Smaller Circuits for Arbitrary n-qubit Diagonal Computations

Stephen S. Bullock; Igor L. Markov

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arXiv · quant-ph/0303039

Smaller Circuits for Arbitrary n-qubit Diagonal Computations

Stephen S. Bullock·Igor L. Markov

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Abstract

A unitary operator U=\sum u_{j,k} |k> ; 0 <= j <= 2^n-1}. These relative phases are often required in applications. Constructing quantum circuits for diagonal computations using standard techniques requires either O(n^2 2^n) controlled-not gates and one-qubit Bloch sphere rotations or else O (n 2^n) such gates and a work qubit. This work provides a recursive, constructive procedure which inputs the matrix coefficients of U and outputs such a diagram containing 2^{n+1}-3 alternating controlled-not gates and one-qubit z-axis Bloch sphere rotations. Up to a factor of two, these circuits are the smallest possible. Moreover, should the computation U be a tensor of diagonal one-qubit computations of the form R_z(α)=e^{-i α/2}|0><0|+ e^{i α/2} |1><1|, then a cancellation of controlled-not gates reduces our circuit to that of an n-qubit tensor.

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Stephen S. Bullock, Igor L. Markov. 2003-08-08. Smaller Circuits for Arbitrary n-qubit Diagonal Computations. https://arxiv.org/abs/quant-ph/0303039

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