat. No.98CB36247).

[21]  Peter W. Shor,et al.  Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer , 1995, SIAM Rev..

[22]  Lov K. Grover,et al.  Quantum computation , 1999, Proceedings Twelfth International Conference on VLSI Design. (Cat. No.PR00013).

[23]  Eric Bach,et al.  Space-bounded quantum computation , 1998 .

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

[25]  Eric Allender,et al.  Relationships Among PL, #L, and the Determinant , 1996, RAIRO Theor. Informatics Appl..

[26]  Lov K. Grover.  Quantum Mechanics Helps in Searching for a Needle in a Haystack , 1997, quant-ph/9706033.

[27]  H. Jung.  Relationships between Probabilistic and Deterministic Tape Complexity , 1981, MFCS.

[28]  Charles H. Bennett.  Time/Space Trade-Offs for Reversible Computation , 1989, SIAM J. Comput..

[29]  Allan Borodin,et al.  On Relating Time and Space to Size and Depth , 1977, SIAM J. Comput..