Solving Sharp Bounded-error Quantum Polynomial Time Problem by Evolution methods

TL;DR

This paper introduces an algorithm that maps a #BQP problem to a ground state problem of a $k$-local Hamiltonian, enabling the use of evolution-based methods to solve complex quantum counting problems.

Contribution

It presents a novel approach to solve #BQP problems by transforming them into ground state searches and demonstrates the use of various evolution methods for practical implementation.

Findings

Successful mapping of #BQP to ground state problems

Application of power, Lanczos, and imaginary time evolution methods

Detection of phase boundaries and quantum fluctuations

Abstract

Counting ground state degeneracy of a $k$-local Hamiltonian is important in many fields of physics. Its complexity belongs to the problem of sharp bounded-error quantum polynomial time (#BQP) class and few methods are known for its solution. Finding ground states of a $k$-local Hamiltonian, on the other hand, is an easier problem of Quantum Merlin Arthur (QMA) class, for which many efficient methods exist. In this work, we propose an algorithm of mapping a #BQP problem into one of finding a special ground state of a $k$-local Hamiltonian. We prove that all traditional methods, which solve the QMA problem by evolution under a function of a Hamiltonian, can be used to find the special ground state from a well-designed initial state, thus can solve the #BQP problem. We combine our algorithm with power method, Lanczos method, and quantum imaginary time evolution method for different systems to illustrate the detection of phase boundaries, competition between frustration and quantum fluctuation, and potential implementations with quantum circuits.