Quantum-inspired encoding enhances stochastic sampling of soft matter systems

Abstract

Quantum advantage in solving physical problems is still hard to assess due to hardware limitations. However, algorithms designed for quantum computers may engender transformative frameworks for modeling and simulating paradigmatically hard systems. Here, we show that the quadratic unconstrained binary optimization encoding enables tackling classical many-body systems that are challenging for conventional Monte Carlo. Specifically, in self-assembled melts of rigid lattice ring polymers, the combination of high density, chain stiffness, and topological constraints results in divergent autocorrelation times for real-space Monte Carlo. Our quantum-inspired encoding overcomes this problem and enables sampling melts of lattice rings with fixed curvature and compactness, unveiling counterintuitive topological effects. Tackling the same problems with the D-Wave quantum annealer leads to substantial performance improvements and advantageous scaling of sampling computational cost with the size of the self-assembled ring melts.

Fig. 1.. QUBO encoding of ring assembly.

(A) Schematic representation of active/inactive binary variables corresponding to lattice sites (${\text{Γ}}_i^{\circ }$), edges (${\text{Γ}}_{ij}^{-}$), and corners (${\text{Γ}}_{ijk}^{⌞}$). (B) Examples of correct (left) and incorrect (right) solutions on a 5 × 3 lattice of the ring-assembly QUBO problem encoded by ℋ_N for N = 12, see also section S2.

Fig. 2.. QUBO-based sampling of rare states.

Probability density and marginal distributions of the number of corner turns and contacts of self-assembled rings on a 4 × 4 × 3 lattice at 2/3 filling fraction, obtained from 10⁴ samples minimizing ℋ_N, as exemplified by the circled configuration on the right, where distinct rings are differently colored. The additional Hamiltonian terms of Eq. 2 enable the direct sampling for atypical, and hence very rare, combinations of curvature and contacts, e.g., at the intersection of the shaded bands, as exemplified by the circled configuration on the left.

Fig. 3.. Ring melts with fixed and free curvature.

(A) Average number of rings and corner turns (curvature) of space-filling rings assembled in cuboids of size N. Data points are averages over 10⁴ states or more. (B) Probability distribution, with smoothed contour lines, computed from >3 × 10⁵ states minimizing ℋ_N on a 5 × 5 × 4 cuboid (N = 100). The marginal curvature distribution (top graph) has SD σ_corners = 4.3. We addressed rare states, from 3 to >10σs from the average (yellow), by minimizing ℋ_N + ℋ_curvature, typically collecting 10⁴ states at given n_corners (blue bands). Green data points and spline show the average number of rings computed using states without (with) fixed curvature close to (far from) the modal value.

Fig. 4.. QUBO-based sampling with and without curvature constraints.

Characteristic run time required by the D-Wave classical solver to generate ring melts filling cuboids of size N with and without the quadratic constraint of minimal curvature. The times refer to the D-Wave classical solver run on standard Intel-based workstations with optimized simulated annealing schedule and coefficients of the potential (see section S8). Estimated relative statistical errors are at most 15%. The indicated scalings are from power-law best fits to the data (solid lines).

Fig. 5.. Linking properties of space-filling ring melts.

(A) Probability that all ring pairs in space-filling ring melts of size N are unconcatenaned, i.e., with zero Gaussian linking number. Single rings, i.e., Hamiltonian cycles, are excluded. (B) Complementary linking probability versus curvature for a 5 × 5 × 4 lattice, N = 100, with blue bands denoting sampling at fixed curvature (the line is a spline to the data points). The yellow point marks the equilibrium ensemble average. Circled are typical configurations at the indicated values of N and n_corners, where distinct rings are differently colored. The rings’ linked state is schematically represented on the side. Counterintuitively, increasing the effective ring stiffness can substantially enhance linking.

Fig. 6.. QUBO-based sampling versus ad hoc real-space MC.

Characteristic run time per independent sample required by QUBO-based and real-space MC schemes to generate minimal-curvature states filling cuboids of size N. The QUBO-based run time is based on the D-Wave neal classical solver with optimized annealing schedule and Hamiltonian coefficients. The real-space MC run time is based on a replica exchange scheme with optimized temperatures and exchange rates and using plaquette-flip moves ad hoc tailored for cubic lattices at complete filling. The run times were measured on a standard Intel-based workstation, and details of the optimized algorithms are provided in section S8. Estimated relative statistical errors are at most 15%. The indicated scalings are from power-law best fits to the data (solid lines).

Fig. 7.. Classical simulated annealing versus hybrid quantum QUBO solver.

Characteristic run times required by D-Wave’s purely classical and hybrid (classical-quantum) solvers to minimize ℋ_N for filled cuboids of size N. The τ_1/2 data for both types of solvers correspond to the run times yielding a 50% hit rate with default parameters. The t_opt data are the same as in Fig. 4 and correspond to the characteristic run times per independent sample required by the classical solver with an optimized annealing schedule. The indicated scalings are from power-law best fits to the data (solid lines).