# Polynomial Time Algorithms for Estimating Spectra of Adiabatic Hamiltonians

## TL;DR

This paper introduces polynomial time algorithms for estimating the spectra of certain adiabatic Hamiltonians with multiple symmetric potential wells, advancing understanding of quantum adiabatic optimization beyond highly symmetric cases.

## Contribution

It presents a polynomial-time exact solution for Hamiltonians with up to three wells and a tight binding approach for analyzing more complex multi-well Hamiltonians in adiabatic quantum computation.

## Key findings

- Exact polynomial-time solutions for two and three well Hamiltonians.
- A tight binding method for analyzing Hamiltonians with more than three wells.
- Applications to adiabatic unstructured search and spin system ground states.

## Abstract

Much research regarding quantum adiabatic optimization has focused on stoquastic Hamiltonians with Hamming symmetric potentials, such as the well studied "spike" example. Due to the large amount of symmetry in these potentials such problems are readily open to analysis both analytically and computationally. However, more realistic potentials do not have such a high degree of symmetry and may have many local minima. Here we present a somewhat more realistic class of problems consisting of many individually Hamming symmetric potential wells. For two or three such wells we demonstrate that such a problem can be solved exactly in time polynomial in the number of qubits and wells. For greater than three wells, we present a tight binding approach with which to efficiently analyze the performance of such Hamiltonians in an adiabatic computation. We provide several basic examples designed to highlight the usefulness of this toy model and to give insight into using the tight binding approach to examining it, including: (1) adiabatic unstructured search with a transverse field driver and a prior guess to the marked item and (2) a scheme for adiabatically simulating the ground states of small collections of strongly interacting spins, with an explicit demonstration for an Ising model Hamiltonian.