A Mathematica Package for Construction of Circuit Matrices for ...

A Mathematica Package for Construction of Circuit Matrices for Quantum Computation V. Gerdt a , R. Kragler b and A. Prokopenya c a b

Joint Institute for Nuclear Research, 141980 Dubna, Russia, [email protected]

University of Applied Sciences, D-88241 Weingarten, Germany, [email protected] c

Brest State Technical University, 224017 Brest, Belarus, [email protected]

Abstract In this talk we present a Mathematica package which provides a user-friendly graphical interface to input a quantum circuit and to construct a unitary matrix defined by the circuit. Having the circuit inputted, the matrix is computed by the linear algebra tools built in Mathematica. If the circuit consists of the Toffoli and Hadamard gates only, then the matrix can also be determined by counting the number of solution in the ground field F2 for the multivariate polynomial system over F2 associated with the circuit.

Quantum computation is a rapidly developing interdisciplinary research field that combines physics, mathematics and computer science [1]. Quantum computation may in some cases polynomially (e.g. Grover’s algorithm for searching in an unsorted data base) or even exponentially (Shor’s algorithm for integer factoring) speed-up classical computation. Among quantum algorithms that are also those oriented to differential equations [2]. Since realistic quantum computers have not yet been built, it is worthwhile to simulate quantum computation on a classical computer, and there are quite a number of such simulators (see, for example, [3]. Among two equivalent models of quantum computation – quantum Turing machine and the circuit model – the last one is more convenient both for simulation and application. Acknowledgements. The contribution of one authors (V.G.) was partially supported by grant No.04-01-00784 from the Russian Foundation for Basic Research and by grant 5362.2006.2 from the Ministry of Education and Science of the Russian Federation.

References [1] Nielsen, M. and Chuang, I.: Quantum Computation and Quantum Information. Cambridge University Press, 2000. [2] Szkopek, T., Roychowdhury, V. and Yablonovitch, E.: Eigenvalue Estimation of Differential Operators. Phys. Rev. A 72, (2005) 062318. arXiv:quant-ph/0408137 a1 a2

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Figure 1: Circuit for 3 qubit quantum Fourier transform

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Figure 2: Matrix defined by circuit of Fig. 3

a1

H

a2

H

a3

x1

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x3
x2 x4

b1

x2 b2 a3
x1 x2

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x4 b3

Figure 3: Circuit taken from [5]

[3] De Raedt, H. and Michielsen, K.: Computational Methods for Simulating Quantum Computers. In: Handbook of Theoretical and Computational Nanotechnology, Vol.3, M.Rieth and W.Schommers (Eds.), Forschungszentrum Karlsruhe, 2006. arXiv:quant-ph/0406210 [4] Buchberger, B.: Gr¨ obner Bases: an Algorithmic Method in Polynomial Ideal Theory, In: Recent Trends in Multidimensional System Theory, N.K. Bose (ed.), Reidel, Dordrecht (1985) 184–232. [5] Dowson, C.M. et al.: Quantum computing and polynomial equations over the finite field Z2 . arXiv:quant-ph/0408129 [6] Gerdt, V.P. and Severyanov, V.M.: An Algorithm for Constructing Polynomial Systems Whose Solution Space Characterizes Quantum Circuits. In: “Quantum Informatics 2005”, Yu.I.Ozhigov (Ed.), SPIE Proceedings, Volume 6264, 2006.

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