Quantum Algorithms with Fixed Points: The Case of Database Search

TL;DR

This paper introduces two novel quantum search algorithms that incorporate fixed points, enabling monotonic convergence towards the solution, which enhances robustness and error correction capabilities.

Contribution

The paper presents two variations of quantum search algorithms with fixed points, overcoming a key limitation of standard quantum search methods.

Findings

Reduced probability of non-target states from ε to ε^{2q+1}

Achieved asymptotically optimal convergence rate

Potential for robust quantum algorithms and error correction schemes

Abstract

The standard quantum search algorithm lacks a feature, enjoyed by many classical algorithms, of having a fixed-point, i.e. a monotonic convergence towards the solution. Here we present two variations of the quantum search algorithm, which get around this limitation. The first replaces selective inversions in the algorithm by selective phase shifts of $\frac{\pi}{3}$. The second controls the selective inversion operations using two ancilla qubits, and irreversible measurement operations on the ancilla qubits drive the starting state towards the target state. Using $q$ oracle queries, these variations reduce the probability of finding a non-target state from $\epsilon$ to $\epsilon^{2q+1}$, which is asymptotically optimal. Similar ideas can lead to robust quantum algorithms, and provide conceptually new schemes for error correction.