113 research outputs found
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
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
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
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
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