![]() ![]() Indeed, we describe depart from the square lattice, to go to the honeycomb and triangular lattice- which can be seen as trivalent graphs. The present work falls within the second class. These schemes provide: a/ numerical schemes that are stable even for classical computers- from which one can derive convergence 8 b/simple toy models of the target physical phenomena, with most symmetries conserved (homogeneity, causality, unitarity… sometimes even Lorentz-covariance 9, 10. ![]() In QW-based quantum simulation schemes, the quantum walker propagates on a grid, and a spacetime continuum limit towards some well-known target physics equation is taken. Whilst full-blown Quantum Computers remain out-of-reach at the experimental level, a number of special-purpose Quantum Simulation devices are appearing, whose architecture is often directly inspired by QWs 6, 7. Quantum simulation was Feynman’s initial motivation to invent Quantum Computing 5. The second is that a number of novel Quantum Simulation schemes, to be run on quantum simulation devices, were first expressed as QWs 3, 4, which seems to be the natural language for doing so. No continuous spacetime limit is taken in these works. Typically in these quantum algorithms, the QW explores a graph, whose shape encodes the instance of the problem. The first is that a number of novel Quantum Computation algorithms, to be run on Quantum Computers, were discovered via QWs 1, 2, or were elegantly expressed using QWs (the Grover search for instance). QWs are blossoming, for two good reasons. it is ‘non-signalling’), meaning that information has a bounded speed of propagation. to a single ‘walker’) (ii) a discrete spacetime (iii) the unitarity of its evolution (iv) the homogeneity of its evolution, meaning its translation-invariance and time-independence, and (v) its causality (i.e. QWs are quantum dynamical systems characterized by: (i) a state space which is restricted to the one-particle sector (i.e. 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. ![]() 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. 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. 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. In a previous paper, we showed that QWs over the honeycomb and triangular lattices can be used to simulate the Dirac equation. A discrete-time Quantum Walk (QW) is an operator driving the evolution of a single particle on the lattice, through local unitaries. ![]()
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