Solution methods on the quantum computer often have misleading computational complexity. For example, wavefunction preparation can be very costly, eliminating quantum advantage or the ability for the quantum computer to outperform all classical methods. A curious feature of quantum systems is that the thermalize, or achieve an equilibrium statistical ensemble value after a long-time. This is known as thermalization and it is well understood that this occurs in classical systems. In quantum mechanics, we suppose that thermalization also occurs in some systems. Using thermalization can provide an equal superposition of eigenstates in a time-average. Using the quantum phase estimation subroutine on the quantum computer allows for a weighting of the eventual expectation value that appears to be applicable in a wide variety of cases. This strategy circumvents the need for wavefunction preparation on the quantum computer. I discuss the implication of allowing a quantum circuit to thermalize and how to use it in a variety of use cases.

This research was undertaken, in part, thanks to funding from the Canada Research Chairs Program (CRC-2021-00257). This work has been supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC) under grants RGPIN-2023-05510 and DGECR-2023-00026.