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

TL;DR

The study tackles combinatorial optimization with non-local couplings by formulating problems as Ising/QUBO Hamiltonians and solving them via matrix product state (MPS) based imaginary-time evolution (ITE). A key innovation is using spectral ordering based on the Laplacian to map logical qubits to MPS sites and employing rectangular or triangular SWAP networks to realize non-local interactions with local gates, achieving linear-depth routing. Across MaxCut benchmarks and a 180-qubit portfolio optimization, spectral ordering combined with the triangular SWAP network yields up to >20x reductions in error and higher entanglement that correlates with improved solutions. The work demonstrates that exploiting problem structure in both qubit mapping and routing enhances tensor-network optimization, with implications for scalable quantum-inspired solvers and HPC implementations, and provides open-source code to enable further exploration.

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.

Figures (9)

• Figure 1: (a) An $N$-qubit quantum state for which the probability amplitudes are encoded in a rank-$N$ tensor $\psi_{i_1...i_N}$ can be decomposed as an MPS, representing a product of lower-rank tensors contracted in a 1D-chain. (b--c) A time evolution operator of commuting long-range interactions $U(t) = \bigotimes_{\langle i, j\rangle}U_{i,j}(t)$ can be exactly factorized as a mesh of local interactions, as shown in (c). Each operator $G_{i,j}$ in the grid applies an interaction $U_{i,j}$ between a logical qubit pair $(i,j)$ followed by a SWAP-gate, as depicted in (b). This way, the order of the logical qubits on the sites of the MPS is permuted in each layer of the mesh.
• Figure 2: (a) Rectangular and (b) triangular SWAP networks used to factorize an imaginary time evolution operator $U(t)=\exp{(-\Delta\tau H)}$ generated by a non-local Ising Hamiltonian $H = \sum_{\langle i, j\rangle}J_{i,j}Z_iZ_j$. In the depicted circuits, we let $G_{i,j} = \text{SWAP}_{i,j}\circ U_{i,j}$ where $U_{i,j} = \exp{(-\Delta\tau J_{i,j} Z_i Z_j)}$. The rectangular and triangular decompositions are both exact factorizations of $U(t)$ but permute logical qubits differently within the networks.
• Figure 3: A depiction of the optimization loop. The initial state $\ket{\psi_0} = \ket{+}^{\otimes N}$ state is prepared by initializing the MPS in the $\ket{0}^{\otimes N}$ state followed by a contraction of a Hadamard gate on each physical index. The optimization loop recursively updates the state as $\ket{\psi_{s+1}}= \exp(-\Delta\tau (H-\braket{H}_s)\ket{\psi_s}$ with renormalization and sampling following each state update. The lowest energy sample of each iteration is kept as the output.
• Figure 4: The error $\epsilon = 1 - \mathrm{AR}$ when solving MaxCut on (a) 3-regular, (b) Erdős–Rényi, and (c) Sherrington–Kirkpatrick graphs as a function of the bond dimension $\chi$. Each curve shows the mean error over all considered instances. Solid lines correspond to shuffled mappings of logical qubits to MPS sites, while dashed lines correspond to spectral ordering.
• Figure 5: Average runtime (s) per optimization step as a function of bond dimension $\chi$, separated by graph type. Values are reported as mean $\pm$ standard deviation across all network architectures and qubit orderings.