Testing a quantum algorithm on the Lipkin model.

Quantum algorithms promise the ability to tackle problems that scale exponentially on a classical computer.  ADAPT-VQE [1] is a quantum algorithm capable of obtaining exact eigenvalues of many-body Hamiltonians by repeatedly optimizing the parameter specifying the many-particle-hole excitation of a reference state as one slowly increases the number excitations.  The method has been applied so far mainly to molecules but can also be useful in nuclear physics.  Here we examine the method's efficiency in the Lipkin model, which, unlike molecular systems, exhibits both a phase transition and spontaneous symmetry breaking.  Though these features reduce the efficiency of the algorithm, relatively simple modifications of the reference state restore it.  In the end, neither a nearby phase transition nor symmetry breaking is a significant obstacle.

[1] Harper R. Grimsley, Sophia E. Economou, Edwin Barnes & Nicholas J. Mayhall. Nature Communications 10(1), 1-9.