Cluster-projected matrix product state: framework for engineering exact quantum many-body ground states in one and two dimensions

Abstract: We propose a framework to design concurrently a frustration-free quantum many-body Hamiltonian and its numerically exact ground states on a sufficiently large finite-size cluster in one and two dimensions using an elementary matrix product state (MPS) representation. Our approach strategically chooses a local cluster Hamiltonian, which is arranged to overlap with neighboring clusters on a designed lattice. The frustration-free Hamiltonian is given as the sum of the cluster Hamiltonians by ensuring that there exists a state that has its local submanifolds as the lowest-energy eigenstate of every cluster. The key to find such a solution is a systematic protocol, which projects out excited states on every cluster using MPS and effectively entangles the cluster states. The protocol offers several advantages, including the ability to achieve exact many-body ground-state solutions at nearly equal cost in one and two dimensions, those belonging to gapless or long-range entangled classes of ground states, flexibility in designing Hamiltonians unbiasedly across various forms of models, and numerically feasible validation through energy calculations. Our protocol offers the exact ground state for general frustration-free Hamiltonian, and enables the exploration of exact phase boundaries and the analysis of even a spatially nonuniform random system, providing platforms for quantum simulations and benchmarks.