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Quantum loop programs

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Abstract

Loop is a powerful program construct in classical computation, but its power is still not exploited fully in quantum computation. The exploitation of such power definitely requires a deep understanding of the mechanism of quantum loop programs. In this paper, we introduce a general scheme of quantum loops and describe its computational process. The function computed by a quantum loop is defined, and a denotational semantics and a weakest precondition semantics of a quantum loop are given. The notions of termination and almost termination are proposed for quantum loops. This paper only consider the case of finite-dimensional state spaces. Necessary and sufficient conditions for termination and almost termination of a general quantum loop on any mixed input state are presented. A quantum loop is said to be (almost) terminating if it (almost) terminates on any input state. We show that a quantum loop is almost terminating if and only if it is uniformly almost terminating. It is observed that a small disturbance either on the unitary transformation in the loop body or on the measurement in the loop guard can make any quantum loop (almost) terminating, provided that some dimension restriction is satisfied. Moreover, a representation of the function computed by a quantum loop is given in terms of finite summations of matrices. To illustrate the notions and results obtained in this paper, two simple classes of quantum loop programs, one qubit quantum loops, and two qubit quantum loops defined by controlled gates, are carefully examined, and to show their expressive power, quantum loops are applied in describing quantum walks.

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Authors and Affiliations

  1. Center of Quantum Computation and Intelligent Systems, Faculty of Engineering and Information Technology, University of Technology, Sydney, NSW, 2007, Australia

    Mingsheng Ying & Yuan Feng

  2. Department of Computer Science and Technology, State Key Laboratory of Intelligent Technology and Systems, Tsinghua National Laboratory for Information Science and Technology, Tsinghua University, Beijing, 100084, China

    Mingsheng Ying & Yuan Feng

Authors
  1. Mingsheng Ying
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  2. Yuan Feng
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Corresponding author

Correspondence to Mingsheng Ying.

Additional information

M. Ying and Y. Feng: This work was partly supported by the National Foundation of Natural Sciences of China (Grant No: 60736011, 60621062) and the National Key Project for Fundamental Research of China (Grant No: 2007CB807901).

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Ying, M., Feng, Y. Quantum loop programs. Acta Informatica 47, 221–250 (2010). https://doi.org/10.1007/s00236-010-0117-4

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