Quantum walking in curved spacetime
A discrete-time quantum walk (QW) is essentially a unitary operator driving the evolution of a single particle on the lattice. Some QWs admit a continuum limit, leading to familiar PDEs (e.g., the Dirac equation). In this paper, we study the continuum limit of a wide class of QWs and show that it leads to an entire class of PDEs, encompassing the Hamiltonian form of the massive Dirac equation in (\(1+1\)) curved spacetime. Therefore, a certain QW, which we make explicit, provides us with a unitary discrete toy model of a test particle in curved spacetime, in spite of the fixed background lattice. Mathematically, we have introduced two novel ingredients for taking the continuum limit of a QW, but which apply to any quantum cellular automata: encoding and grouping.
KeywordsPaired QWs Lattice quantum field theory Quantum simulation
- 13.Lloyd, S.: A theory of quantum gravity based on quantum computation. arXiv:quant-ph/0501135 (2005)
- 24.Arrighi, P., Facchini, S., Forets, M.: Three discrete models for the \((1+1)\) curved Dirac equation. Unpublished manuscript (2015)Google Scholar
- 27.London, D.: A note on matrices with positive definite real part. In: Proceedings of the American Mathematical Society. pp. 322–324 (1981)Google Scholar
- 28.http://pageperso.lif.univ-mrs.fr/~pablo.arrighi/publis/CurvedSpacetimeDiracEquation.sws. Also available as SageMathCloud worksheet at https://goo.gl/mDLwoL