Non-Boolean Quantum Amplitude Amplification and Quantum Mean Estimation

TL;DR

This paper extends quantum amplitude amplification and estimation algorithms to non-boolean oracles that apply state-dependent phase shifts, enabling more flexible quantum algorithms with demonstrated simulations and potential applications.

Contribution

It introduces non-boolean amplitude amplification and quantum mean estimation algorithms, generalizing prior methods to handle phase-shift oracles and showing quadratic speedup over classical approaches.

Findings

Non-boolean amplitude amplification preferentially amplifies states based on phase values.

Quantum mean estimation achieves quadratic speedup over classical methods.

Algorithms are demonstrated through simulations on toy examples.

Abstract

This paper generalizes the quantum amplitude amplification and amplitude estimation algorithms to work with non-boolean oracles. The action of a non-boolean oracle $U_\varphi$ on an eigenstate $|x\rangle$ is to apply a state-dependent phase-shift $\varphi(x)$. Unlike boolean oracles, the eigenvalues $\exp(i\varphi(x))$ of a non-boolean oracle are not restricted to be $\pm 1$. Two new oracular algorithms based on such non-boolean oracles are introduced. The first is the non-boolean amplitude amplification algorithm, which preferentially amplifies the amplitudes of the eigenstates based on the value of $\varphi(x)$. Starting from a given initial superposition state $|\psi_0\rangle$, the basis states with lower values of $\cos(\varphi)$ are amplified at the expense of the basis states with higher values of $\cos(\varphi)$. The second algorithm is the quantum mean estimation algorithm, which uses quantum phase estimation to estimate the expectation $\langle\psi_0|U_\varphi|\psi_0\rangle$, i.e., the expected value of $\exp(i\varphi(x))$ for a random $x$ sampled by making a measurement on $|\psi_0\rangle$. It is shown that the quantum mean estimation algorithm offers a quadratic speedup over the corresponding classical algorithm. Both algorithms are demonstrated using simulations for a toy example. Potential applications of the algorithms are briefly discussed.