# From curved spacetime to spacetime-dependent local unitaries over the honeycomb and triangular Quantum Walks

## TL;DR

This paper demonstrates that by applying spacetime coordinate transformations to quantum walks on honeycomb and triangular lattices, one can simulate Dirac equations in curved spacetime using local unitaries, revealing a duality between geometry and unitaries.

## Contribution

It introduces a novel duality between geometric transformations and local unitaries in quantum walks, enabling simulation of curved spacetime physics on specific lattices.

## Key findings

- Spacetime-dependent quantum walks simulate Dirac equations in curved spacetime.
- The duality relies on the non-linear independence of lattice directions.
- This approach enables simulating field theories on curved manifolds.

## Abstract

A discrete-time Quantum Walk (QW) is an operator driving the evolution of a single particle on the lattice, through local unitaries. Some QW admit, as their continuum limit, a well-known equation of Physics. In arXiv:1803.01015 the QW is over the honeycomb and triangular lattices, and simulates the Dirac equation. We apply a spacetime coordinate transformation upon the lattice of this QW, and show that it is equivalent to introducing spacetime-dependent local unitaries --- whilst keeping the lattice fixed. By exploiting this duality between changes in geometry, and changes in local unitaries, we show that the spacetime-dependent QW simulates the Dirac equation in $(2+1)$ - dimensional curved spacetime. Interestingly, the duality crucially relies on the non linear-independence of the three preferred directions of the honeycomb and triangular lattices: The same construction would fail for the square lattice. At the practical level, this result opens the possibility to simulate field theories on curved manifolds, via the quantum walk on different kinds of lattices.