Unbiased Quantum Simulation with Feynman's iη Prescription

Quantum phase estimation is a paradigm of unbiased quantum simulation designed to yield exact results for intractable models. However, this approach is, in practice, hindered by high demand for deep circuits and error-correcting codes. Biased quantum simulation, including variational quantum eigensolvers, was developed as an alternative approach using shallow circuits, but only provides approximate ground-state energy estimates. In this work, we revisit unbiased quantum simulation in the context of quantum phase estimation while introducing Feynman's iη prescription. We show that this prescription relaxes the otherwise stringent conditions on circuit depth. As a specific application, we develop a hybrid quantum gap estimation algorithm to estimate the energy gap of the transverse-field Ising model. We show that the gap estimation algorithm tolerates a common noise channel, one and two-qubit depolarizing noise. We demonstrate the result of noisy quantum simulation using IBM quantum hardware, and discuss the impact of noise channels on gap estimation.