Path-Dependent QUBO Annealing Simulation of Mesoscale Thermal Aging in Porous Sintered Silver

Each site i of a periodic N_x × N_y lattice carries a one-hot grain label X_ig over Q states and a unary composition ladder U_im over M bits,

Thus, the register holds N(Q+M) logical qubits. The reference run uses 60 × 60, Q = 2, and M = 2, giving 14,400 variables. Collecting all bits into z ∈ {0,1}^N(Q+M), the energy splits into a part assembled once and a part rebuilt at every stage n,

The five static terms are a Potts term rewarding matching neighbour labels over axial and diagonal pairs, the two-phase bulk chemical energy coupling phase to composition, a Cahn–Hilliard gradient energy penalising sharp composition steps, a material-balance penalty, and the one-hot and unary-ordering constraints.

The stage term is purely linear and carries the history. A breadth-first sweep returns the distance d_i from every site to the nearest interface in O(N) time, and the bias rewards retention of the reference label g_i^ref decoded at stage n−1,

Setting b_bulk = 20 above the largest Potts gain available to a bulk flip, against b_intf = 3.974 at d_sat = 2, anchors the bulk and leaves interfacial sites free. Consequently, each solve returns a low-energy state near its predecessor rather than the global minimum.

Material balance is imposed block-locally. Overlapping D × D windows at stride S taper across their overlaps through complementary B-spline ramps w_b, which form a partition of unity,

Thus, the local targets tile exactly to the global silver budget while transport per stage is bounded by the block scale.

Equations (1)–(4) define a standard QUBO that carries no solver-specific structure, so the identical Hamiltonian is submittable without modification to any Ising machine. The register is the set of N(Q+M) binary variables from Eq. (1), and encoding validity is enforced by the penalty terms in H_cons rather than by hardware.

The current microstructure is never prepared as a state: it enters through the linear coefficients of Eq. (3) alone, which keeps the quadratic part reusable. Extraction consists of a single decoding of the lowest-energy sample into labels and composition, which then becomes the reference for stage n+1.

We demonstrate the method through what the call describes as a fully worked-out formulation, supported by evidence from execution on a simulator. Equations (1)–(4) specify every coefficient of H_total⁽ⁿ⁾ in closed form as a function of the free-energy parameters, the block geometry, and the previous microstructure. The optimisation problem is therefore fully determined at every stage.

Complete 50-stage trajectories are then sampled from it. We assemble the QUBO using the Fixstars Amplify SDK and solve it on the Amplify Annealing Engine, a GPU annealer that runs parallel simulated-annealing dynamics on a complete graph, with a 3000 ms budget per stage. Because it samples the same H_total⁽ⁿ⁾ without connectivity restrictions, it plays the role that a state-vector simulator plays in gate-model work. Execution on superconducting annealing hardware has not yet been performed and is the immediate next step.

Four checks establish that the construction works as specified. First, the samples decode to valid encodings: the per-stage one-hot check reports no violations, the decoded energy falls monotonically, and the soft material-balance residual remains within 4.2% in area fraction of the initial porosity.

Second, the observed physics is consistent with coarsening: small pores vanish, neighbouring pores coalesce, and the surviving pores become rounder. The pore count follows N_p(n) ∝ n^−0.58 over the active regime, while the total boundary length falls by 67% at a porosity maintained near 9.7%.

Third, two ablation studies show that the kinetic-memory and block-local terms perform the roles attributed to them. Label changes are confined to sites already located on an interface and are indistinguishable from zero one pixel away, even though the interfacial shell contracts from 40% of the lattice to 16% as the structure coarsens.

Removing the memory term produces correlated bulk patches, a non-monotonic porosity history, and no extractable exponent. Replacing the block-local penalty of Eq. (4) with a single global square-using the same seed, physics, and solver budget-widens the porosity band from 0.22 to 1.58 percentage points and allows distant pores to exchange material. The mean pore-pair separation peaks at 15.7 px, compared with a stable range of 7.1–7.5 px for the block-local formulation.

Fourth, the trajectory is checked against an independent update rule. Kawasaki dynamics, evolving the same free energy from the identical initial microstructure [Hu2025SimulationModel], converts 286 small pores into 25. The staged solver reaches 24 pores and tracks the reference trajectory in porosity, pore count, and boundary length throughout. Two scalar amplitudes were calibrated for this comparison.

The workflow is hybrid by construction. A classical host computes the distance field, assembles the linear bias of Eq. (3), decodes the returned sample, and promotes it to the next reference, all in O(N) time per stage. The annealer performs one minimisation per stage.

Because the quadratic physics is cached and only the linear field is rebuilt, successive stages use the same coupling graph and differ only in their linear terms. This separation makes extended trajectories practical.

The problem size grows linearly. At fixed block geometry,

This gives 1.8 × 10⁶ quadratic terms for a 60 × 60 lattice, compared with 1.3 × 10⁷ terms for the global square replaced by the blocks. Extending the formulation to three dimensions multiplies the variable count by N_z while leaving the coupling count linear in volume, which is the regime occupied by a real die-attach layer.

In terms of iteration count, the staged solver resolves in approximately eight stages the early coalescence burst that the Kawasaki reference covers in 300 sweeps.

The logical encoding contains no solver-specific structure. The identical Hamiltonian can therefore be submitted without modification to any Ising machine, subject to its connectivity and coefficient-range requirements. Execution on superconducting quantum-annealing hardware is the direct continuation of this work.

The construction can also be extended to other porous polycrystalline systems governed by conserved redistribution, provided that their free energies admit quadratic binary representations.