# Quantum walks as mathematical foundation for quantum gates

## TL;DR

This paper establishes that quantum walks serve as a fundamental mathematical framework for describing quantum gates in gate-based quantum computing, applicable across various field-matter interactions and qubit designs.

## Contribution

It introduces quantum walks as a universal mathematical description for quantum gates, extending beyond specific qubit architectures and including the effects of physical interactions.

## Key findings

- Quantum walks naturally describe quantum gates in various architectures.
- Resonant approximations simplify the walks to continuous-time models.
- Examples include descriptions of single- and two-qubit gates.

## Abstract

It is demonstrated that in gate-based quantum computing architectures quantum walk is a natural mathematical description of quantum gates. It originates from field-matter interaction driving the system, but is not attached to specific qubit designs and can be formulated for very general field-matter interactions. It is shown that, most generally, gates are described by a set of coined quantum walks. Rotating wave and resonant approximations for field-matter interaction simplify the walks, factorizing the coin, and leading to pure continuous time quantum walk description. The walks reside on a graph formed by the Hilbert space of all involved qubits and auxiliary states, if present. Physical interactions between different parts of the system necessary to propagate entanglement through such graph -- quantum network -- enter via reduction of symmetries in graph edges. Description for several single- and two-qubit gates are given as examples.