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arXiv:2411.00480

Fusing matrix-product states with quantum Monte Carlo: reducing entanglement and sign problem at the same time

Gunnar Bollmark, Sam Mardazad, Johannes Stephan Hofmann, Adrian Kantian

Published Nov 1, 2024• 0 citations
arXivHTMLPDF

Abstract

Systems of correlated quantum matter can be a steep challenge to any would-be method of solution. Matrix-product state (MPS)-based methods can describe 1D systems quasiexactly, but often struggle to retain sufficient bipartite entanglement to accurately approximate 2D systems already. Conversely, Quantum Monte Carlo (QMC) approaches, based on sampling a probability distribution, can generally approximate 2D and 3D systems with an error that decays systematically with growing sampling size. However, QMC can suffer from the so-called sign problem, that makes the approach prohibitively costly for many systems of interest, such as repulsively interacting fermions away from commensurate densities and frustrated systems. In this article, we introduce a new hybrid approach, that combines auxiliary-field QMC (AFQMC) with MPS-based algorithms. This hybrid technique removes or reduces the sign problem (depending on the specific model) while also needing to retain much lower bipartite entanglement than brute-force application of a MPS-algorithm, without the use of uncontrolled approximations. We present two use-cases of the algorithm that would be challenging or impossible to address with any other approach, and quantify the extent of any remaining sign problem.

DOI
10.48550/arxiv.2411.00480

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Systems of correlated quantum matter can be a steep challenge to any would-be method of solution. Matrix-product state (MPS)-based methods can describe 1D systems quasiexactly, but often struggle to retain sufficient bipartite entanglement to accurately approximate 2D systems already. Conversely, Quantum Monte Carlo (QMC) approaches, based on sampling a probability distribution, can generally approximate 2D and 3D systems with an error that decays systematically with growing sampling size. However, QMC can suffer from the so-called sign problem, that makes the approach prohibitively costly for many systems of interest, such as repulsively interacting fermions away from commensurate densities and frustrated systems. In this article, we introduce a new hybrid approach, that combines auxiliary-field QMC (AFQMC) with MPS-based algorithms. This hybrid technique removes or reduces the sign problem (depending on the specific model) while also needing to retain much lower bipartite entanglement than brute-force application of a MPS-algorithm, without the use of uncontrolled approximations. We present two use-cases of the algorithm that would be challenging or impossible to address with any other approach, and quantify the extent of any remaining sign problem.
We marry tensor network states (TNS) and projector quantum Monte Carlo (PMC) to overcome the high computational scaling of TNS and the sign problem of PMC. Using TNS as trial wave functions provides a route to systematically improve the sign structure and to eliminate the bias in fixed-node and constrained-path PMC. As a specific example, we describe phaseless auxiliary-field quantum Monte Carlo with matrix product states (MPS-AFQMC). MPS-AFQMC improves significantly on the density-matrix renormalization group (DMRG) ground-state energy. For the ${J}_{1}\ensuremath{-}{J}_{2}$ model on two-dimensional square lattices, we observe with MPS-AFQMC an order of magnitude reduction in the error for all couplings, compared to DMRG. The improvement is independent of walker bond dimension, and we therefore use bond dimension 1 for the walkers. The computational cost of MPS-AFQMC is then quadratic in the bond dimension of the trial wave function, which is lower than the cubic scaling of DMRG. The error due to the constrained-path bias is proportional to the variational error of the trial wave function. We show that for the ${J}_{1}\ensuremath{-}{J}_{2}$ model on two-dimensional square lattices, a linear extrapolation of the MPS-AFQMC energy with the discarded weight from the DMRG calculation allows us to remove the constrained-path bias. Extensions to other tensor networks are briefly discussed.
In this work, we report, for the first time, an implementation of fermionic auxiliary-field quantum Monte Carlo (AFQMC) using matrix product state (MPS) trial wavefunctions, dubbed MPS-AFQMC. Calculating overlaps between an MPS trial and arbitrary Slater determinants up to a multiplicative error, a crucial subroutine in MPS-AFQMC, is proven to be #P-hard. Nonetheless, we tested several promising heuristics in successfully improving fermionic phaseless AFQMC energies. We also proposed a way to evaluate local energy and force bias evaluations free of matrix product operators. This allows for larger basis set calculations without significant overhead. We showcase the utility of our approach on one- and two-dimensional hydrogen lattices, even when the MPS trial itself struggles to obtain high accuracy. Our work offers a new set of tools that can solve currently challenging electronic structure problems with future improvements.
to develop quantum Monte Carlo (QMC) approaches for excited states and for more systematic and accurate treatment of ground states of strongly correlated sytems and heavier elements. We use the auxiliary-field quantum Monte Carlo (AFQMC) framework, which is a many-body total energy method that looks like many coupled mean-field calculations. I will discuss this method’s relations and dierences with traditional density-functional and quantum chemistry methods, and with other forms of QMC. The AFQMC method uses importance-sampled random walks in the space of nonorthogonal Slater determinants to project out the many-body ground state, or excited state. The random walks are constrained by a trial wave function (TWF) to remove the so-called sign problem, giving approximate (but non-perturbative) results. With singledeterminant TWFs from Hartree-Fock or density-functional theory (DFT), this approach can be thought of as a post-DFT method with no parameter. In this form, the accuracy is typically comparable to the quantum chemistry coupled-cluster CCSD(T) method near equilibrium geometries, and better than CCSD(T) when bonds are stretched. The computational cost of AFQMC scales as N 3 with characteristic size of the system. Its accuracy can be further improved at the cost of increased computing (releasing the constraint) or better TWF. With AFQMC, we can systematically target selected orbitals to correlate. This allows many-body calculations in solids with a frozen core, with down-folded Hamiltonians, or possibly embeded in a DFT calculations to extend scales. Results from a few on-going applications will be presented, including band structures in simple solids, magnetic phases in MnO, and Cobalt adsoption on graphene.
Quantum Monte Carlo (QMC) simulations involving fermions have a notorious sign problem. Some well-known exceptions to the auxiliary field QMC algorithm rely on the factorizibility of the fermion determinant. Recently, a fermionic QMC algorithm [C. Wu, J. Hu, and S. Zhang, Phys. Rev. Lett. 91, 186402 (2003)] has been found in which the fermion determinant may not necessarily be factorizable, but can instead be expressed as a product of complex conjugate pairs of eigenvalues, thus eliminating the sign problem for a much wider class of models. In this paper, we present the general conditions for the applicability of this algorithm and point out that it is deeply related to the time-reversal symmetry of the fermion matrix. We apply this method to various models of strongly correlated systems at all doping levels and lattice geometries, and show that many phases can be simulated without the sign problem.
Quantum Monte Carlo simulations of fermions are hampered by the notorious sign problem whose most striking manifestation is an exponential growth of sampling errors with the number of particles. With the sign problem known to be an NP-hard problem and any generic solution thus highly elusive, the Monte Carlo sampling of interacting many-fermion systems is commonly thought to be restricted to a small class of model systems for which a sign-free basis has been identified. Here we demonstrate that entanglement measures, in particular the so-called R\'enyi entropies, can intrinsically exhibit a certain robustness against the sign problem in auxiliary-field quantum Monte Carlo approaches and possibly allow for the identification of global ground-state properties via their scaling behavior even in the presence of a strong sign problem. We corroborate these findings via numerical simulations of fermionic quantum phase transitions of spinless fermions on the honeycomb lattice at and below half filling.
Reliable simulations of correlated quantum systems, including high-temperature superconductors and frustrated magnets, are increasingly desired nowadays to further our understanding of essential features in such systems. Quantum Monte Carlo (QMC) is a unique numerically exact and intrinsically unbiased method to simulate interacting quantum many-body systems. More importantly, when QMC simulations are free from the notorious fermion sign problem, they can reliably simulate interacting quantum models with large system size and low temperature to reveal low-energy physics such as spontaneously broken symmetries and universal quantum critical behaviors. Here, we concisely review recent progress made in developing new sign-problem-free QMC algorithms, including those employing Majorana representation and those utilizing hot-spot physics. We also discuss applications of these novel sign-problem-free QMC algorithms in simulations of various interesting quantum many-body models. Finally, we discuss possible future directions of designing sign-problem-free QMC methods.