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Computing ground states of local Hamiltonians is a fundamental problem incondensed matter physics. We give the first randomized polynomial-timealgorithm for finding ground states of gapped one-dimensional Hamiltonians: itoutputs an (inverse-polynomial) approximation, expressed as a matrix productstate (MPS) of polynomial bond dimension. The algorithm combines manyingredients, including recently discovered structural features of gapped 1Dsystems, convex programming, insights from classical algorithms for 1Dsatisfiability, and new techniques for manipulating and bounding the complexityof MPS. Our result provides one of the first major classes of Hamiltonians forwhich computing ground states is provably tractable despite the exponentialnature of the objects involved.

A polynomial-time algorithm for the ground state of 1D gapped local Hamiltonians