Quantum Monte Carlo methods are powerful tools for studying quantum many-body systems but face difficulties in accessing excited states and in treating sign problems. We present a continuous-time path-integral Monte Carlo method for computing the low-lying spectrum of generic quantum Hamiltonians within a projection subspace. The method projects the thermal density matrix onto a subspace spanned by a chosen set of linearly independent states. It is free of Trotter discretization errors and systematically converges to the low-energy states which have finite overlap with the projection subspace as the $β$ parameter increases. While most effective for systems without a sign problem, the method also yields information about low-energy spectra when sign problems are present. We illustrate the approach on two problems. For the sign-free case, we compute the first four low-energy levels in the scaling limit of the one-dimensional Ising model with both transverse and longitudinal fields, demonstrating the flow from the conformal limit to the massive $E_8$ quantum field theory. For the sign-problem case, we apply the method to the frustrated Shastry-Sutherland model and benchmark it against exact diagonalization on small lattices. We also present results for larger systems beyond the lattice sizes accessible to exact diagonalization, while limited to small $β$ where sign problems occur. Our method provides a general route toward quantum Monte Carlo spectroscopy for lattice Hamiltonians.Quantum Monte Carlo methods are powerful tools for studying quantum many-body systems but face difficulties in accessing excited states and in treating sign problems. We present a continuous-time path-integral Monte Carlo method for computing the low-lying spectrum of generic quantum Hamiltonians within a projection subspace. The method projects the thermal density matrix onto a subspace spanned by a chosen set of linearly independent states. It is free of Trotter discretization errors and systematically converges to the low-energy states which have finite overlap with the projection subspace as the $β$ parameter increases. While most effective for systems without a sign problem, the method also yields information about low-energy spectra when sign problems are present. We illustrate the approach on two problems. For the sign-free case, we compute the first four low-energy levels in the scaling limit of the one-dimensional Ising model with both transverse and longitudinal fields, demonstrating the flow from the conformal limit to the massive $E_8$ quantum field theory. For the sign-problem case, we apply the method to the frustrated Shastry-Sutherland model and benchmark it against exact diagonalization on small lattices. We also present results for larger systems beyond the lattice sizes accessible to exact diagonalization, while limited to small $β$ where sign problems occur. Our method provides a general route toward quantum Monte Carlo spectroscopy for lattice Hamiltonians.

arXiv:2511.19209Nov 24, 20250 citations

Estimating eigenvalues for lattice hamiltonian models

Chris J. Hamer

Physics Letters BMar 1, 197927 citations

Monte Carlo simulation of quantum statistical lattice models

Hans A. De Raedt, Ad Lagendijk

Physics ReportsOct 1, 1985188 citations

Sample-based Hamiltonian and Lindbladian simulation: non-asymptotic analysis of sample complexity

Byeongseon Go, Hyukjoon Kwon, Siheon Park +2 more

Abstract Density matrix exponentiation (DME) is a quantum algorithm that processes multiple copies of a program state σ to realize the Hamiltonian evolution e − i σ t . Wave matrix Lindbladization (WML) similarly processes multiple copies of a program state ψ L in order to realize a Lindbladian evolution. Both algorithms are prototypical sample-based quantum algorithms and can be used for various quantum information processing tasks, including quantum principal component analysis, Hamiltonian simulation, and Lindbladian simulation. In this work, we present detailed sample complexity analyses for DME and sample-based Hamiltonian simulation, as well as for WML and sample-based Lindbladian simulation. In particular, we prove that the sample complexity of DME is no larger than 4 t 2 / ε , where t is the desired evolution time and ɛ is the desired imprecision level, as quantified by the normalized diamond distance. We also establish a fundamental lower bound on the sample complexity of sample-based Hamiltonian simulation, which matches our DME sample complexity bound up to a constant multiplicative factor, thereby proving that DME is optimal for sample-based Hamiltonian simulation. Additionally, we prove that the sample complexity of WML is no larger than 3 t 2 d 2 / ε , where d is the dimension of the space on which the Lindblad operator acts nontrivially, and we prove a lower bound of 10 − 4 t 2 / ε on the sample complexity of sample-based Lindbladian simulation. These results prove that WML is optimal for sample-based Lindbladian simulation whenever the Lindblad operator acts nontrivially on a constant-sized system. Finally, we point out that the DME sample complexity analysis in Kimmel et al(2017 npj Quantum Inf. 3 13) and the WML sample complexity analysis in Patel and Wilde (2023 Open Syst. Inf. Dyn. 30 2350010) appear to be incomplete, highlighting the need for the results presented here.Abstract Density matrix exponentiation (DME) is a quantum algorithm that processes multiple copies of a program state σ to realize the Hamiltonian evolution e − i σ t . Wave matrix Lindbladization (WML) similarly processes multiple copies of a program state ψ L in order to realize a Lindbladian evolution. Both algorithms are prototypical sample-based quantum algorithms and can be used for various quantum information processing tasks, including quantum principal component analysis, Hamiltonian simulation, and Lindbladian simulation. In this work, we present detailed sample complexity analyses for DME and sample-based Hamiltonian simulation, as well as for WML and sample-based Lindbladian simulation. In particular, we prove that the sample complexity of DME is no larger than 4 t 2 / ε , where t is the desired evolution time and ɛ is the desired imprecision level, as quantified by the normalized diamond distance. We also establish a fundamental lower bound on the sample complexity of sample-based Hamiltonian simulation, which matches our DME sample complexity bound up to a constant multiplicative factor, thereby proving that DME is optimal for sample-based Hamiltonian simulation. Additionally, we prove that the sample complexity of WML is no larger than 3 t 2 d 2 / ε , where d is the dimension of the space on which the Lindblad operator acts nontrivially, and we prove a lower bound of 10 − 4 t 2 / ε on the sample complexity of sample-based Lindbladian simulation. These results prove that WML is optimal for sample-based Lindbladian simulation whenever the Lindblad operator acts nontrivially on a constant-sized system. Finally, we point out that the DME sample complexity analysis in Kimmel et al(2017 npj Quantum Inf. 3 13) and the WML sample complexity analysis in Patel and Wilde (2023 Open Syst. Inf. Dyn. 30 2350010) appear to be incomplete, highlighting the need for the results presented here.

arXiv:2412.02134Quantum Science and TechnologySep 17, 20254 citations

Density matrix analysis of systems with periodic Hamiltonians

Soham Sen, Manjari Dutta, Sunandan Gangopadhyay

arXiv:2507.06355The European Physical Journal DSep 1, 20250 citations

Perfect Born Sampling of Symmetric Thermal Tensor Network States

Jianxin Gao, Qiaoyi Li, Yuan Gao +5 more

Accurate calculations of correlated fermions at low temperatures constitute a major challenge in quantum many-body physics, most notably in the studies of high- $T_c$ superconductivity and ultracold atom optical lattices. Stochastic sampling of tensor-network states offers a promising route to tackle this problem; however, existing schemes have long faced a fundamental dilemma---sampling efficiency and symmetry acceleration \textit{cannot} be achieved simultaneously. Here we propose a perfect Born sampling approach for thermal tensor networks, which performs importance sampling directly from the purified density matrix via the Born rule and incorporates Abelian and non-Abelian symmetries by sampling symmetry quantum numbers. We benchmark the method, realized as both stochastic matrix product states (stoMPS) and stochastic projected entangled pair states (stoPEPS), on large-scale quantum lattice models. Using stoMPS, we accurately simulate the square-lattice Hubbard model on cylinders up to width $W=10$ , and study the triangular-lattice Hubbard model down to $T/t = 1/64$ . We reveal scalar chiral order at half filling and kinetic ferromagnetism upon electron doping, with the latter directly compared to ultracold atom experiments. We further extend the stoMPS method to compute finite-temperature quantum dynamics, as demonstrated by the optical conductivity of the Hubbard model. We showcase the generalization of stoPEPS on up to $20\times20$ square-lattice quantum Ising model, where accurate results are obtained near its quantum critical point. Our method combines high sampling efficiency with full symmetry acceleration. The resulting state-of-the-art framework captures both equilibrium and dynamical properties down to ultralow temperatures, thereby providing a powerful theoretical tool for correlated quantum systems, particularly fermion Hubbard quantum gas microscopy.Accurate calculations of correlated fermions at low temperatures constitute a major challenge in quantum many-body physics, most notably in the studies of high-$T_c$ superconductivity and ultracold atom optical lattices. Stochastic sampling of tensor-network states offers a promising route to tackle this problem; however, existing schemes have long faced a fundamental dilemma---sampling efficiency and symmetry acceleration \textit{cannot} be achieved simultaneously. Here we propose a perfect Born sampling approach for thermal tensor networks, which performs importance sampling directly from the purified density matrix via the Born rule and incorporates Abelian and non-Abelian symmetries by sampling symmetry quantum numbers. We benchmark the method, realized as both stochastic matrix product states (stoMPS) and stochastic projected entangled pair states (stoPEPS), on large-scale quantum lattice models. Using stoMPS, we accurately simulate the square-lattice Hubbard model on cylinders up to width $W=10$, and study the triangular-lattice Hubbard model down to $T/t = 1/64$. We reveal scalar chiral order at half filling and kinetic ferromagnetism upon electron doping, with the latter directly compared to ultracold atom experiments. We further extend the stoMPS method to compute finite-temperature quantum dynamics, as demonstrated by the optical conductivity of the Hubbard model. We showcase the generalization of stoPEPS on up to $20\times20$ square-lattice quantum Ising model, where accurate results are obtained near its quantum critical point. Our method combines high sampling efficiency with full symmetry acceleration. The resulting state-of-the-art framework captures both equilibrium and dynamical properties down to ultralow temperatures, thereby providing a powerful theoretical tool for correlated quantum systems, particularly fermion Hubbard quantum gas microscopy.

arXiv:2609.35754Sep 28, 20260 citations

β

parameter increases. While most effective for systems without a sign problem, the method also yields information about low-energy spectra when sign problems are present. We illustrate the approach on two problems. For the sign-free case, we compute the first four low-energy levels in the scaling limit of the one-dimensional Ising model with both transverse and longitudinal fields, demonstrating the flow from the conformal limit to the massive

E_8

quantum field theory. For the sign-problem case, we apply the method to the frustrated Shastry-Sutherland model and benchmark it against exact diagonalization on small lattices. We also present results for larger systems beyond the lattice sizes accessible to exact diagonalization, while limited to small

β

where sign problems occur. Our method provides a general route toward quantum Monte Carlo spectroscopy for lattice Hamiltonians.

Accurate calculations of correlated fermions at low temperatures constitute a major challenge in quantum many-body physics, most notably in the studies of high-

T_c

superconductivity and ultracold atom optical lattices. Stochastic sampling of tensor-network states offers a promising route to tackle this problem; however, existing schemes have long faced a fundamental dilemma---sampling efficiency and symmetry acceleration \textit{cannot} be achieved simultaneously. Here we propose a perfect Born sampling approach for thermal tensor networks, which performs importance sampling directly from the purified density matrix via the Born rule and incorporates Abelian and non-Abelian symmetries by sampling symmetry quantum numbers. We benchmark the method, realized as both stochastic matrix product states (stoMPS) and stochastic projected entangled pair states (stoPEPS), on large-scale quantum lattice models. Using stoMPS, we accurately simulate the square-lattice Hubbard model on cylinders up to width

W=10

, and study the triangular-lattice Hubbard model down to

T/t = 1/64

. We reveal scalar chiral order at half filling and kinetic ferromagnetism upon electron doping, with the latter directly compared to ultracold atom experiments. We further extend the stoMPS method to compute finite-temperature quantum dynamics, as demonstrated by the optical conductivity of the Hubbard model. We showcase the generalization of stoPEPS on up to

20\times20

square-lattice quantum Ising model, where accurate results are obtained near its quantum critical point. Our method combines high sampling efficiency with full symmetry acceleration. The resulting state-of-the-art framework captures both equilibrium and dynamical properties down to ultralow temperatures, thereby providing a powerful theoretical tool for correlated quantum systems, particularly fermion Hubbard quantum gas microscopy.