Model Checking Matrix Product States against Linear Chain Logic

2026-05-14

AI Summary

This work addresses the lack of a systematic verification framework for ground states of one-dimensional quantum many-body systems by introducing Linear Chain Logic (LCL) to characterize the physical behavior of periodic matrix product states (MPS) as system size increases. Leveraging completely positive maps induced by MPS, we develop efficient algorithms for inner product computation and approximate model checking, thereby introducing model checking—traditionally used in temporal verification—to the spatial verification of MPS for the first time. Our approach enables scalable asymptotic analysis of large systems without explicitly expanding the quantum state. The method successfully automates the verification of nontrivial properties of canonical MPS families and identifies their asymptotic spatial structures, significantly outperforming conventional numerical techniques.

Abstract

Matrix product states (MPS) are a standard tensor-network representation for ground states of one-dimensional quantum many-body systems, and they underpin widely used simulation tools such as DMRG. However, while quantum model checking has been developed mainly for quantum programs and communication protocols (with properties expressed along a time axis), there is still no comparable framework for systematically verifying \emph{spatial} and \emph{size-dependent} properties of physical many-body states, where the key parameter is the system size. This paper takes a step toward bridging the gap. We propose \emph{Linear Chain Logic} (LCL), a spatial logic designed to specify physically meaningful properties of periodic MPS families as the system size grows, such as nontriviality on rings and large-size asymptotic patterns. Our approach builds on a simple but powerful connection: every periodic MPS naturally induces a completely positive map (a quantum operation) on its virtual space, so many quantitative features of the MPS can be analysed through the repeated application of the operation. Using this perspective, we derive an effective procedure to compute the inner products of an MPS at a given size and to support richer LCL specifications, without relying on brute-force state expansion. We then develop approximate model-checking algorithms that combine sound bounding with asymptotic structural analysis, enabling scalable reasoning about large system sizes. Experiments on representative MPS families illustrate that our method can automatically verify nontriviality and detect asymptotic spatial regimes in a way that complements traditional numerical techniques.

Problem

• matrix product states
• spatial logic
• system size
• quantum many-body systems
• model checking

Innovation

• Matrix Product States
• Linear Chain Logic
• Spatial Model Checking
• Completely Positive Maps
• Asymptotic Analysis