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arXiv:2603.10948

Generalized Reduced-Density-Matrix Quantum Monte Carlo Gives Access to More

Zhiyan Wang, Zhe Wang, Bin-Bin Mao, Zheng Yan

cond-mat.str-elcond-mat.stat-mechphysics.comp-phquant-phSubmitted Mar 11, 2026• Announced Mar 12, 2026• 0 citations
v1 · Mar 11, 2026v2 · Mar 12, 2026
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Abstract

In quantum Monte Carlo (QMC), what can be measured efficiently is largely determined by what is sampled. When the sampled object is the partition function, a broad class of observables, including general off-diagonal operators, is typically unavailable as direct estimators. In this article, we introduce a paradigm shift by replacing the partition function with a generalized reduced density matrix (GRDM) as the simulated object. This reformulation removes the measurement bottleneck at its source and extends the dimensional-reduction advantage of reduced descriptions from static quantities to dynamical observables, thereby enabling much richer information extraction. As substantial demonstrations, the framework allows the directed-loop algorithm to measure both equal-time and imaginary-time off-diagonal observables, with the latter giving direct access to dynamical spectra. It also enables measurements of Rényi-1 correlators that diagnose strong-to-weak symmetry breaking in mixed states. This work establishes a unified framework for holographic characterization within QMC.

Comments
12 pages, 8 figures

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In quantum Monte Carlo (QMC), what can be measured efficiently is largely determined by what is sampled. When the sampled object is the partition function, a broad class of observables, including general off-diagonal operators, is typically unavailable as direct estimators. In this article, we introduce a paradigm shift by replacing the partition function with a generalized reduced density matrix (GRDM) as the simulated object. This reformulation removes the measurement bottleneck at its source and extends the dimensional-reduction advantage of reduced descriptions from static quantities to dynamical observables, thereby enabling much richer information extraction. As substantial demonstrations, the framework allows the directed-loop algorithm to measure both equal-time and imaginary-time off-diagonal observables, with the latter giving direct access to dynamical spectra. It also enables measurements of Rényi-1 correlators that diagnose strong-to-weak symmetry breaking in mixed states. This work establishes a unified framework for holographic characterization within QMC.
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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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Several algorithms have been proposed to calculate the spatial entanglement spectrum from high order R\'enyi entropies and maximum entropy techniques. We present an alternative approach for computing the entanglement spectrum with quantum Monte Carlo for both continuum and lattice Hamiltonians. This method provides direct access to the matrix elements of the spatially reduced density matrix and we determine an estimator that can be used in variational Monte Carlo as well as other Monte Carlo methods. The algorithm is based on using a generalization of the swap operator, which can be extended to calculate a general class of density matrices that can include combinations of spin, space, particle, and even momentum degrees of freedom. We demonstrate the method by applying it to the H2{\mathrm{H}}_{2}H2​ and N2{\mathrm{N}}_{2}N2​ molecules and describe how the spatial entanglement spectrum encodes a covalent bond that includes all the many body correlations.
We extend Quantum Computing Quantum Monte Carlo (QCQMC) beyond ground-state energy estimation by systematically constructing the quantum circuits used for state preparation. Replacing the original Variational Quantum Eigensolver (VQE) prescription with task-adapted unitaries, we show that QCQMC can address excited-state spectra via Variational Fast Forwarding and the Variational Unitary Matrix Product Operator (VUMPO), combinatorial optimization via a symmetry-preserving VQE ansatz, and finite-temperature observables via Haar-random unitaries. Benchmarks on molecular, condensed-matter, nuclear-structure, and graph-optimization problems demostrate that the QMC diffusion step consistently improves the energy accuracy of the underlying state-preparation method across all tested domains. For weakly correlated systems, VUMPO achieves near-exact energies with significantly shallower circuits by offloading optimization to a classical tensor-network pre-training step, while for strongly correlated systems, the QMC correction becomes essential. We further provide a proof-of-concept demonstration that Haar-random basis state preparation within QCQMC yields finite-temperature estimates from pure-state dynamics.