Approximate Phase Search and Eigen Estimation using Modified Grover's Algorithm

TL;DR

This paper proposes modifications to Grover's Algorithm with new operators and ancilla qubits to approximate solutions near a target cost, and suggests a method for eigenvalue and eigenstate estimation of Hamiltonians.

Contribution

It introduces novel controlled oracle and diffusion operators, along with a strategy for eigen estimation, extending Grover's Algorithm for approximate phase search.

Findings

Modified Grover's Algorithm effectively approximates solutions near target costs.

New operators improve the search process for specific cost values.

Eigenvalue and eigenstate estimation methods are feasible with the proposed approach.

Abstract

An attempt has been made in this paper to modify Grover's Algorithm to find the binary string solutions approximating a target cost value. In that direction, new Controlled Oracle and the Local Diffusion Operator are suggested, apart from incorporating suitable ancilla qubits. A possible strategy to estimate eigenvalues and eigenstates of a given cost Hamiltonian, extending the reasoning of the methodology, is also pointed out. Typical results and relevant discussions are captured to support the propositions.