Quantum conservative many-valued computing

We extend the basic principles and results of conservative logic to include the main features of many-valued logics with a finite number of truth values. Different approaches to many-valued logics are examined in order to determine some possible functionally complete sets of logic connectives. As a result, we describe some possible finite-valued gates which realize a functionally complete set of fundamental connectives. One of the purposes of this work is to show that the framework of reversible and conservative computation can be extended toward some non-classical ''reasoning environments'', originally proposed to deal with propositions which embed imprecise and/or uncertain information, that are usually based upon many-valued and modal logics. We also describe a possible quantum realization of the proposed gates, using creation and annihilation operators. In such realization the gates are expressed as formulas that are obtained using three techniques: a ''brute force'' technique, an extension of the Conditional Quantum Control method introduced by Barenco, Deutsch, Ekert and Jozsa [Conditional quantum control and logic gates, Phys. Rev. Lett. 74 (1995) 4083], and a new technique that we call the Constants method. We show the Constants method allows one to reduce the number of local operators in the formulas which correspond to the proposed gates.

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

[2]  C. Petri.  Grundsätzliches zur Beschreibung Diskreter Prozesse , 1967 .

[3]  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.

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

[5]  Atwell R. Turquette,et al.  On the Many-Valued Logics , 1941 .

[6]  A. Leporati,et al.  Quantum Conservative Gates for Finite-Valued Logics , 2004 .

[7]  R. Feynman.  Quantum mechanical computers , 1986 .

[8]  Gianpiero Cattaneo,et al.  Fredkin gates for finite-valued reversible and conservative logics , 2002 .

[9]  R Cuykendall.  Three-port reversible logic. , 1988, Applied optics.

[10]  P. Benioff.  Quantum Mechanical Models of Turing Machines That Dissipate No Energy , 1982 .

[11]  R. Jozsa,et al.  Quantum Computation and Shor's Factoring Algorithm , 1996 .

[12]  D. Gottesman.  Fault-Tolerant Quantum Computation with Higher-Dimensional Systems , 1998, quant-ph/9802007.

[13]  Paul Benioff,et al.  Quantum mechanical Hamiltonian models of discrete processes , 1981 .

[14]  Colin P. Williams,et al.  Ultimate zero and one - computing at the quantum frontier , 2012 .

[15]  C. Chang,et al.  Algebraic analysis of many valued logics , 1958 .

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

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

[18]  C. Chang,et al.  A new proof of the completeness of the Łukasiewicz axioms , 1959 .

[19]  Xue Fei,et al.  Realization of Fredkin Gate by Three Transition Pulses in NMR Quantum Information Processor , 2002 .

[20]  Artur Ekert,et al.  Basic Concepts in Quantum Computation (量子情報理論とその応用論文小特集) , 1998 .

[21]  Barenco,et al.  Conditional Quantum Dynamics and Logic Gates. , 1995, Physical review letters.

[22]  Paul Benioff,et al.  Quantum Mechanical Hamiltonian Models of Computers a , 1986 .

[23]  Milburn,et al.  Quantum optical Fredkin gate. , 1989, Physical review letters.

[24]  T. Toffoli,et al.  Conservative logic , 2002, Collision-Based Computing.

[25]  N. Rescher.  Many Valued Logic , 1969 .

[26]  W. Wootters,et al.  A single quantum cannot be cloned , 1982, Nature.

[27]  Colin P. Williams,et al.  Ultimate Zero and One , 1999, Springer New York.