SWAP-Network Routing and Spectral Qubit Ordering for MPS Imaginary-Time Optimization

Abstract

We propose a quantum-inspired combinatorial solver that performs imaginary-time evolution (ITE) on a matrix product state (MPS), incorporating non-local couplings through structured SWAP networks and spectral qubit mapping of logical qubits. The SWAP networks, composed exclusively of local two-qubit gates, effectively mediate non-local qubit interactions. We investigate two distinct network architectures based on rectangular and triangular meshes of SWAP gates and analyze their performance in combination with spectral qubit ordering, which maps logical qubits to MPS sites based on the Laplacian of the logical qubit connectivity graph. The proposed framework is evaluated on synthetic MaxCut instances with varying graph connectivity, as well as on a dynamic portfolio optimization problem based on real historical asset data involving 180 qubits. On certain problem configurations, we observe an over 20$\times$ reduction in error when combining spectral ordering and triangular SWAP networks compared to optimization with shuffled qubit ordering. Furthermore, an analysis of the entanglement entropy during portfolio optimization reveals that spectral qubit ordering not only improves solution quality but also enhances the total and spatially distributed entanglement within the MPS. These findings demonstrate that exploiting problem structure through spectral mapping and efficient routing networks can substantially enhance the performance of tensor-network-based optimization algorithms.