Sam Slezak | Searcharxiv

arXiv2024

Quantum Approximate Counting with Additive Error: Hardness and Optimality

Quantum counting is the task of determining the dimension of the subspace of states that are accepted by a quantum verifier circuit. It is the quantum analog of counting the number of valid solutions to NP problems -- a problem well-studied in theoretical computer science with far-reaching implications in computational complexity. The complexity of solving the class #BQP of quantum counting problems, either exactly or within suitable approximations, is related to the hardness of computing many-body physics quantities arising in algebraic combinatorics. Here, we address the complexity of quantum approximate counting under additive error. First, we show that computing additive approximations to #BQP problems to within an error exponential in the number of witness qubits in the corresponding verifier circuit is as powerful as polynomial-time quantum computation. Next, we show that returning an estimate within error that is any smaller is #BQP-hard. Finally, we show that additive approximations to a restricted class of #BQP problems are equivalent in computational hardness to the class DQC1. Our work parallels results on additively approximating #P and GapP functions.

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arXiv2024

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

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.

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