Amplitude Amplification for Optimization via Subdivided Phase Oracle

TL;DR

This paper introduces a quantum algorithm that uses a subdivided phase oracle to efficiently amplify the probability of finding optimal solutions in combinatorial optimization problems, achieving near-deterministic success for certain distributions.

Contribution

The paper presents a novel subdivided phase oracle method that assigns unique phase shifts based on objective values, enhancing amplitude amplification for optimization tasks.

Findings

Effective amplification of optimal solutions probability across various distributions.

Near-deterministic success for skew normal and exponential distributions.

Significant quantum speedup over classical algorithms.

Abstract

We propose an algorithm using a modified variant of amplitude amplification to solve combinatorial optimization problems via the use of a subdivided phase oracle. Instead of dividing input states into two groups and shifting the phase equally for all states within the same group, the subdivided phase oracle changes the phase of each input state uniquely in proportion to their objective value. We provide visualization of how amplitudes change after each iteration of applying the subdivided phase oracle followed by conventional Grover diffusion in the complex plane. We then show via numerical simulation that for normal, skew normal, and exponential distribution of objective values, the algorithm can be used to amplify the probability of measuring the optimal solution to a significant degree independent of the search space size. In the case of skew normal and exponential distributions, this probability can be amplified to be close to unity, making our algorithm near deterministic. We then modify our algorithm in order to demonstrate how it can be extended to a broader set of objective value distributions. Finally, we discuss the speedup compared to classical schemes using the query complexity model, and show that our algorithm offers a significant advantage over these classical approaches.