  In this work, we introduce a quantum-control-inspired method for the
characterization of variational quantum circuits using the rank of the
dynamical Lie algebra associated with the hermitian generator(s) of the
individual layers. Layer-based architectures in variational algorithms for the
calculation of ground-state energies of physical systems are taken as the focus
of this exploration. A promising connection is found between the Lie rank, the
accuracy of calculated energies, and the requisite depth to attain target
states via a given circuit architecture, even when using a lot of parameters
which is appreciably below the number of separate terms in the generators. As
the cost of calculating the dynamical Lie rank via an iterative process grows
exponentially with the number of qubits in the circuit and therefore becomes
prohibitive quickly, reliable approximations thereto are desirable. The
rapidity of the increase of the dynamical Lie rank in the first few iterations
of the calculation is found to be a viable (lower bound) proxy for the full
calculation, balancing accuracy and computational expense. We, therefore,
propose the dynamical Lie rank and proxies thereof as a useful design metric
for layer-structured quantum circuits in variational algorithms.
