From curved spacetime to spacetime-dependent local unitaries over the honeycomb and triangular Quantum Walks. (arXiv:1812.02601v1 [quant-ph])

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.