  The quantum Zeno effect asserts that quantum measurements inhibit
simultaneous unitary dynamics when the "collapse" events are sufficiently
strong and frequent. This applies in the limit of strong continuous measurement
or dissipation. It is possible to implement a dissipative control that is known
as "Zeno Dragging", by dynamically varying the monitored observable, and hence
also the eigenstates which are attractors under Zeno dynamics. This is similar
to adiabatic processes, in that the Zeno dragging fidelity is highest when the
rate of eigenstate change is slow compared to the measurement rate. We
demonstrate here two theoretical methods for using such dynamics to achieve
control of quantum systems. The first, which we shall refer to as "shortcut to
Zeno" (STZ), is analogous to the shortcuts to adiabaticity (counterdiabatic
driving) that are frequently used to accelerate unitary adiabatic evolution. In
the second approach we apply the Chantasri Dressel Jordan (2013, CDJ)
stochastic action, and demonstrate that the extremal-probability readout paths
derived from this are well suited to setting up a Pontryagin-style optimization
of the Zeno dragging schedule. A fundamental contribution of the latter
approach is to show that an action suitable for measurement-driven control
optimization can be derived quite generally from statistical arguments.
Implementing these methods on the Zeno dragging of a qubit, we find that both
approaches yield the same solution, namely, that the optimal control is a
unitary that matches the motion of the Zeno-monitored eigenstate. We then show
that such a solution can be more robust than a unitary-only operation, and
comment on solvable generalizations of our qubit example embedded in larger
systems. These methods open up new pathways toward systematically developing
dynamic control of Zeno subspaces to realize dissipatively-stabilized quantum
operations.
