# Practical designs for permutation symmetric problem Hamiltonians on hypercubes

## TL;DR

This paper introduces a practical method to implement large-scale permutation-symmetric Hamiltonians on hypercubes for quantum protocols, enabling efficient quantum search and benchmarking in Rydberg atomic systems.

## Contribution

The authors propose a perturbative, second-order implementation of permutation-symmetric Hamiltonians on hypercubes suitable for experimental realization in Rydberg systems.

## Key findings

- Implementation at second order in perturbation theory regardless of system size
- Enables encoded quantum search on hypercube graphs with 2^n vertices
- Provides a scalable approach for benchmarking quantum hardware

## Abstract

We present a method to experimentally realize large-scale permutation-symmetric Hamiltonians for continuous-time quantum protocols such as quantum walk and adiabatic quantum computation. In particular, the method can be used to perform an encoded continuous-time quantum search on a hypercube graph with 2^n vertices encoded into 2n qubits. We provide details for a realistically achievable implementation in Rydberg atomic systems. Although the method is perturbative, the realization is always achieved at second order in perturbation theory, regardless of the size of the mapped system. This highly efficient mapping provides a natural set of problems which are tractable both numerically and analytically, thereby providing a powerful tool for benchmarking quantum hardware and experimentally investigating the physics of continuous-time quantum protocols.