30 research outputs found
Dynamic Homotopy and Landscape Dynamical Set Topology in Quantum Control
We examine the topology of the subset of controls taking a given initial
state to a given final state in quantum control, where "state" may mean a pure
state |\psi>, an ensemble density matrix \rho, or a unitary propagator U(0,T).
The analysis consists in showing that the endpoint map acting on control space
is a Hurewicz fibration for a large class of affine control systems with vector
controls. Exploiting the resulting fibration sequence and the long exact
sequence of basepoint-preserving homotopy classes of maps, we show that the
indicated subset of controls is homotopy equivalent to the loopspace of the
state manifold. This not only allows us to understand the connectedness of
"dynamical sets" realized as preimages of subsets of the state space through
this endpoint map, but also provides a wealth of additional topological
information about such subsets of control space.Comment: Minor clarifications, and added new appendix addressing scalar
control of 2-level quantum system
Characterization of the Critical Sets of Quantum Unitary Control Landscapes
This work considers various families of quantum control landscapes (i.e.
objective functions for optimal control) for obtaining target unitary
transformations as the general solution of the controlled Schr\"odinger
equation. We examine the critical point structure of the kinematic landscapes
J_F (U) = ||(U-W)A||^2 and J_P (U) = ||A||^4 - |Tr(AA'W'U)|^2 defined on the
unitary group U(H) of a finite-dimensional Hilbert space H. The parameter
operator A in B(H) is allowed to be completely arbitrary, yielding an objective
function that measures the difference in the actions of U and the target W on a
subspace of state space, namely the column space of A. The analysis of this
function includes a description of the structure of the critical sets of these
kinematic landscapes and characterization of the critical points as maxima,
minima, and saddles. In addition, we consider the question of whether these
landscapes are Morse-Bott functions on U(H). Landscapes based on the intrinsic
(geodesic) distance on U(H) and the projective unitary group PU(H) are also
considered. These results are then used to deduce properties of the critical
set of the corresponding dynamical landscapes.Comment: 15 pages, 3 figure
Zeno effect for quantum computation and control
It is well known that the quantum Zeno effect can protect specific quantum
states from decoherence by using projective measurements. Here we combine the
theory of weak measurements with stabilizer quantum error correction and
detection codes. We derive rigorous performance bounds which demonstrate that
the Zeno effect can be used to protect appropriately encoded arbitrary states
to arbitrary accuracy, while at the same time allowing for universal quantum
computation or quantum control.Comment: Significant modifications, including a new author. To appear in PR
Volume Fractions of the Kinematic "Near-Critical" Sets of the Quantum Ensemble Control Landscape
An estimate is derived for the volume fraction of a subset in the neighborhood
of the critical set
of the kinematic quantum ensemble control landscape J(U) = Tr(U\rho U' O),
where represents the unitary time evolution operator, {\rho} is the initial
density matrix of the ensemble, and O is an observable operator. This estimate
is based on the Hilbert-Schmidt geometry for the unitary group and a
first-order approximation of . An upper bound on these
near-critical volumes is conjectured and supported by numerical simulation,
leading to an asymptotic analysis as the dimension of the quantum system
rises in which the volume fractions of these "near-critical" sets decrease to
zero as increases. This result helps explain the apparent lack of influence
exerted by the many saddles of over the gradient flow.Comment: 27 pages, 1 figur
Optimized pulses for the control of uncertain qubits
Constructing high-fidelity control fields that are robust to control, system,
and/or surrounding environment uncertainties is a crucial objective for quantum
information processing. Using the two-state Landau-Zener model for illustrative
simulations of a controlled qubit, we generate optimal controls for \pi/2- and
\pi-pulses, and investigate their inherent robustness to uncertainty in the
magnitude of the drift Hamiltonian. Next, we construct a quantum-control
protocol to improve system-drift robustness by combining environment-decoupling
pulse criteria and optimal control theory for unitary operations. By
perturbatively expanding the unitary time-evolution operator for an open
quantum system, previous analysis of environment-decoupling control pulses has
calculated explicit control-field criteria to suppress environment-induced
errors up to (but not including) third order from \pi/2- and \pi-pulses. We
systematically integrate this criteria with optimal control theory,
incorporating an estimate of the uncertain parameter, to produce improvements
in gate fidelity and robustness, demonstrated via a numerical example based on
double quantum dot qubits. For the qubit model used in this work, post facto
analysis of the resulting controls suggests that realistic control-field
fluctuations and noise may contribute just as significantly to gate errors as
system and environment fluctuations.Comment: 38 pages, 15 figures, RevTeX 4.1, minor modifications to the previous
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