Mitsuru Hamada
A recently obtained fundamental lemma on Euler angles and its application to some issue on universal gates for quantum computation [M. Hamada, “Overlooked restrictions on Euler angles in quantum computation,” APS 2013 March Meeting, Baltimore, USA, 2012, http://meetings.aps.org/link/BAPS.2013.MAR.H1.318 (abstract)] are described in detail. The fundamental lemma in the unpublished work is as follows. Let X,Y , and Z denote the Pauli matrices. For any three-dimensional real unit vector n = (nx, ny , nz) and θ ∈ R, put Rn(θ) = cos(θ/2)I − i sin(θ/2)(nxX + nyY + nzZ). Put Ry(θ) = R(0,1,0)T (θ) and Rz(θ) = R(0,0,1)T (θ). Lemma: Assume α, γ, θ ∈ R and n = (nx, ny , nz) is a real unit vector; then, there exists some β, δ ∈ R satisfying Rn(θ) = e Rz(β)Ry(γ)Rz(δ) if and only if e ∈ {1,−1} and √ 1− n2z| sin(θ/2)| = | sin(γ/2)|. By means of this lemma, a widespread fallacy on universal gates has been pointed out. Mathematical details of the lemma and its application, together with self-contained expositions of underlying notions, are given so as to help us dispel the fallacy.
•Jan 1, 2013•1 citation