Smaller Circuits for Arbitrary n-qubit Diagonal Computations

Details Smaller Circuits for Arbitrary n-qubit Diagonal Computations Smaller Circuits for Arbitrary n-qubit Diagonal Computations

2003-03-07

arxiv.org/abs/quant-ph/0303039v4

Quantum Information & Computation, vol 4, no 1, 027-047 (2004)

Physics

Quantum Physics

v3 improves the results in v1 and achieves asymptotically optimal gate counts v4 makes dimension counting argument rigorous

Scientific paper

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(\alpha)=e^{-i \alpha/2}|0><0|+ e^{i \alpha/2} |1><1|, then a cancellation of controlled-not gates reduces our circuit to that of an n-qubit tensor.

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