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Mathematical models of quantum computation

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Abstract

In this paper, we introduce two mathematical models of realistic quantum computation. First, we develop a theory of bulk quantum computation such as NMR (Nuclear Magnetic Resonance) quantum computation. For this purpose, we define bulk quantum Turing machine (BQTM for short) as a model of bulk quantum computation. Then, we define complexity classes EBQP, BBQP and ZBQP as counterparts of the quantum complexity classes EQP, BQP and ZQP, respectively, and show that EBQP=EQP, BBQP=BQP and ZBQP=ZQP. This implies that BQTMs are polynomially related to ordinary QTMs as long as they are used to solve decision problems. We also show that these two types of QTMs are also polynomially related when they solve a function problem which has a unique solution. Furthermore, we show that BQTMs can solve certain instances of NP-complete problems efficiently.

On the other hand, in the theory of quantum computation, only feed-forward quantum circuits are investigated, because a quantum circuit represents a sequence of applications of time evolution operators. But, if a quantum computer is a physical device where the gates are interactions controlled by a current computer such as laser pulses on trapped ions, NMR and most implementation proposals, it is natural to describe quantum circuits as ones that have feedback loops if we want to visualize the total amount of the necessary hardware. For this purpose, we introduce a quantum recurrent circuit model, which is a quantum circuit with feedback loops. LetC be a quantum recurrent circuit which solves the satisfiability problem for a blackbox Boolean function includingn variables with probability at least 1/2. And lets be the size ofC (i.e. the number of the gates inC) andt be the number of iterations that is needed forC to solve the satisfiability problem. Then, we show that, for those quantum recurrent circuits, the minimum value ofmax(s, t) isO(n22n/3).

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Authors and Affiliations

  1. Department of Information and Communication Engineering, The University of Electro-Communications, Chofu, 182-8585, Tokyo, Japan

    Tetsuro Nishino

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  1. Tetsuro Nishino
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Correspondence to Tetsuro Nishino.

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Tetsuro Nishino, D.Sc.: He is presently an Associate Professor in the Department of Information and Communication Engineering, The University of Electro-Communications. He received the B.S., M.S. and D.Sc degrees in mathematics from Waseda University, in 1982, 1984 and 1991 respectively. From 1984 to 1987, he joined Tokyo Research Laboratory, IBM Japan. From 1987 to 1992, he was a Research Associate of Tokyo Denki University, and from 1992 to 1994, he was an Associate Professor of Japan Advanced Institute of Science and Technology, Hokuriku. His main interests are circuit complexity theory, computational learning theory and quantum complexity theory.

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Nishino, T. Mathematical models of quantum computation. New Gener Comput 20, 317–337 (2002). https://doi.org/10.1007/BF03037370

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