# Penalty models for bitstrings of constant Hamming weight

## TL;DR

This paper proves that the standard penalty model for enforcing a fixed Hamming weight in bitstrings is optimal in maximizing the objective value gap, aiding quantum annealer programming.

## Contribution

It establishes the optimality of the standard penalty model for fixed Hamming weight constraints in quantum annealing.

## Key findings

- Standard penalty model is proven optimal for fixed Hamming weight constraints.
- Maximizes the objective value gap between valid and invalid solutions.
- Supports better encoding of constraints in quantum annealing.

## Abstract

To program a quantum annealer, one must construct objective functions whose minima encode hard constraints imposed by the underlying problem. For such "penalty models," one desires the additional property that the gap in the objective value between such minima and states that fail the constraints is maximized amongst the allowable objective functions. In this short note, we prove the standard penalty model for the constraint that a bitstring has given Hamming weight is optimal with respect to objective value gap.