Post-measurement Quantum Monte Carlo

Abstract

We show how the effects of large numbers of measurements on many-body quantum ground and thermal states can be studied using Quantum Monte Carlo (QMC). Density matrices generated by measurement in this setting feature products of many local non-unitary operators, and by expanding these density matrices as sums over operator strings we arrive at a generalized stochastic series expansion (SSE). Our `post-measurement SSE' is based on importance sampling of operator strings contributing to a measured thermal density matrix. We demonstrate our algorithm by probing the effects of measurements on the spin-$1/2$ Heisenberg antiferromagnet on the square lattice. Thermal states of this system have \SU{2} symmetry, and at first we preserve this symmetry by measuring \SU{2} symmetric observables. We identify classes of post-measurement states for which correlations can be calculated efficiently, as well as states for which \SU{2} symmetric measurements generate a QMC sign problem when working in any site-local basis. For the first class, we show how deterministic loop updates can be leveraged. Using our algorithm we demonstrate the creation of long-range Bell pairs and symmetry-protected topological order, as well as the measurement-induced enhancement of antiferromagnetic correlations. We then study the effects of measuring the system in a basis where the standard (unmeasured) SSE is sign-free: for measurement schemes with this property, we can calculate correlations in all post-measurement states without a sign problem. The method developed in this work opens the door to scalable experimental probes of measurement-induced collective phenomena, which require numerical estimates for the effects of measurements.