Counting the Ground State Degeneracy by Evolution Methods.

Abstract

Counting ground state degeneracy of a k-local Hamiltonian is important in many fields of physics. Its complexity class is harder than that of finding the ground state of a k-local Hamiltonian. Very few methods can efficiently count the degeneracy of ground state while many are known for finding the ground state. This Letter proposes an algorithm for mapping the problem of counting ground state degeneracy into one for finding a special ground state of a k-local super Hamiltonian in an enlarged system with twice as many qubits or spins. We show that all traditional methods, designed for finding the ground state by evolution under a function of a Hamiltonian, can be employed to find the special ground state of the super Hamiltonian from a well-designed initial state and thus can be used to count the ground degeneracy. We combine our algorithm with the power method, the Lanczos method, and the quantum imaginary time evolution method for selective examples to illustrate the detection of phase boundaries, competition between frustration and quantum fluctuation, and potential implementation with quantum circuits.