Analysis and Experimental Demonstration of Amplitude Amplification for Combinatorial Optimization

TL;DR

This paper extends quantum amplitude amplification to encode cost functions like QUBO, demonstrating its effectiveness for combinatorial optimization through simulations and experiments on IBMQ and IonQ quantum computers.

Contribution

It generalizes the amplitude amplification framework to include cost functions and provides optimal parameter formulas, with experimental validation on real quantum hardware.

Findings

QAA performs well near the global optimum in simulations.

Optimal oracle parameters can be exactly determined for linear cost functions.

Experimental results match theoretical predictions across different hardware.

Abstract

Quantum Amplitude Amplification (QAA), the generalization of Grover's algorithm, is capable of yielding optimal solutions to combinatorial optimization problems with high probabilities. In this work we extend the conventional 2-dimensional representation of Grover's (orthogonal collective states) to oracles which encode cost functions such as QUBO, and show that linear cost functions are a special case whereby an exact formula exists for determining optimal oracle parameter settings. Using simulations of problem sizes up to 40 qubits we demonstrate QAA's algorithmic performance across all possible solutions, with an emphasis on the closeness in Grover-like performance for solutions near the global optimum. We conclude with experimental demonstrations of generalized QAA on both IBMQ (superconducting) and IonQ (trapped ion) qubits, showing that the observed probabilities of each basis state match our equations as a function of varying the free parameters in the oracle and diffusion operators.