Simulation of bilayer Hamiltonians based on monitored quantum trajectories

In the study of open quantum systems it is often useful to treat mixed states as pure states of a fictitious doubled system. In this work we explore the opposite approach: mapping isolated bilayer systems to open monolayer systems. Specifically, we show that arbitrary bilayer Hamiltonians possessing an antiunitary layer exchange symmetry, and subject to a constraint on the sign of interlayer couplings, can be mapped to Lindbladians on a monolayer system with some of the jump operators postselected on a fixed outcome ("monitored"). Low-energy states of the bilayer Hamiltonian then correspond to late-time states of the monolayer dynamics. Simulating the latter by quantum trajectory methods has the potential of substantially reducing the computational cost of estimating low-energy observables in the bilayer Hamiltonian by effectively halving the system size. The overhead due to sampling quantum trajectories can be controlled by a suitable importance sampling scheme. We show that, when the quantum trajectories exhibit free fermion dynamics, our approach reduces to the auxiliary field quantum Monte Carlo (AFQMC) method. This provides a physically transparent interpretation of the AFQMC sign-free criteria in terms of properties of quantum dynamics. Finally, we benchmark our approach on the 1D quantum Ashkin-Teller model.