Quantum algorithms for compact polymer thermodynamics

Efficient sampling from ensembles of Hamiltonian cycles is critical for predicting the thermodynamic properties of compact polymers, with applications including modeling protein and RNA folding and designing soft materials. Although classical Monte Carlo methods are widely regarded as the standard approach, their efficiency is strongly limited when applied to compact polymers. In this work, we enable a quadratic speedup in the estimation of thermodynamic properties of maximally compact polymers and heteropolymers by quantum computation. To this end, we encode the target thermodynamic ensemble into the amplitudes of a quantum state, i.e., a quantum sample, which can be processed via amplitude amplification. Using quantum equational reasoning, we construct a local parent Hamiltonian whose unique ground state realizes a quantum sample of all Hamiltonian cycles. This state can be prepared on quantum hardware using ground-state preparation methods, such as quantum annealing, and subsequently manipulated to generate quantum samples of polymers and heteropolymers at a target temperature. Finally, we approximate the quantum sample as a tensor network, revealing an entanglement area law. For fixed-width rectangular lattices, we obtain a time-efficient and compact encoding of the full ensemble of Hamiltonian cycles, enabling the efficient evaluation of expectation values, partition functions, and configuration probabilities via tensor contractions, without resorting to sampling.