2 research outputs found
Measurement-Induced Boolean Dynamics and Controllability for Quantum Networks
In this paper, we study dynamical quantum networks which evolve according to
Schr\"odinger equations but subject to sequential local or global quantum
measurements. A network of qubits forms a composite quantum system whose state
undergoes unitary evolution in between periodic measurements, leading to hybrid
quantum dynamics with random jumps at discrete time instances along a
continuous orbit. The measurements either act on the entire network of qubits,
or only a subset of qubits. First of all, we reveal that this type of hybrid
quantum dynamics induces probabilistic Boolean recursions representing the
measurement outcomes. With global measurements, it is shown that such resulting
Boolean recursions define Markov chains whose state-transitions are fully
determined by the network Hamiltonian and the measurement observables.
Particularly, we establish an explicit and algebraic representation of the
underlying recursive random mapping driving such induced Markov chains. Next,
with local measurements, the resulting probabilistic Boolean dynamics is shown
to be no longer Markovian. The state transition probability at any given time
becomes dependent on the entire history of the sample path, for which we
establish a recursive way of computing such non-Markovian probability
transitions. Finally, we adopt the classical bilinear control model for the
continuous Schr\"odinger evolution, and show how the measurements affect the
controllability of the quantum networks.Comment: 26 pages, 7 figure
Structural Controllability on Graphs for Drifted Bilinear Systems over Lie Groups
In this paper, we study graphical conditions for structural controllability
and accessibility of drifted bilinear systems over Lie groups. We consider a
bilinear control system with drift and controlled terms that evolves over the
special orthogonal group, the general linear group, and the special unitary
group. Zero patterns are prescribed for the drift and controlled dynamics with
respect to a set of base elements in the corresponding Lie algebra. The drift
dynamics must respect a rigid zero-pattern in the sense that the drift takes
values as a linear combination of base elements with strictly non-zero
coefficients; the controlled dynamics are allowed to follow a free zero pattern
with potentially zero coefficients in the configuration of the controlled term
by linear combination of the controlled base elements. First of all, for such
bilinear systems over the special orthogonal group or the special unitary
group, the zero patterns are shown to be associated with two undirected or
directed graphs whose connectivity and connected components ensure structural
controllability/accessibility. Next, for bilinear systems over the special
unitary group, we introduce two edge-colored graphs associated with the drift
and controlled zero patterns, and prove structural controllability conditions
related to connectivity and the number of edges of a particular color