Variational quantum and neural quantum states algorithms for the linear complementarity problem

TL;DR

DEM-C granular contact simulations require solving large sparse linear systems from time-discretized LCPs, which bottlenecks traditional solvers. The paper evaluates variational quantum methods by comparing VQLS and VNLS as black-box solvers within a Newton-minimum-map framework for the LCP, including Ising-inspired baselines and a 100-sphere frictionless DEM-C test. VNLS demonstrates practical integration and solution quality within the DEM-C loop, while VQLS encounters encoding and measurement bottlenecks due to Pauli-string decompositions; truncation can mitigate some cost but may affect accuracy. Overall, the work provides a concrete, physics-backed testbed for quantum-inspired linear algebra in granular physics and outlines concrete routes to scale and stabilize quantum-classical hybrids in time-stepped contact simulations.

Abstract

Variational quantum algorithms (VQAs) are promising hybrid quantum-classical methods designed to leverage the computational advantages of quantum computing while mitigating the limitations of current noisy intermediate-scale quantum (NISQ) hardware. Although VQAs have been demonstrated as proofs of concept, their practical utility in solving real-world problems -- and whether quantum-inspired classical algorithms can match their performance -- remains an open question. We present a novel application of the variational quantum linear solver (VQLS) and its classical neural quantum states-based counterpart, the variational neural linear solver (VNLS), as key components within a minimum map Newton solver for a complementarity-based rigid body contact model. We demonstrate using the VNLS that our solver accurately simulates the dynamics of rigid spherical bodies during collision events. These results suggest that quantum and quantum-inspired linear algebra algorithms can serve as viable alternatives to standard linear algebra solvers for modeling certain physical systems.

Figures (5)

• Figure 1: The local coordinate frame for the collision between two spheres, and the linearization of the corresponding Coulomb friction cone.
• Figure 2: Comparisons between the global cost VQLS (left), local cost VQLS (center) and the VNLS (right) on the $\kappa=10$ Ising-inspired problem. Loss curves are de-noised using a Savitzky--Golay filter; the true loss curves are also presented here, with lower opacity.
• Figure 3: Snapshots from a simulation of one hundred spheres sedimenting under gravity inside a spherical enclosure. The inter-particle collisions are resolved using the LCP formulation presented in Section \ref{['sec:lcp']}. Color represents the magnitude of the contact force, with red indicating high and blue indicating zero.
• Figure 4: Demonstration of a complex RBM VNLS ansatz solving linear systems derived from the DEM-C LCP at several moments in time of a 100-sphere system. Matrices were generated at intermediate steps of the minimum map Newton solver, at points in time separated by 40,000 Euler time step updates. The dimension of each system must be padded to the nearest power of two to make it amenable to VNLS/VQLS.
• Figure 5: Pauli-string decomposition of frictionless LCP linear systems over the course of three different DEM-C simulations. We plot the number of terms in the Pauli expansions as a function of number of qubits needed to represent the matrices in blue. We also overlay the number of terms when the Pauli expansion is truncated by discarding coefficients with absolute values smaller than $10^{-6}$ (red) and $10^{-3}$ (green). The gray stair-plot corresponds to the theoretically maximum number of terms in the Pauli expansion. The numbers above this stair-plot indicate how many of the linear systems over the course of the simulation were expressed as $n$-qubit systems and the box-plot levels represent the corresponding minimum, maximum, 25% and 75% quantiles, and average number of Pauli terms.