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Hamiltonian engineering has been shown to be a powerful technique, which
can be applied to many different problems that involve steering a
quantum system to achieve a desirable outcome, and a particularly
promising approach to Hamiltonian engineering is the optimal control
approach, i.e., formulating the problem as an optimization problem.
However, the problem formulation is important, and although optimization
is a well-established field, the solution of the resulting optimization
problems is usually not trivial, in part because the search space is
usually infinite dimensional. To overcome this obstacle the controls
must be parametrized, and the parametrization is critical. The most
common approach is to approximate the controls using piecewise constant
functions. While adequate for some problems, such a parametrization
inevitably leads to high bandwidth solutions due to the discontinuities
of the fields. We demonstrate that using more natural parameterizations
we can significantly reduce the bandwidth of the fields, although at the
expense of having to solve more complex optimization problems. Another
crucial variable is the problem formulation itself. Often, optimal
control problems are formulated using Hamiltonians that incorporate many
approximations, e.g., RWA, off-resonant excitations and fixed couplings
negligible, etc, which inevitably limit what can be achieved by optimal
control. We show that we can in principle speed up the implementation
of quantum gates several orders of magnitude compared to conventional
frequency-selective geometric control pulses for certain systems by
avoiding such approximations and taking advantage of the full range of
off-resonant excitations and couplings available in the optimal control
framework. Another problem with Hamiltonian engineering is that the
most effective approaches are model-based, i.e., we require a model of
the system, especially its response to external fields, or the
functional dependence on the controls. In some cases this isn't a
problem and optimal controls can be designed to be robust with regard to
model uncertainties. For other problems, however, such as information
transfer through spin networks using simple local actuators, it can be
shown that the optimal switching sequences are highly model-dependent,
while the exact network topology and precise couplings for such systems
are usually not known. Such problems call for closed loop optimization.
We show that we can effectively solve problems such as finding optimal
switching time sequences for such networks by adapting gradient-based
optimization algorithms even for problems where the standard
evolutionary algorithms fail completely to find acceptable solutions.
Finally, there are certain types of problems that Hamiltonian
engineering, although an extremely powerful tool for quantum
engineering, cannot solve. One such problem is stabilization in the
presence of environmental interactions. This problem can in principle
be addressed using reservoir engineering. We consider a variant of
Markovian reservoir engineering using direct feedback from an indirect
measurement such as homodyne detection. We show that if the control and
feedback Hamiltonians in this setting are unrestricted and we have some
degree of control over the type of measurement we can perform, then any
state can be in principle be stabilized.
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