# Bang-bang control as a design principle for classical and quantum optimization algorithms

## TL;DR

This paper demonstrates that bang-bang control strategies significantly improve the performance of classical and quantum optimization algorithms, including simulated annealing and QAOA, on complex problem instances.

## Contribution

It introduces a new bang-bang control variant for simulated annealing and shows its superiority, along with constructing low-depth QAOA protocols for symmetric cost functions.

## Key findings

- Bang-bang control outperforms quasistatic methods exponentially.
- Both classical and quantum algorithms benefit from bang-bang strategies.
- Low-depth QAOA protocols are effective for symmetric cost functions.

## Abstract

Physically motivated classical heuristic optimization algorithms such as simulated annealing (SA) treat the objective function as an energy landscape, and allow walkers to escape local minima. It has been argued that quantum properties such as tunneling may give quantum algorithms advantage in finding ground states of vast, rugged cost landscapes. Indeed, the Quantum Adiabatic Algorithm (QAO) and the recent Quantum Approximate Optimization Algorithm (QAOA) have shown promising results on various problem instances that are considered classically hard. Here, we argue that the type of control strategy used by the optimization algorithm may be crucial to its success. Along with SA, QAO and QAOA, we define a new, bang-bang version of simulated annealing, BBSA, and study the performance of these algorithms on two well-studied problem instances from the literature. Both classically and quantumly, the successful control strategy is found to be bang-bang, exponentially outperforming the quasistatic analogues on the same instances. Lastly, we construct O(1)-depth QAOA protocols for a class of symmetric cost functions, and provide an accompanying physical picture.