Rapidly mixing loop representation quantum Monte Carlo for Heisenberg models on star-like bipartite graphs

Abstract

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.

Figures (10)

• Figure 1: a) The "Heisenberg star", or star graph with $N-1$ qubits interacting antiferromagnetically with a single central qubit, is the archetypical example of the bipartite AFM we analyze. b) More generally we allow a constant number of qubits $M = O(1)$ to interact antiferromagnetically with $N - M$ qubits. Our analysis also extends to include staggered local fields and ferromagnetic interactions within each bipartite component.
• Figure 2: An example configuration from the SPE Markov chain we analyze. The loop configuration (top) is an example of a configuration with $2B=8$ operators and 6 line segments ("loops") for a Heisenberg model on a 4-arm star graph (bottom).
• Figure 3: a) The consistent spin configurations when an $I$ operator appears inserted in the inner product \ref{['eq:ConsistentCount']}. The consistent configurations equivalently described by the blue and purple directed loops that make "u-turns" when they encounter an operator. b) The consistent spin configurations and associated directed loops for when an $S$ operator appears in \ref{['eq:ConsistentCount']}. c) Consistent configurations can be switched between by reversing the direction of a directed loop, and then interchanging all of the operators that the loop touches from $I\leftrightarrow S$. Note that when the loops are stitched together as in e), if a loop touches both sides of an operator then that operator must always be $I$ and does not interchange when the direction of the loop is reversed. d) Given that the total number of consistent configurations is given by $2$ raised to the power of the number of loops, we shift to an representation where all of the possible configurations in a) and b) are represented by un-directed loops and no distinction is made between the $I$ and $S$ operators. e) For a full operator string \ref{['eq:ConsistentSeq']}, the local loops are joined together. The example here is for 4 qubits on a line, in the configuration $x = ((1,2)(1,2),(2,3),(3,4),(2,3))$. Since there are $5$ loops in $x$ the steady state weight of this configuration is proportional to $2^{L(x)} = 2^5$.
• Figure 4: a) The possible spin configurations and associated directed loops for the $I^F$ operator coming from a ferromagnetic term. b) The possible spin configurations and associated directed loops for the $S$ operator coming from a ferromagnetic term. c) The ferromagnetic term in the un-directed loop representation.
• Figure 5: a) The possible spin configurations and associated directed loops for the $\mathbbm 1$ operator coming from a transverse field term. b) The possible spin configurations and associated directed loops for the $X$ operator coming from a transverse field term. c) The transverse field term in the un-directed loop representation.

Theorems & Definitions (4)

• Lemma 1
• proof
• Lemma 2
• proof