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 $\sharp$P $\supseteq$ NP-complete counting problem $\sharp$3-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.