Smaller Circuits for Arbitrary n-qubit Diagonal Computations
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A unitary operator U=\sum u_{j,k} |k><j| is called diagonal when u_{j,k}=0 unless j=k. The definition extends to quantum computations, where j and k vary over the 2^n binary expressions for integers 0,1 ..., 2^n-1, given n qubits. Such operators do not affect outcomes of the projective measurement {<j| ; 0 <= j <= 2^n-1} but rather create arbitrary relative phases among the computational basis states {|j> ; 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.
v3 improves the results in v1 and achieves asymptotically optimal gate counts v4 makes dimension counting argument rigorous
v3 improves the results in v1 and achieves asymptotically optimal gate counts v4 makes dimension counting argument rigorous