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An algorithmic approach to solving polynomial equations associated with quantum circuits

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Abstract

In this paper we present two algorithms for reducing systems of multivariate polynomial equations over the finite field F 2 to the canonical triangular form called lexicographical Gröbner basis. This triangular form is the most appropriate for finding solutions of the system. On the other hand, the system of polynomials over F 2 whose variables also take values in F 2 (Boolean polynomials) completely describes the unitary matrix generated by a quantum circuit. In particular, the matrix itself can be computed by counting the number of solutions (roots) of the associated polynomial system. Thereby, efficient construction of the lexicographical Gröbner bases over F 2 associated with quantum circuits gives a method for computing their circuit matrices that is alternative to the direct numerical method based on linear algebra. We compare our implementation of both algorithms with some other software packages available for computing Gröbner bases over F 2.

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Authors and Affiliations

  1. Laboratory of Information Technologies Joint Institute for Nuclear Research, 141980, Dubna, Russia

    V. P. Gerdt & M. V. Zinin

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  1. V. P. Gerdt
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  2. M. V. Zinin
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Correspondence to V. P. Gerdt.

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The article is published in the original.

The research presented in this paper was partially supported by grant 07-01-00660 from the Russian Foundation for Basic Research and by grant 5362.2006.2 from the Ministry of Education and Science of the Russian Federation.

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Gerdt, V.P., Zinin, M.V. An algorithmic approach to solving polynomial equations associated with quantum circuits. Phys. Part. Nuclei Lett. 6, 521–525 (2009). https://doi.org/10.1134/S154747710907005X

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