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Di Molfetta, G., & Perez, A. (2016). Quantum walks as simulators of neutrino oscillations in a vacuum and matter. New J. Phys., 18, 103038–8pp.
Abstract: We analyze the simulation of Dirac neutrino oscillations using quantum walks, both in a vacuum and in matter. We show that this simulation, in the continuum limit, reproduces a set of coupled Dirac equations that describe neutrino flavor oscillations, and we make use of this to establish a connection with neutrino phenomenology, thus allowing one to fix the parameters of the simulation for a given neutrino experiment. We also analyze how matter effects for neutrino propagation can be simulated in the quantum walk. In this way, important features, such as the MSW effect, can be incorporated. Thus, the simulation of neutrino oscillations with the help of quantum walks might be useful to illustrate these effects in extreme conditions, such as the solar interior or supernovae.
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Hatifi, M., Di Molfetta, G., Debbasch, F., & Brachet, M. (2019). Quantum walk hydrodynamics. Sci Rep, 9, 2989–7pp.
Abstract: A simple Discrete-Time Quantum Walk (DTQW) on the line is revisited and given an hydrodynamic interpretation through a novel relativistic generalization of the Madelung transform. Numerical results show that suitable initial conditions indeed produce hydrodynamical shocks and that the coherence achieved in current experiments is robust enough to simulate quantum hydrodynamical phenomena through DTQWs. An analytical computation of the asymptotic quantum shock structure is presented. The non-relativistic limit is explored in the Supplementary Material (SM).
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Arrighi, P., Di Molfetta, G., Marquez-Martin, I., & Perez, A. (2019). From curved spacetime to spacetime-dependent local unitaries over the honeycomb and triangular Quantum Walks. Sci Rep, 9, 10904–10pp.
Abstract: A discrete-time Quantum Walk (QW) is an operator driving the evolution of a single particle on the lattice, through local unitaries. In a previous paper, we showed that QWs over the honeycomb and triangular lattices can be used to simulate 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.
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Arnault, P., Di Molfetta, G., Brachet, M., & Debbasch, F. (2016). Quantum walks and non-Abelian discrete gauge theory. Phys. Rev. A, 94(1), 012335–6pp.
Abstract: A family of discrete-time quantum walks (DTQWs) on the line with an exact discrete U(N) gauge invariance is introduced. It is shown that the continuous limit of these DTQWs, when it exists, coincides with the dynamics of a Dirac fermion coupled to usual U(N) gauge fields in two-dimensional spacetime. A discrete generalization of the usual U(N) curvature is also constructed. An alternate interpretation of these results in terms of superimposed U(1) Maxwell fields and SU(N) gauge fields is discussed in the Appendix. Numerical simulations are also presented, which explore the convergence of the DTQWs towards their continuous limit and which also compare the DTQWs with classical (i.e., nonquantum) motions in classical SU(2) fields. The results presented in this paper constitute a first step towards quantum simulations of generic Yang-Mills gauge theories through DTQWs.
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Arnault, P., Macquet, A., Angles-Castillo, A., Marquez-Martin, I., Pina-Canelles, V., Perez, A., et al. (2020). Quantum simulation of quantum relativistic diffusion via quantum walks. J. Phys. A, 53(20), 205303–39pp.
Abstract: Two models are first presented, of a one-dimensional discrete-time quantum walk (DTQW) with temporal noise on the internal degree of freedom (i.e., the coin): (i) a model with both a coin-flip and a phase-flip channel, and (ii) a model with random coin unitaries. It is then shown that both these models admit a common limit in the spacetime continuum, namely, a Lindblad equation with Dirac-fermion Hamiltonian part and, as Lindblad jumps, a chirality flip and a chirality-dependent phase flip, which are two of the three standard error channels for a two-level quantum system. This, as one may call it, Dirac Lindblad equation, provides a model of quantum relativistic spatial diffusion, which is evidenced both analytically and numerically. This model of spatial diffusion has the intriguing specificity of making sense only with original unitary models which are relativistic in the sense that they have chirality, on which the noise is introduced: the diffusion arises via the by-construction (quantum) coupling of chirality to the position. For a particle with vanishing mass, the model of quantum relativistic diffusion introduced in the present work, reduces to the well-known telegraph equation, which yields propagation at short times, diffusion at long times, and exhibits no quantumness. Finally, the results are extended to temporal noises which depend smoothly on position.
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