We study the impact of different coordinate systems and measurement schemes to reduce the measurement overhead in calculating anharmonic vibrational wavefunctions on quantum computers.

We present equations of motion (EOMs) for general time-dependent wave functions with exponentially parameterized biorthogonal basis sets. The equations are fully bivariational in the sense of the time-dependent bivariational principle and offer an alternative, constraint-free formulation of adaptive basis sets for bivariational wave functions. We simplify the highly non-linear basis set equations using Lie algebraic techniques and show that the computationally intensive parts of the theory are, in fact, identical to those that arise with linearly parameterized basis sets. Thus, our approach offers easy implementation on top of existing code in the context of both nuclear dynamics and time-dependent electronic structure. Computationally tractable working equations are provided for single and double exponential parametrizations of the basis set evolution. The EOMs are generally applicable for any value of the basis set parameters, unlike the approach of transforming the parameters to zero at each evaluation of the EOMs. We show that the basis set equations contain a well-defined set of singularities, which are identified and removed by a simple scheme. The exponential basis set equations are implemented in conjunction with the time-dependent modals vibrational coupled cluster (TDMVCC) method, and we investigate the propagation properties in terms of the average integrator step size. For the systems we test, the exponentially parameterized basis sets yield slightly larger step sizes compared to the linearly parameterized basis set.

We derive general bivariational equations of motion (EOMs) for time-dependent wave functions with biorthogonal time-dependent basis sets. The time-dependent basis functions are linearly parametrized and their fully variational time evolution is ensured by solving a set of so-called constraint equations, which we derive for arbitrary wave function expansions. The formalism allows the division of the basis set into an active basis and a secondary basis, ensuring a flexible and compact wave function. We show how the EOMs specialize to a few common wave function forms, including coupled cluster (CC) and linearly expanded wave functions. It is demonstrated, for the first time, that the propagation of such wave functions is not unconditionally stable when a secondary basis is employed. The main signature of the instability is a strong increase in non-orthogonality that eventually causes the calculation to fail; specifically, the biorthogonal active bra and ket bases tend towards spanning different spaces. Although formally allowed, this causes severe numerical issues. We identify the source of the problem by reparametrizing the time-dependent basis set through polar decomposition. The subsequent analysis allows us to remove the instability by setting appropriate matrix elements to zero. Although this solution is not fully variational, we find essentially no deviation in terms of autocorrelation functions relative to the variational formulation. We expect that the results presented here will be useful for the formal analysis of bivariational time-dependent wave functions for electronic and nuclear dynamics in general and for the practical implementation of time-dependent CC wave functions in particular.