Computing the Degenerate Ground Space of Gapped Spin Chains in Polynomial Time

Given a gapped Hamiltonian of a spin chain, we give a polynomial-time algorithm for finding the degenerate ground space projector. The output is an orthonormal set of matrix product states that approximate the true ground space projector up to an inverse polynomial error in any Schatten norm, with a runtime exponential in the degeneracy. Our algorithm is an extension of the recent algorithm of Landau, Vazirani, and Vidick [1] for the nondegenerate case, and it includes the recent improvements due to Huang [2]. The main new idea is to incorporate the local distinguishability of ground states on the half-chain to ensure that the algorithm returns a complete set of global ground states. Strongly correlated quantum systems are at the heart of such diverse physical phenomena as frustrated magnets, high-temperature superconductors, and topological quantum phases, to name but a few, and understanding their low-temperature behaviour is a grand challenge at the interface of physics and computer science. The computational tools to study these systems numerically are however inevitably stymied by the exponential growth of the Hilbert space of an n-particle system. Many numerical tools have been developed to surmount the computational difficulties surrounding the exponential growth of the state space. Tools like quantum Monte Carlo [3, 4], coupled cluster methods [5], and the density-matrix renormalisation group (DMRG) [6] all exploit some underlying physical structure to perform heuristically efficient simulations of interacting quantum systems. However, like all heuristics they often lack basic theoretical guarantees except in certain special cases, and some of them have failure modes that are difficult to diagnose or characterise. This makes it hard to precisely determine which physical features enable efficient simulation and which features might be responsible in the cases where the heuristic fails. Very recently, a success story has emerged in the rigorous understanding of gapped 1D spin chains. Hastings’ proof of the 1D area law [7] established that approximating the ground state of a gapped spin chain is in NP. The proof technique crucially used the idea of matrix product states (MPS) [8], an ansatz for quantum systems that exploits the area law structure of entanglement [9]. The first rigorous nontrivial algorithm for finding ground states, due independently to Schuch and Cirac [10] and Aharonov, Arad, and Irani [11], used dynamic programming to find MPS ground states. The runtime of the algorithm was exponential in the bond dimension, and since Hastings’ result used a polynomial bond dimension to achieve a sufficiently accurate approximation, the runtime could only be guaranteed to be exponential for general gapped 1D systems. The first subexponentialtime algorithm was found by Arad, Kitaev, Landau, and Vazirani [12]. They used the