Smaller Circuits for Arbitrary n-qubit Diagonal Computations

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dc.creatorBullock, Stephen S.
dc.creatorMarkov, Igor L.
dc.date2003-03-07
dc.date2003-08-08
dc.date.accessioned2026-07-07T06:06:16Z
dc.date.available2026-07-07T06:06:16Z
dc.descriptionA 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.
dc.descriptionv3 improves the results in v1 and achieves asymptotically optimal gate counts v4 makes dimension counting argument rigorous
dc.identifierhttps://arxiv.org/abs/quant-ph/0303039
dc.identifierhttp://arxiv.org/abs/quant-ph/0303039
dc.identifierQuantum Information & Computation, vol 4, no 1, 027-047 (2004)
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/90918
dc.subjectQuantum Physics
dc.titleSmaller Circuits for Arbitrary n-qubit Diagonal Computations
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