  Adaptive variational quantum eigensolvers (ADAPT-VQEs) are promising
candidates for simulations of strongly correlated systems on near-term quantum
hardware. To further improve the noise resilience of these algorithms, recent
efforts have been directed towards compactifying, or layering, their ansatz
circuits. Here, we broaden the understanding of the algorithmic layering
process in three ways. First, we investigate the non-commutation relations
between the different elements that are used to build ADAPT-VQE ans\"atze.
Doing so, we develop a framework for studying and developing layering
algorithms, which produce shallower circuits. Second, based on this framework,
we develop a new subroutine that can reduce the number of quantum-processor
calls by optimizing the selection procedure with which a variational quantum
algorithm appends ansatz elements. Third, we provide a thorough numerical
investigation of the noise-resilience improvement available via layering the
circuits of ADAPT-VQE algorithms. We find that layering leads to an improved
noise resilience with respect to amplitude-damping and dephasing noise, which,
in general, affect idling and non-idling qubits alike. With respect to
depolarizing noise, which tends to affect only actively manipulated qubits, we
observe no advantage of layering.
