Entanglement Barriers from Computational Complexity: Matrix-Product-State Approach to Satisfiability

We approach the 3-SAT satisfiability problem with the quantum-inspired method of imaginary time propagation (ITP) applied to matrix product states (MPS) on a classical computer. This ansatz is fundamentally limited by a quantum entanglement barrier that emerges in imaginary time, reflecting the exponential hardness expected for this NP-complete problem. Strikingly, we argue based on careful analysis of the structure imprinted onto the MPS by the 3-SAT instances that this barrier arises from classical computational complexity. To reveal this connection, we elucidate with stochastic models the specific relationship between the classical hardness of the sharpP supseteq NP-complete counting problem sharp3-SAT and the entanglement properties of the quantum state. Our findings illuminate the limitations of this quantum-inspired approach and demonstrate how purely classical computational complexity can manifest in quantum entanglement. Furthermore, we present estimates of the non-stabilizerness required by the protocol, finding a similar resource barrier. Specifically, the necessary amount of non-Clifford operations scales superlinearly in system size, thus implying extensive resource requirements of ITP on different architectures such as Clifford circuits or gate-based quantum computers.