  We introduce the concept of \textit{Quantum Dynamical Emulation}, a
constructive method for mapping the solutions of non-unitary dynamics to a
weighted set of unitary operations. This allows us to derive a new
correspondence between real and imaginary time, which we term Imaginary Time
Quantum Dynamical Emulation (ITQDE). This enables an imaginary time evolution
to be constructed from the overlaps of states evolved in opposite directions in
real time. We show that a single trajectory evolved using ITQDE can be used not
only to infer ground and thermal states, but also to resolve information about
the complete Hamiltonian spectrum. We further employ ITQDE to derive novel
thermodynamic results, including a generalisation of the Hubbard-Stratonovich
transform. We go on to develop a quantum algorithm for computing the spectra of
quantum systems that is based on this premise. We demonstrate the utility of
this method through numerical simulation, as well as quantum hardware
implementations.
