# Duality in Quantum Quenches and Classical Approximation Algorithms: Pretty Good or Very Bad

## TL;DR

This paper explores a duality property in classical and quantum algorithms related to solution quality, providing guarantees for finding improvements or identifying poor solutions, with specific analysis on MAX-$K$-LIN$2$ problems.

## Contribution

It introduces a duality-based framework for analyzing classical and quantum algorithms, offering new guarantees and insights, especially through a dequantized classical approach.

## Key findings

- Quantum algorithm based on Hamiltonian quench analyzed for MAX-$K$-LIN$2$.
- Classical dequantized algorithm achieves similar guarantees as quantum.
- Quantum perspective aids in analyzing and potentially improving classical algorithms.

## Abstract

We consider classical and quantum algorithms which have a duality property: roughly, either the algorithm provides some nontrivial improvement over random or there exist many solutions which are significantly worse than random. This enables one to give guarantees that the algorithm will find such a nontrivial improvement: if few solutions exist which are much worse than random, then a nontrivial improvement is guaranteed. The quantum algorithm is based on a sudden of a Hamiltonian; while the algorithm is general, we analyze it in the specific context of MAX-$K$-LIN$2$, for both even and odd $K$. The classical algorithm is a "dequantization of this algorithm", obtaining the same guarantee (indeed, some results which are only conjectured in the quantum case can be proven here); however, the quantum point of view helps in analyzing the performance of the classical algorithm and might in some cases perform better.