Beyond quantum Shannon decomposition: Circuit construction for n-qubit gates based on block-ZXZ decomposition

Abstract

This paper proposes an optimized quantum block-$ZXZ$ decomposition method that results in more optimal quantum circuits than the quantum Shannon decomposition, which was presented in 2005 by M. Möttönen, and J. J. Vartiainen [in Trends in quantum computing research, edited by S. Shannon (Nova Science Publishers, 2006) Chap. 7, p. 149, arXiv:quant-ph/0504100]. The decomposition is applied recursively to generic quantum gates, and can take advantage of existing and future small-circuit optimizations. Because our method uses only single-qubit gates and uniformly controlled rotation-Z gates, it can easily be adapted to use other types of multi-qubit gates. With the proposed decomposition, a general three-qubit gate can be decomposed using 19 cnot gates (rather than 20). For general $n$-qubit gates, the proposed decomposition generates circuits that have $\frac{22}{48}{4}^n-\frac{3}{2}{2}^n+\frac{5}{3}$ cnot gates, which is less than the best-known exact decomposition algorithm by $(4^{n-2}-1)/3$ cnot gates.