Model Checking Matrix Product States against Linear Chain Logic
This work provides a first framework for spatial model checking of many-body quantum states, addressing a gap for physicists and quantum simulation tool users who need to verify properties of MPS families as system size grows.
The paper introduces Linear Chain Logic (LCL) for specifying spatial and size-dependent properties of periodic matrix product states (MPS) and develops approximate model-checking algorithms that avoid brute-force expansion, enabling verification of properties like nontriviality and asymptotic patterns for large system sizes.
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.