Quantum algorithms for compact polymer thermodynamics

Abstract

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.

Figures (11)

• Figure 1: Hamiltonian cycles and multiloop configurations over lattices and dual lattices.Panel (a): Hamiltonian cycle over a rectangular lattice. Panel (b): configuration of the dual lattice encoding the Hamiltonian cycle in Panel (a). Panel (c): configuration of the dual lattice encoding a multiloop violating the Hamiltonian condition: some vertex is never visited, and some vertex is visited two times. Panel (d): configuration of the dual lattice encoding a 2-factor violating the single-loop condition, as it consists of two disjoint loops.
• Figure 2: Schematic construction of the parent Hamiltonian $\hat{H}_\text{HC}$. The Hamiltonian is expressed as the sum of three local positive semidefinite Hamiltonians, each enforcing a distinct condition on the states in its kernel. The ground state of $\hat{H}_\text{HC}$ simultaneously lies in these three kernels.
• Figure 3: Local constraints and transformation rules involved in the definition of the Hamiltonian $\hat{H}_\text{HC}$.Panels $C_1$, $C_2$, $C_3$, $C_4$: local configurations forbidden in 2-factors. Panels $E_1$, $E_3$, $E_5$, $E_9$: local transformations connecting the complete set of topologically equivalent 2-factors. The red arrow indicates the swap between two different color plaquettes, while the state of the other plaquettes remains unchanged. The missing transformations $\{E_2,E_4\}$ are obtained by rotating transformations $E_1$ and $E_3$ by 90 degrees. The missing transformations $\{E_6,E_7,E_8,E_{10},E_{11},E_{12}\}$ are obtained by rotating transformations $E_5$ and $E_9$ by 90, 180 and 270 degrees. Plaquettes shown in parentheses are not part of the corresponding rule or constraint, but are fixed in the displayed configuration as a consequence of the constraints $C_1$, $C_2$, $C_3$, and $C_4$. Panels $L_1$, $L_2$: local configurations encoding local cycles.
• Figure 4: Contraction of a loop to a local loop. Rules in $E$ can be used to contract a 2-factor with more than one cycle to a 2-factor where innermost cycles enclose a single plaquette of the dual lattice.
• Figure 5: Dressing Hamiltonian cycles with a monomer sequence. Monomers are represented by the characters P and H, and the sequence is PPPPPPHHHHHH. Panel (a): A qudit is assigned to each lattice vertex, with its internal state encoding either a monomer character, the empty character, or the terminal character. The qudits are initially prepared in a $W$ state, representing a superposition of all possible placements of the first monomer type of the sequence. Panel (b): Action of the operators $\hat{K}_{i,j}$, $\hat{M}_{i,j}$, $\hat{W}_{c,i,j}$, $\hat{V}_{c,i,j}$, and $\hat{Z}_{c,i,j}$ on local configurations. The action of $\hat{K}_{i,j}$ and $\hat{M}_{i,j}$ is identical for rotated configurations and for configurations in which plaquettes in state $0$ are exchanged with plaquettes in state $1$, and vice versa. For all other local configurations, the operators act as the identity. Panel (c): Action of $\prod_{i,j}\hat{K}_{i,j}\hat{M}_{i,j}$ on a configuration containing a single monomer. Panel (d): Repeated application of the sequence-extension operators $\prod_{ij}\hat{K}_{ij}\hat{M}_{i,j}$ and the final action of $\prod_{ij}\hat{Z}_{c_{mn},i,j}$, completing the dressing of the Hamiltonian cycle with the full monomer sequence.

Theorems & Definitions (1)

• Theorem 1