AF: Small: Ground State Complexity in Quantum Many-Body Systems

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One of the central goals of quantum information theory is to understand quantum systems from the standpoint of computational complexity. Physicists have been using computers for decades to understand various aspects of quantum systems, but these methods are typically heuristic and achieve success on only limited classes of systems. Understanding quantum systems through the lens of computational and information-theoretic complexity has already lead to new powerful computational methods in physics and deeper insight into what causes these methods to fail once we step outside specific classes. Meanwhile hardness results often have important implications for quantum computation. If computing a property of a system is shown to be as difficult as computing the output of an arbitrary quantum circuit, that system becomes a candidate for a quantum computer.This proposal focuses specifically on the complexity of ground states, the lowest energy state of a system. How hard is it to compute the ground energy of a quantum system? What are the properties of a system that give rise to provably efficient algorithms to compute the ground state? What is the structure of the ground state? Under what circumstances does the ground state have a succinct representation? The PI will pursue these questions in the context of one-dimensional systems and will begin work on two-dimensional systems about which much less is known.