A simpler algorithm to mark the unknown eigenstates

TL;DR

This paper introduces a simplified quantum algorithm for marking unknown eigenstates that eliminates the need for ancilla qubits by using fixed-point quantum search, trading off some query complexity.

Contribution

The paper demonstrates that majority-voting is unnecessary for eigenstate marking and replaces it with fixed-point quantum search, simplifying implementation and reducing ancilla qubit requirements.

Findings

Eliminates ancilla qubits needed for majority-voting

Uses fixed-point quantum search to mark eigenstates

Increases the number of U applications by a logarithmic factor

Abstract

For an unknown eigenstate $|\psi\rangle$ of a unitary operator $U$, suppose we have an estimate of the corresponding eigenvalue which is separated from all other eigenvalues by a minimum gap of magnitude $\Delta$. In the eigenstate-marking problem (EMP), the goal is to implement a selective phase transformation of the $|\psi\rangle$ state (known as \emph{marking} the $|\psi\rangle$ state in the language of the quantum search algorithms). The EMP finds important applications in the construction of several quantum algorithms. The best known algorithm for the EMP combines the ideas of the phase estimation algorithm and the majority-voting. It uses $\Theta(\frac{1}{\Delta}\ln \frac{1}{\epsilon})$ applications of $U$ where $\epsilon$ is the tolerable error. It needs $\Theta\left(\ln \frac{1}{\Delta}\right)$ ancilla qubits for the phase estimation and another $\Theta\left(\ln \frac{1}{\epsilon}\right)$ ancilla qubits for the majority-voting. In this paper, we show that the majority-voting is not a crucial requirement for the EMP and the same purpose can also be achieved using the fixed-point quantum search algorithm which does not need any ancilla qubits. In the case of majority-voting, these ancilla qubits were needed to do controlled transformations which are harder to implement physically. Using fixed-point quantum search, we get rid off these $\Theta\left(\ln \frac{1}{\epsilon}\right)$ ancilla qubits and same number of controlled transformations. Thus we get a much simpler algorithm for marking the unknown eigenstates. However, the required number of applications of $U$ increases by the factor of $\Theta \left(\ln \frac{1}{\epsilon}\right)$. This tradeoff can be beneficial in typical situations where spatial resources are more constrained or where the controlled transformations are very expensive.