The Canonical Coset Decomposition of Unitary Matrices Through Householder Transformations

This paper reveals the relation between the canonical coset decomposition of unitary matrices and the corresponding decomposition via Householder reflections. These results can be used to parametrize unitary matrices via Householder reflections

Fidelity Between Unitary Operators and the Generation of Gates Robust Against Off-Resonance Perturbations

We perform a functional expansion of the fidelity between two unitary matrices in order to find the necessary conditions for the robust implementation of a target gate. Comparison of these conditions with those obtained from the Magnus expansion and Dyson series shows that they are equivalent in first order. By exploiting techniques from robust design optimization, we account for issues of experimental feasibility by introducing an additional criterion to the search for control pulses. This search is accomplished by exploring the competition between the multiple objectives in the implementation of the NOT gate by means of evolutionary multi-objective optimization

A geometric algebra approach ton-qubit systems

A geometric formalism for the treatment of n-qubit systems is presented in erms of the Clifford\u27s Geometric algebra Cfm, as an alternative to the tra­ itional matrix formulation. This objective is accomplished by generalizing he one-qubit system formulated in terms of the bivector space of Cfg. This ormulation is based in the well known isomorphism between the so(3) and u(2) Lie algebras. It is known that a quantum system with N orthogonal states (levels) is ontrollable with the SU(N) group. However, a system with an even number of orthogonal states may also be state-controllable with the USp(N) group, hich is a subgroup of SU(N) with a Lie algebra isomorphic with the Lie lgebra of the symplectic group Sp(N). The isomorphism between the sp(4) and spin(5) Lie algebras allows the ormulation of a two-qubit system in terms of the the bivector space of Cfs, as a natural instance of the spin(5) Lie algebra. Another isomorphism be­ ween the sp( 4) Lie algebra and the anti-Hermitian space of Cf4 is revealed, herefore allowing the formulation of a two-qubit system in terms of Cf4 as ell. More isomorphisms are exposed between some subspaces of higher di­ ensional Clifford algebras with the Lie algebras of the USp(N) and SU(N) roups. The immediate consequence is the possibility to represent an n-qubit iv A geometric formalism for the treatment of n-qubit systems is presented in erms of the Clifford\u27s Geometric algebra Cfm, as an alternative to the tra­ itional matrix formulation. This objective is accomplished by generalizing he one-qubit system formulated in terms of the bivector space of Cfg. This ormulation is based in the well known isomorphism between the so(3) and u(2) Lie algebras. It is known that a quantum system with N orthogonal states (levels) is ontrollable with the SU(N) group. However, a system with an even number of orthogonal states may also be state-controllable with the USp(N) group, hich is a subgroup of SU(N) with a Lie algebra isomorphic with the Lie lgebra of the symplectic group Sp(N). The isomorphism between the sp(4) and spin(5) Lie algebras allows the ormulation of a two-qubit system in terms of the the bivector space of Cfs, as a natural instance of the spin(5) Lie algebra. Another isomorphism be­ ween the sp( 4) Lie algebra and the anti-Hermitian space of Cf4 is revealed, herefore allowing the formulation of a two-qubit system in terms of Cf4 as ell. More isomorphisms are exposed between some subspaces of higher di­ ensional Clifford algebras with the Lie algebras of the USp(N) and SU(N) roups. The immediate consequence is the possibility to represent an n-qubit i

Making Distinct Dynamical Systems Appear Spectrally Identical

We show that a laser pulse can always be found that induces a desired optical response from an arbitrary dynamical system. As illustrations, driving fields are computed to induce the same optical response from a variety of distinct systems (open and closed, quantum and classical). As a result, the observed induced dipolar spectra without detailed information on the driving field is not sufficient to characterize atomic and molecular systems. The formulation may also be applied to design materials with specified optical characteristics. These findings reveal unexplored flexibilities of nonlinear optics.Comment: 9 pages, 5 figure

Analytic Solutions to Coherent Control of the Dirac Equation

A simple framework for Dirac spinors is developed that parametrizes admissible quantum dynamics and also analytically constructs electromagnetic fields, obeying Maxwell's equations, which yield a desired evolution. In particular, we show how to achieve dispersionless rotation and translation of wave packets. Additionally, this formalism can handle control interactions beyond electromagnetic. This work reveals unexpected flexibility of the Dirac equation for control applications, which may open new prospects for quantum technologies

Dirac open quantum system dynamics: formulations and simulations

We present an open system interaction formalism for the Dirac equation. Overcoming a complexity bottleneck of alternative formulations, our framework enables efficient numerical simulations (utilizing a typical desktop) of relativistic dynamics within the von Neumann density matrix and Wigner phase space descriptions. Employing these instruments, we gain important insights into the effect of quantum dephasing for relativistic systems in many branches of physics. In particular, the conditions for robustness of Majorana spinors against dephasing are established. Using the Klein paradox and tunneling as examples, we show that quantum dephasing does not suppress negative energy particle generation. Hence, the Klein dynamics is also robust to dephasing

The landscape of quantum transitions driven by single-qubit unitary transformations with implications for entanglement

This paper considers the control landscape of quantum transitions in multi-qubit systems driven by unitary transformations with single-qubit interaction terms. The two-qubit case is fully analyzed to reveal the features of the landscape including the nature of the absolute maximum and minimum, the saddle points and the absence of traps. The results permit calculating the Schmidt state starting from an arbitrary two-qubit state following the local gradient flow. The analysis of multi-qubit systems is more challenging, but the generalized Schmidt states may also be located by following the local gradient flow. Finally, we show the relation between the generalized Schmidt states and the entanglement measure based on the Bures distance

Operational Dynamical Modeling of spin 1/2 relativistic particles: the Dirac equation and its classical limit

The formalism of Operational Dynamical Modeling [Phys. Rev. Lett. {\bf 109}, 190403 (2012)] is employed to analyze dynamics of spin half relativistic particles. We arrive at the Dirac equation from specially constructed relativistic Ehrenfest theorems by assuming that the coordinates and momenta do not commute. Forbidding creation of antiparticles and requiring the commutativity of the coordinates and momenta lead to classical Spohn's equation [Ann. Phys. {\bf 282}, 420 (2000)]. Moreover, Spohn's equation turns out to be the classical Koopman-von Neumann theory underlying the Dirac equation