Post-measurement quantum Monte Carlo

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 nonunitary 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 $\mathrm{SU}(2)$ symmetry, and at first we preserve this symmetry by measuring $\mathrm{SU}(2)$ symmetric observables. We identify classes of post-measurement states for which correlations can be calculated efficiently, as well as states for which $\mathrm{SU}(2)$ 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.