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arXiv:2011.08457 · Physical review. B./Physical review. B

Global scheme of sweeping cluster algorithm to sample among topological sectors

Zheng Yan

Published May 31, 2022• 36 citations
arXivHTMLPDF

Abstract

Local constraint is closely related to the gauge field, so constrained models are usually effective low energy descriptions and important in condensed matter physics. On the other hand, local restriction hinders the application of numerical algorithms. In addition to the computational difficulties of the constraints, the various topological sectors which cannot be connected through local operators are also one of the key computational difficulties. Taking a quantum dimer model as an example in this paper, we construct a global scheme based on a sweeping cluster Monte Carlo method, which can sample among different topological sectors. In principle, this method can be generalized to other models.

DOI
10.1103/physrevb.105.184432

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Local constraint is closely related to the gauge field, so constrained models are usually effective low energy descriptions and important in condensed matter physics. On the other hand, local restriction hinders the application of numerical algorithms. In addition to the computational difficulties of the constraints, the various topological sectors which cannot be connected through local operators are also one of the key computational difficulties. Taking a quantum dimer model as an example in this paper, we construct a global scheme based on a sweeping cluster Monte Carlo method, which can sample among different topological sectors. In principle, this method can be generalized to other models.
Quantum dimer models~(QDMs) featured by strong geometric constraint are effective low energy descriptions of many quantum spin systems. The geometric restriction described by local gauge field, hinders the application of numerical algorithms. Before sweeping cluster method was applied in world-line quantum Monte Carlo (QMC) algorithm, there is only projector QMC which obey the constraints and could be used for calculation on QDMs. However, the projector QMC for QDMs has some drawbacks, e.g., it is not effective when the parameter interval away from Rokhsar-Kivelson (RK) point. That's because the projector method still lacks a cluster update to improve its efficiency. Although sweeping cluster algorithm improves the update for these systems, it also only works in one winding (topological) sector. In this paper, we improve the sweeping cluster method to sample in different winding sectors.
Abstract Simulating lattice gauge theory (LGT) Hamiltonian and its nontrivial states by programmable quantum devices has attracted numerous attention in recent years. Rydberg atom arrays constitute one of the most rapidly developing arenas for quantum simulation and quantum computing. The Z2 LGT and topological order has been realized in experiments while the U(1) LGT is being worked hard on the way. States of LGT have local constraint and are fragmented into several winding sectors with topological protection. It is therefore difficult to prepare a state in certain sector for experiments, and it is also an important task for quantum topological memory. Here, we propose a protocol of sweeping quantum annealing (SQA) for searching the state within target topological sector. With the Monte Carlo method, we show that this SQA has linear time complexity of size with applications to the antiferromagnetic transverse field Ising model, which has emergent U(1) gauge fields. This SQA protocol can be realized easily on quantum simulation platforms such as Rydberg atoms and superconducting circuits. We expect this approach would provide a generic recipe for resolving the topological hindrances in quantum optimization and the preparation of quantum topological state.
Quantum spin systems with strong geometric restrictions give rise to rich quantum phases such as valence bond solids and spin liquid states. However, the geometric restrictions often hamper the application of sophisticated numerical approaches. Based on the stochastic series expansion method, we develop an efficient and exact quantum Monte Carlo ``sweeping cluster'' algorithm which automatically satisfies the geometrical restrictions. Here we use the quantum dimer model as a benchmark to demonstrate the reliability and power of this algorithm. Comparing to existing numerical methods, we can obtain higher accuracy results for a wider parameter region and much more substantial system sizes.
The sign problem remains a major obstacle to extending the success of Monte Carlo methods to generic quantum many-body systems. A powerful alternative relies on cluster expansions, which provide convergent representations of free energies and related quantities, with applications ranging from tensor network contractions to quantum dynamics and circuits. Within their convergence regime, these expansions often yield deterministic polynomial-time algorithms. However, such methods require exhaustive enumeration of connected clusters, leading to a computational overhead of (Nε−1)O(log⁡Δ)(N\varepsilon^{-1})^{\mathcal{O}(\logΔ)}(Nε−1)O(logΔ), where NNN is the system size, ε\varepsilonε the desired precision, and ΔΔΔ the interaction-graph degree. This becomes particularly severe in long-range systems, where Δ≡NΔ\equiv NΔ≡N, resulting in quasi-polynomial runtimes. Here, we introduce a Markov chain algorithm that samples cluster expansions directly in the space of connected clusters, replacing deterministic enumeration by Monte Carlo sampling. Because sampling occurs in an abstract cluster space rather than a physical configuration space, the conventional exponential sign problem is absent within the convergence regime. For general polymer models, we establish an FPRAS with runtime O~(N4ε−3)\tilde{\mathcal{O}}(N^4\varepsilon^{-3})O~(N4ε−3). For quantum partition functions at temperatures β≤βsampling≡O(1)β\leqβ_{\mathrm{sampling}}\equiv\mathcal{O}(1)β≤βsampling​≡O(1), this improves to O~(N3+1/Dε−3)\tilde{\mathcal{O}}(N^{3+1/D}\varepsilon^{-3})O~(N3+1/Dε−3), where DDD is the lattice dimension. The framework directly applies to long-range quantum systems with interactions ∝1/rα\propto1/r^α∝1/rα, including 222-local models with α>2Dα>2Dα>2D.
Lattice gauge theories are an important class of physical models with high-dimensional structured distributions that underpin first-principles calculations in particle, nuclear, and condensed-matter physics. Recent advances in generative modeling have opened new avenues for sampling Boltzmann distributions in lattice gauge theories, offering the potential to alleviate limitations of traditional Monte Carlo methods, including critical slowing down and topological freezing. However, generative samplers for lattice gauge theories are typically constructed in link space, where correlations become increasingly long-ranged toward weak coupling, posing a challenge for learning. Plaquettes provide a more natural representation, as the action is local in these variables. However, exact Bianchi constraints restrict them to a lower-dimensional manifold, complicating direct generative modeling. We introduce a multilevel normalizing-flow construction that samples directly in plaquette space while satisfying these constraints exactly. The key idea is a coarse-to fine factorization that transforms a globally coupled constraint problem into a sequence of local solves: at each refinement, every determined plaquette depends on at most four newly generated variables, while the size of the constraint problem remains independent of the lattice size. We validate the construction for U(1)U(1)U(1) in two and four dimensions and for SU(2)SU(2)SU(2) in two dimensions. Our multilevel Plaquette-Space Sampler (PSS) substantially outperforms link-space baselines, with the advantage increasing toward weak coupling, where the gauge coupling becomes small. This regime is particularly challenging for generative sampling and, in asymptotically free gauge theories, is relevant to continuum studies.
Abstract As a typical quantum computing algorithm, quantum annealing is widely used in the optimization of glass-like problems to find the best solution. However, the optimization problems in constrained complex systems usually involve topological structures, and the performance of the quantum annealing algorithm is still largely unknown. Here, we take an Ising system as a typical example with local constraints accompanied by intrinsic topological properties that can be implemented on quantum computing platforms such as the D-wave machine, and study the effectiveness of the quantum annealing algorithm in its optimization and compare it with that of the thermal annealing. We find that although conventional quantum annealing is difficult for the optimization of constrained topological problems, a generalized algorithm --- the sweeping quantum annealing method --- can be designed and solve the problem with better efficiency than both conventional quantum and thermal annealing algorithms. The sweeping quantum annealing algorithm, therefore, opens up a promising avenue for quantum computing of constrained problems and can be readily employed on the optimizations in quantum material design, engineering, and even social sciences.