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

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