On Cayley’s Factorization of 4D Rotations and Applications

Pérez-Gracia, Alba; Thomas, Federico

doi:10.1007/s00006-016-0683-9

Cited by 27 publications

(21 citation statements)

References 15 publications

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“…A closed-form matrix solution for this problem can be found in [11]. Nevertheless, it involves four alternative mappings and the best one has to be chosen according to a voting scheme.…”

Section: Cayley's Factorizationmentioning

confidence: 99%

On Cayley’s Factorization with an Application to the Orthonormalization of Noisy Rotation Matrices

Sarabandi

Pérez-Gracia

Thomas

2019

Adv. Appl. Clifford Algebras

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A real orthogonal matrix representing a rotation in four dimensions can be decomposed into the commutative product of a leftand a right-isoclinic rotation matrix. This operation, known as Cayley's factorization, directly provides the double quaternion representation of rotations in four dimensions. This factorization can be performed without divisions, thus avoiding the common numerical issues attributed to the computation of quaternions from rotation matrices. In this paper, it is shown how this result is particularly useful, when particularized to three dimensions, to re-orthonormalize a noisy rotation matrix by converting it to quaternion form and then obtaining back the corresponding proper rotation matrix. This re-orthonormalization method is commonly implemented using the Shepperd-Markley method, but the method derived here is shown to outperform it by returning results closer to those obtained using the Singular Value Decomposition which are known to be optimal in terms of the Frobenius norm.Mathematics Subject Classification (2010). Primary 99Z99; Secondary 15A66.

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“…A closed-form matrix solution for this problem can be found in [11]. Nevertheless, it involves four alternative mappings and the best one has to be chosen according to a voting scheme.…”

Section: Cayley's Factorizationmentioning

confidence: 99%

On Cayley’s Factorization with an Application to the Orthonormalization of Noisy Rotation Matrices

Sarabandi

Pérez-Gracia

Thomas

2019

Adv. Appl. Clifford Algebras

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“…An isoclinic rotation can be left-or right-isoclinic (depending on whether α=β or α=-β) [30]. According to Cayley's factorization [31,32], any 4D rotation matrix can be decomposed into the product of a right-and a left-isoclinic matrix. This decomposition is also conveniently expressed in terms of quaternions, as discussed in the following subsection.…”

Section: D Rotationsmentioning

confidence: 99%

“…a formula due to Van Elfrinkhof [30,31]. M L and M R are isoclinic matrices [33,34], so R = M L M R = M R M L is a 4D rotation matrix without loss of generality.…”

Section: B Isoclinic Rotations and Quaternionsmentioning

confidence: 99%

“…II A, an arbitrary rotation R in E 4 of a 4-vector C can be represented by the product qCp, associated with left and right isoclinic rotations with rotation angles γ 1 and γ 2 . R also corresponds to a product of rotations in two mutually orthogonal planes [30,31,33,34]. If u = ±v, R rotates the plane spanned by u + v and uv − 1 through the angle |γ 1 + γ 2 |, and the plane spanned by v − u and uv + 1 through the angle |γ 1 − γ 2 |, respectively [45].…”

Section: Appendix A: Quaternions and 4d Rotationsmentioning

confidence: 99%

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Hamiltonian design to prepare arbitrary states of four-level systems

Martínez-Cercós

Martínez-Garaot

et al. 2018

Phys. Rev. A

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We propose a method to manipulate, possibly faster than adiabatically, four-level systems with time-dependent couplings and constant energy shifts (detunings in quantum-optical realizations). We inversely engineer the Hamiltonian, in ladder, tripod, or diamond configurations, to prepare arbitrary states using the geometry of four-dimensional rotations to set the state populations, specifically we use Cayley's factorization of a general rotation into right-and left-isoclinic rotations.

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“…To engineer H r (t) for a specific 4D rotation, it is convenient to express first a general 4D rotation matrix as a product of two isoclinic rotation matrices [22,23]: where q i and p j are components of two unit quaternions q=q w +q x i+q y j+q z k and p=p w +p x i+p y j+p z k. We shall parameterize them in terms of generalized 4D spherical angles [24,25], where 0f 1,2 2π, 0θ 1,2 , γ 1,2 π. Thus by using…”

mentioning

confidence: 99%

Qubit gates with simultaneous transport in double quantum dots

Chen

Muga

et al. 2018

New J. Phys.

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A single electron spin in a double quantum dot in a magnetic field is considered in terms of a four-level system. By describing the electron motion between the potential minima via spin-conserving tunneling and spin flip caused by a spin-orbit coupling, we inversely engineer faster-than-adiabatic state manipulation operations based on the geometry of four-dimensional rotations. In particular, we show how to transport a qubit among the quantum dots performing simultaneously required spin rotations. IntroductionDevice architecture based on electrons confined in coupled quantum dots [1-4] is considered as a potential and significant candidate for quantum computing and quantum information processing. The advantages of this architecture rely on the facts that electron spin is a natural qubit with spin-up and spin-down states, mature semiconductor technology may be used, and long coherence times on the scale of microseconds have been achieved in these systems [5,6]. Laboratories use electric, microwave or magnetic fields to manipulate spin states, performing 10 3 -10 5 operations in the spin dephasing time [5-10].Scalability of quantum information devices is a major challenge for any architecture, and it is associated with the capability to transport qubits. In this paper we theoretically explore a four-level model for a spin in a double quantum dot (DQD) aiming at the possibilities to implement fast qubit transport with simultaneous qubit rotations. We achieve this goal for arbitrary rotations by controlling the synchronized time dependences of interdot tunneling and spin-orbit coupling (SOC). We inverse-engineer these time dependences based on our recent work [11] on the control of four-level systems. The method separates population control from control of the phases of the bare state basis [12]. Populations can be mapped onto a four-dimensional (4D) sphere such that their evolution amounts to 4D transformation controlled by the (4D-)rotation Hamiltonian that may be engineered from the target state (in our case via isoclinic rotations and quaternions). A full Hamiltonian can then be constructed from the rotation Hamiltonian to realize the desired phase changes. Arbitrary state manipulations require full flexibility in the Hamiltonian, i.e. the possibility to implement the different Hamiltonian matrix elements with specific time-dependences. In the systems of interest, however, there are constraints that hinder certain manipulations and transitions. In particular, in this paper we examine the Hamiltonian structure that corresponds to combined tunneling and SOC controllable couplings, and deduce the possible transformations.Spin-orbit coupling in semiconductors consists of two main contributions due to the Dresselhaus-and the Bychkov-Rashba-effect. The former is due to the bulk inversion asymmetry of the material and the latter results from the structure inversion asymmetry, produced, e.g. by the confining potential or an external electric field [13]. The practical advantage of the Rashba coupling is the ability to man...

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On Cayley’s Factorization of 4D Rotations and Applications

Cited by 27 publications

References 15 publications

On Cayley’s Factorization with an Application to the Orthonormalization of Noisy Rotation Matrices

On Cayley’s Factorization with an Application to the Orthonormalization of Noisy Rotation Matrices

Hamiltonian design to prepare arbitrary states of four-level systems

Qubit gates with simultaneous transport in double quantum dots

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