Group theory based synthesis of binary reversible circuits

  • Authors:

  • Guowu Yang;Xiaoyu Song;William N. N. Hung;Fei Xie;Marek A. Perkowski

  • Affiliations:

  • Dept. of Computer Science, Portland State University, Portland, OR;Dept. of Electrical and Computer Engineering, Portland State University, Portland, OR;Dept. of Electrical and Computer Engineering, Portland State University, Portland, OR;Dept. of Computer Science, Portland State University, Portland, OR;Dept. of Electrical and Computer Engineering, Portland State University, Portland, OR

  • Venue:
  • TAMC'06 Proceedings of the Third international conference on Theory and Applications of Models of Computation
  • Year:
  • 2006

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Abstract

This paper presents an important result addressing a fundamental question in synthesizing binary reversible logic circuits for quantum computation. We show that any even-reversible-circuit of n (n3) qubits can be realized using NOT gate and Toffoli gate (‘2’-Controlled-Not gate), where the numbers of Toffoli and NOT gates required in the realization are bounded by $(n + \lfloor \frac{n}{3} \rfloor)(3 \times 2^{2n-3}-2^{n+2})$ and $4n(n + \lfloor \frac{n}{3} \rfloor)2^n$, respectively. A provable constructive synthesis algorithm is derived. The time complexity of the algorithm is $\frac{10}{3}n^2 \cdot 2^n$. Our algorithm is exponentially lower than breadth-first search based synthesis algorithms with respect to space and time complexities.