We establish a general framework for developing fast sampling and counting algorithms for stoquastic spin systems at high temperature. Our framework is based on a rapidly mixing Markov chain for polymer models and a subcritical percolation process for sampling individual polymers. We apply our framework to obtain fast algorithms for approximating the partition function and sampling from the thermal distribution of (1) general stoquastic spin systems, (2) ferromagnetic Heisenberg models, and (3) antiferromagnetic Heisenberg models on bipartite graphs. For the Heisenberg models, we obtain an improved bound on the inverse temperature by using their respective cycle and loop representations.
We establish a general framework for developing fast sampling and counting algorithms for stoquastic spin systems at high temperature. Our framework is based on a rapidly mixing Markov chain for polymer models and a subcritical percolation process for sampling individual polymers. We apply our framework to obtain fast algorithms for approximating the partition function and sampling from the thermal distribution of (1) general stoquastic spin systems, (2) ferromagnetic Heisenberg models, and (3) antiferromagnetic Heisenberg models on bipartite graphs. For the Heisenberg models, we obtain an improved bound on the inverse temperature by using their respective cycle and loop representations.
We establish a general framework for developing approximation algorithms for a class of counting problems. Our framework is based on the cluster expansion of the abstract polymer model formalism of Kotecký and Preiss. We apply our framework to obtain efficient algorithms for (1) approximating probability amplitudes of a class of quantum circuits close to the identity, (2) approximating expectation values of a class of quantum circuits with operators close to the identity, (3) approximating partition functions of a class of quantum spin systems at high temperature, and (4) approximating thermal expectation values of a class of quantum spin systems at high temperature with positive-semidefinite operators. Further, we obtain hardness of approximation results for approximating probability amplitudes of quantum circuits and partition functions of quantum spin systems. This establishes a computational complexity transition for these problems and shows that our algorithmic conditions are optimal under complexity-theoretic assumptions. Finally, we show that our algorithmic condition is almost optimal for expectation values and optimal for thermal expectation values in the sense of zero freeness. Published by the American Physical Society 2024
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(logΔ), where N is the system size, ε the desired precision, and Δ the interaction-graph degree. This becomes particularly severe in long-range systems, where Δ≡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). For quantum partition functions at temperatures β≤βsampling≡O(1), this improves to O~(N3+1/Dε−3), where D is the lattice dimension. The framework directly applies to long-range quantum systems with interactions ∝1/rα, including 2-local models with α>2D.
The computational equivalence between approximate counting and sampling is well established for polynomial-time algorithms. The most efficient general reduction from counting to sampling is achieved via simulated annealing, where the counting problem is formulated in terms of estimating the ratio Q=Z(βmax)/Z(βmin) between partition functions Z(β)=∑x∈Ωexp(βH(x)) of Gibbs distributions μβ over Ω with Hamiltonian H, given access to a sampling oracle that produces samples from μβ for β∈[βmin,βmax]. The best bound achieved by known annealing algorithms with relative error ε is O(qlogh/ε2), where q,h are parameters which respectively bound lnQ and H. However, all known algorithms attaining this near-optimal complexity are inherently sequential, or *adaptive*: the queried parameters β depend on previous samples. We develop a simple non-adaptive algorithm for approximate counting using O(qlog2h/ε2) samples, as well as an algorithm that achieves O(qlogh/ε2) samples with just two rounds of adaptivity, matching the best sample complexity of sequential algorithms. These algorithms naturally give rise to work-efficient parallel (RNC) counting algorithms. We discuss applications to RNC counting algorithms for several classic models, including the anti-ferromagnetic 2-spin, monomer-dimer and ferromagnetic Ising models.
Quantum Monte Carlo (QMC) methods have proven invaluable in condensed matter physics, particularly for studying ground states and thermal equilibrium properties of quantum Hamiltonians without a sign problem. Over the past decade, significant progress has also been made on their rigorous convergence analysis. Heisenberg antiferromagnets (AFM) with bipartite interaction graphs are a popular target of computational QMC studies due to their physical importance, but despite the apparent empirical efficiency of these simulations it remains an open question whether efficient classical approximation of the ground energy is possible in general. In this work we introduce a ground state variant of the stochastic series expansion QMC method, and for the special class of AFM on interaction graphs with an O(1)-bipartite component (star-like), we prove rapid mixing of the associated QMC Markov chain (polynomial time in the number of qubits) by using Jerrum and Sinclair's method of canonical paths. This is the first Markov chain analysis of a practical class of QMC algorithms with the loop representation of Heisenberg models. Our findings contribute to the broader effort to resolve the computational complexity of Heisenberg AFM on general bipartite interaction graphs.
We present quantum algorithms for solving two problems regarding stochastic processes. The first algorithm prepares the thermal Gibbs state of a quantum system and runs in time almost linear in Nβ/Z and polynomial in log(1/ε), where N is the Hilbert space dimension, β is the inverse temperature, Z is the partition function, and ε is the desired precision of the output state. Our quantum algorithm exponentially improves the dependence on 1/ε and quadratically improves the dependence on β of known quantum algorithms for this problem. The second algorithm estimates the hitting time of a Markov chain. For a sparse stochastic matrix P, it runs in time almost linear in 1/(εΔ3/2), where ε is the absolute precision in the estimation and Δ is a parameter determined by P, and whose inverse is an upper bound of the hitting time. Our quantum algorithm quadratically improves the dependence on 1/ε and 1/Δ of the analog classical algorithm for hitting-time estimation. Both algorithms use tools recently developed in the context of Hamiltonian simulation, spectral gap amplification, and solving linear systems of equations.