© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

A Lemma on Euler Angles

Mitsuru Hamada

Published Jan 1, 2013• 1 citation

Abstract

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 The fundamental lemma in the unpublished work is as follows.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.

Related papers

Overlooked restrictions on Euler angles in quantum computation

Mitsuru Hamada

Bulletin of the American Physical Society•Mar 19, 2013•2 citations

Hybrid quantum magnetism in circuit-QED: from spin-photon waves to many-body spectroscopy

A. Bermúdez, Andreas Kurcz, Juan José García‐Ripoll

APS March Meeting 201, Denver, Colorado, March 3–7, 2014; Bulletin of the American Physical Society, Volume 59, Number 1. http://meetings.aps.org/link/BAPS.2014.MAR.A34.9

Bulletin of the American Physical Society•Mar 3, 2014•0 citations

Improving quantum gate fidelities using optimized Euler angles

K. Ch. Chatzisavvas, Goce Chadzitaskos, C. Daskaloyannis +1 more

An explicit algorithm for calculating the optimized Euler angles for both qubit state transfer and gate engineering given two arbitrary fixed Hamiltonians is presented. It is shown how the algorithm enables us to efficiently implement single qubit gates even if the control is severely restricted and the experimentally accessible Hamiltonians are far from orthogonal. It is further shown that using the optimized Euler angles can significantly improve the fidelity of quantum operations even for systems where the experimentally accessible Hamiltonians are nearly orthogonal. Unlike schemes such as composite pulses, the proposed scheme does not significantly increase the number of local operations or gate operation times.

arXiv:0805.2894•Physical Review A•Nov 23, 2009•4 citations

Exact Solutions of Schrödinger Equation with Improved Ring-Shaped Non-Spherical Harmonic Oscillator and Coulomb Potential

A. N. Ikot, Ita O. Akpan, Tamunoimi Michael Abbey +1 more

Abstract We propose improved ring shaped like potential of the form, V(r, θ) = V(r) + (ħ 2 /2Mr 2 )[(β sin 2 θ + γ cos 2 θ + λ) / sin θ cos θ] 2 and its exact solutions are presented via the Nikiforov–Uvarov method. The angle dependent part V(θ) = (ħ 2 / 2 Mr 2 )[(β sin 2 θ + γ cos 2 θ + λ) / sin θ cos θ] 2 , which is reported for the first time embodied the novel angle dependent (NAD) potential and harmonic novel angle dependent potential (HNAD) as special cases. We discuss in detail the effects of the improved ring shaped like potential on the radial parts of the spherical harmonic and Coulomb potentials.

Communications in Theoretical Physics•May 1, 2016•25 citations

General Multiphase Matching for Grover Quantum Search Algorithm

Chen Xiangdong

Since a quantum system is inevitably influenced by some unpredictable perturbations,we thereby conclude that all the experimental realizations of Grover quantum search algorithm reported were,in fact,achieved in a three-dimensional complex subspace.We also prove that in a two-dimensional complex subspace,for any given initial superposition of basis states |γ0〉=cos β0|α〉+sin β0eiζ|β〉(β0 is a small positive real number,ζ is an arbitrary real number),there exists a set of solutions Fj={(θj,θj-1,…,θ1),(φj,φj-1,…,φ1)} such that a desired state can be found with certainty for some positive integer j≥2,where the phase rotation angles θl and φl are real numbers but not equal to 2k′π,1≤l≤j,k′ is an arbitrary integer.If it is only required that a desired state can be found with high success probability,then as the total number of the desired and undesired states in an unsorted database is sufficiently large the above set of solutions Fj can be written in the form Σji=1θl=Σji=1φl.for a relatively small positive integer j.

Jisuan wuli•Jan 1, 2012•0 citations

Usage of an Euler Angle Parameterization of SU(N) and U(N) for Entanglement Calculations on a Two-Qubit System (Waseda International Symposium on Fundamental Physics--New Perspectives in Quantum Physics) -- (Chapter 4. Quantum Information & Computation)

Todd Tilma, E. C. G. Sudarshan

Journal of the Physical Society of Japan•Jan 1, 2003•0 citations

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.

An explicit algorithm for calculating the optimized Euler angles for both qubit state transfer and gate engineering given two arbitrary fixed Hamiltonians is presented. It is shown how the algorithm enables us to efficiently implement single qubit gates even if the control is severely restricted and the experimentally accessible Hamiltonians are far from orthogonal. It is further shown that using the optimized Euler angles can significantly improve the fidelity of quantum operations even for systems where the experimentally accessible Hamiltonians are nearly orthogonal. Unlike schemes such as composite pulses, the proposed scheme does not significantly increase the number of local operations or gate operation times.

Abstract We propose improved ring shaped like potential of the form, V(r, θ) = V(r) + (ħ 2 /2Mr 2 )[(β sin 2 θ + γ cos 2 θ + λ) / sin θ cos θ] 2 and its exact solutions are presented via the Nikiforov–Uvarov method. The angle dependent part V(θ) = (ħ 2 / 2 Mr 2 )[(β sin 2 θ + γ cos 2 θ + λ) / sin θ cos θ] 2 , which is reported for the first time embodied the novel angle dependent (NAD) potential and harmonic novel angle dependent potential (HNAD) as special cases. We discuss in detail the effects of the improved ring shaped like potential on the radial parts of the spherical harmonic and Coulomb potentials.

Since a quantum system is inevitably influenced by some unpredictable perturbations,we thereby conclude that all the experimental realizations of Grover quantum search algorithm reported were,in fact,achieved in a three-dimensional complex subspace.We also prove that in a two-dimensional complex subspace,for any given initial superposition of basis states |γ0〉=cos β0|α〉+sin β0eiζ|β〉(β0 is a small positive real number,ζ is an arbitrary real number),there exists a set of solutions Fj={(θj,θj-1,…,θ1),(φj,φj-1,…,φ1)} such that a desired state can be found with certainty for some positive integer j≥2,where the phase rotation angles θl and φl are real numbers but not equal to 2k′π,1≤l≤j,k′ is an arbitrary integer.If it is only required that a desired state can be found with high success probability,then as the total number of the desired and undesired states in an unsorted database is sufficiently large the above set of solutions Fj can be written in the form Σji=1θl=Σji=1φl.for a relatively small positive integer j.