Effective temperature in approximate quantum many-body states

In the pursuit of numerically identifying ground states of quantum many-body systems, approximate quantum wave function ansatzes are commonly employed. This study focuses on the spectral decomposition of these approximate quantum many-body states into exact eigenstates of the target Hamiltonian, which reflects the intricate physics at the interplay between quantum systems and numerical algorithms. Here, we examine various parametrized approximate quantum states constructed from neural networks, tensor networks, and quantum circuits, employing differentiable programming to numerically approximate ground states and imaginary-time evolved states. Our findings reveal a consistent exponential pattern in the energy eigenbasis decomposition contributions of approximate states across different ansatzes, optimization objectives, and quantum systems, characterized by remarkably small decay rates, i.e., effective temperatures. This finding is counterintuitive for high-fidelity approximate ground states: While the total contribution from excited states can be made sufficiently small, the residual spectral weight does not decay rapidly with energy. This behavior is an intrinsic property of the variational ansatz, independent of the approximation’s overall accuracy. The effective temperature is related to ansatz expressiveness and accuracy and shows phase transition behaviors in learning imaginary-time evolved states. The universal picture and unique features suggest the significance and potential of the effective temperature metric in characterizing approximate quantum states.