Towards energy-constrained continuous-variable designs

Exactly characterizing and verifying the behavior of multi-qubit devices typically requires sampling from exponentially large sets. Approximate approaches often utilize state and unitary designs --- "evenly distributed" subsets of states and unitaries, respectively, that reduce the computational cost of calculating averages of functions over the aforementioned sets. In the continuous-variable (CV) domain, such tools would help characterize novel optical- and microwave-based quantum devices that are quickly becoming too large for an exact analysis. However, the noncompact nature of the CV phase space presents obstructions to creating analogous tools using simple resources. For example, positive-weighted CV designs cannot be constructed with Gaussian operations [Blume-Kohout, Turner 2014]. In this work, we present various approximate designs formed by relaxing the positive-weight constraint and, in some cases, also using non-Gaussian resources. Our constructions are constrained in either the average or the total occupation number, utilizing displaced squeezed states, displaced Fock states, and combinations of finite-simplex and torus designs.