Quantum Dynamical Emulation of Imaginary Time Evolution

We introduce a constructive method for mapping non-unitary dynamics to a weighted set of unitary operations. We utilize this construction to derive a new correspondence between real and imaginary time, which we term Imaginary Time Quantum Dynamical Emulation (ITQDE). This correspondence enables an imaginary time evolution to be constructed from the overlaps of states evolved in opposite directions. We develop ITQDE as a tool for estimating the ground and thermal state properties associated with a given Hamiltonian. We additionally provide a prescription for leveraging ITQDE to estimate the complete Hamiltonian spectrum. We go on to develop a quantum algorithm for computing Hamiltonian spectra based on ITQDE, which we validate through numerical simulations and quantum hardware implementations. We conclude with a discussion of how ITQDE can be utilized more broadly to derive novel thermodynamic results, including a generalisation of the Hubbard-Stratonovich transformation.