# On the relationships between Z-, C-, and H-local unitaries

## TL;DR

This paper explores the relationships between different notions of local unitaries in quantum walks, providing a method to approximate H-local unitaries with C-local unitaries efficiently, thus clarifying their interconnections.

## Contribution

It establishes a way to approximate H-local unitaries using C-local unitaries with specific error bounds, linking continuous and discrete quantum walk models.

## Key findings

- H-local unitaries can be approximated by C-local unitaries with error δ.
- The approximation complexity is O(1/√δ, √N).
- The work clarifies the relationship between different locality criteria in quantum walks.

## Abstract

Quantum walk algorithms can speed up search of physical regions of space in both the discrete-time [arXiv:quant-ph/0402107] and continuous-time setting [arXiv:quant-ph/0306054], where the physical region of space being searched is modeled as a connected graph. In such a model, Aaronson and Ambainis [arXiv:quant-ph/0303041] provide three different criteria for a unitary matrix to act locally with respect to a graph, called $Z$-local, $C$-local, and $H$-local unitaries, and left the open question of relating these three locality criteria. Using a correspondence between continuous- and discrete-time quantum walks by Childs [arXiv:0810.0312], we provide a way to approximate $N\times N$ $H$-local unitaries with error $\delta$ using $O(1/\sqrt{\delta},\sqrt{N})$ $C$-local unitaries, where the comma denotes the maximum of the two terms.