136 research outputs found
Chow's theorem and universal holonomic quantum computation
A theorem from control theory relating the Lie algebra generated by vector
fields on a manifold to the controllability of the dynamical system is shown to
apply to Holonomic Quantum Computation. Conditions for deriving the holonomy
algebra are presented by taking covariant derivatives of the curvature
associated to a non-Abelian gauge connection. When applied to the Optical
Holonomic Computer, these conditions determine that the holonomy group of the
two-qubit interaction model contains . In particular, a
universal two-qubit logic gate is attainable for this model.Comment: 13 page
Limits of control for quantum systems: kinematical bounds on the optimization of observables and the question of dynamical realizability
In this paper we investigate the limits of control for mixed-state quantum
systems. The constraint of unitary evolution for non-dissipative quantum
systems imposes kinematical bounds on the optimization of arbitrary
observables. We summarize our previous results on kinematical bounds and show
that these bounds are dynamically realizable for completely controllable
systems. Moreover, we establish improved bounds for certain partially
controllable systems. Finally, the question of dynamical realizability of the
bounds for arbitary partially controllable systems is shown to depend on the
accessible sets of the associated control system on the unitary group U(N) and
the results of a few control computations are discussed briefly.Comment: 5 pages, orginal June 30, 2000, revised September 28, 200
Monotonically convergent optimal control theory of quantum systems with spectral constraints on the control field
We propose a new monotonically convergent algorithm which can enforce
spectral constraints on the control field (and extends to arbitrary filters).
The procedure differs from standard algorithms in that at each iteration the
control field is taken as a linear combination of the control field (computed
by the standard algorithm) and the filtered field. The parameter of the linear
combination is chosen to respect the monotonic behavior of the algorithm and to
be as close to the filtered field as possible. We test the efficiency of this
method on molecular alignment. Using band-pass filters, we show how to select
particular rotational transitions to reach high alignment efficiency. We also
consider spectral constraints corresponding to experimental conditions using
pulse shaping techniques. We determine an optimal solution that could be
implemented experimentally with this technique.Comment: 16 pages, 4 figures. To appear in Physical Review
Quantum Adiabatic Brachistochrone
We formulate a time-optimal approach to adiabatic quantum computation (AQC).
A corresponding natural Riemannian metric is also derived, through which AQC
can be understood as the problem of finding a geodesic on the manifold of
control parameters. This geometrization of AQC is demonstrated through two
examples, where we show that it leads to improved performance of AQC, and sheds
light on the roles of entanglement and curvature of the control manifold in
algorithmic performance.Comment: 4 pages, 2 figure
Optimal Control of Underactuated Mechanical Systems: A Geometric Approach
In this paper, we consider a geometric formalism for optimal control of
underactuated mechanical systems. Our techniques are an adaptation of the
classical Skinner and Rusk approach for the case of Lagrangian dynamics with
higher-order constraints. We study a regular case where it is possible to
establish a symplectic framework and, as a consequence, to obtain a unique
vector field determining the dynamics of the optimal control problem. These
developments will allow us to develop a new class of geometric integrators
based on discrete variational calculus.Comment: 20 pages, 2 figure
Search complexity and resource scaling for the quantum optimal control of unitary transformations
The optimal control of unitary transformations is a fundamental problem in
quantum control theory and quantum information processing. The feasibility of
performing such optimizations is determined by the computational and control
resources required, particularly for systems with large Hilbert spaces. Prior
work on unitary transformation control indicates that (i) for controllable
systems, local extrema in the search landscape for optimal control of quantum
gates have null measure, facilitating the convergence of local search
algorithms; but (ii) the required time for convergence to optimal controls can
scale exponentially with Hilbert space dimension. Depending on the control
system Hamiltonian, the landscape structure and scaling may vary. This work
introduces methods for quantifying Hamiltonian-dependent and kinematic effects
on control optimization dynamics in order to classify quantum systems according
to the search effort and control resources required to implement arbitrary
unitary transformations
Optimal Control Theory for Continuous Variable Quantum Gates
We apply the methodology of optimal control theory to the problem of
implementing quantum gates in continuous variable systems with quadratic
Hamiltonians. We demonstrate that it is possible to define a fidelity measure
for continuous variable (CV) gate optimization that is devoid of traps, such
that the search for optimal control fields using local algorithms will not be
hindered. The optimal control of several quantum computing gates, as well as
that of algorithms composed of these primitives, is investigated using several
typical physical models and compared for discrete and continuous quantum
systems. Numerical simulations indicate that the optimization of generic CV
quantum gates is inherently more expensive than that of generic discrete
variable quantum gates, and that the exact-time controllability of CV systems
plays an important role in determining the maximum achievable gate fidelity.
The resulting optimal control fields typically display more complicated Fourier
spectra that suggest a richer variety of possible control mechanisms. Moreover,
the ability to control interactions between qunits is important for delimiting
the total control fluence. The comparative ability of current experimental
protocols to implement such time-dependent controls may help determine which
physical incarnations of CV quantum information processing will be the easiest
to implement with optimal fidelity.Comment: 39 pages, 11 figure
The rolling problem: overview and challenges
In the present paper we give a historical account -ranging from classical to
modern results- of the problem of rolling two Riemannian manifolds one on the
other, with the restrictions that they cannot instantaneously slip or spin one
with respect to the other. On the way we show how this problem has profited
from the development of intrinsic Riemannian geometry, from geometric control
theory and sub-Riemannian geometry. We also mention how other areas -such as
robotics and interpolation theory- have employed the rolling model.Comment: 20 page
Optimal control, geometry, and quantum computing
We prove upper and lower bounds relating the quantum gate complexity of a
unitary operation, U, to the optimal control cost associated to the synthesis
of U. These bounds apply for any optimal control problem, and can be used to
show that the quantum gate complexity is essentially equivalent to the optimal
control cost for a wide range of problems, including time-optimal control and
finding minimal distances on certain Riemannian, subriemannian, and Finslerian
manifolds. These results generalize the results of Nielsen, Dowling, Gu, and
Doherty, Science 311, 1133-1135 (2006), which showed that the gate complexity
can be related to distances on a Riemannian manifoldComment: 7 Pages Added Full Names to Author
A Riemannian approach to randers geodesics
In certain circumstances tools of Riemannian geometry are sufficient to address questions arising in the more general Finslerian context. We show that one such instance presents itself in the characterisation of geodesics in Randers spaces of constant flag curvature. To achieve a simple, Riemannian derivation of this special family of curves, we exploit the connection between Randers spaces and the Zermelo problem of time-optimal navigation in the presence of background fields. The characterisation of geodesics is then proven by generalising an intuitive argument developed recently for the solution of the quantum Zermelo problem
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