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arXiv:2410.13844 · Physical review. B./Physical review. B

Post-measurement quantum Monte Carlo

Kriti Baweja, David J. Luitz, Samuel J. Garratt

Published Nov 13, 2025• 1 citation
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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 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.

DOI
10.1103/2tx2-7z18

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Quantum Monte Carlo (QMC) methods are essential for the numerical study of large-scale quantum many-body systems, yet their utility has been significantly hampered by the difficulty in computing key quantities such as off-diagonal operators and entanglement. This Letter introduces Bell-QMC, a novel QMC framework leveraging Bell sampling, a two-copy measurement protocol in the transversal Bell basis. We demonstrate that Bell-QMC enables an efficient and unbiased estimation of both challenging classes of observables, offering a significant advantage over previous QMC approaches. Notably, the entanglement across all system partitions can be computed in a single Bell-QMC simulation. We implement this method within the stochastic series expansion, where we design an efficient update scheme for sampling the configurations in the Bell basis. We demonstrate our algorithm in the one-dimensional transverse-field Ising model and the two-dimensional Z 2 lattice gauge theory, extracting universal quantum features using only simple diagonal measurements. This Letter establishes Bell-QMC as a powerful framework that significantly expands the accessible quantum properties in QMC simulations, providing a substantial advantage over conventional QMC.
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The Stochastic Series Expansion (SSE) technique is a quantum Monte Carlo method that is especially efficient for many quantum spin systems and boson models. It was the first generic method free from the discretization errors affecting previous path integral based approaches. These lecture notes give a brief overview of the SSE method and its applications. In the introductory section, the representation of quantum statistical mechanics by the power series expansion of e−βH{\rm e}^{-βH}e−βH will be compared with path integrals in discrete and continuous imaginary time. Extensions of the SSE approach to ground state projection and quantum annealing in imaginary time will also be briefly discussed. The later sections introduce efficient sampling schemes (loop and cluster updates) that have been developed for many classes of models. A summary of generic forms of estimators for important observables are also given. Applications are discussed in the last section.
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