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arXiv:1902.11079 · Condensed Matter

Discrete Geometry from Quantum Walks

Fabrice Debbasch

Published Apr 11, 2019• 11 citations
arXivHTMLPDF

Abstract

A particular family of Discrete Time Quantum Walks (DTQWs) simulating fermion propagation in 2D curved space-time is revisited. Usual continuous covariant derivatives and spin-connections are generalized into discrete covariant derivatives along the lattice coordinates and discrete connections. The concepts of metrics and 2-beins are also extended to the discrete realm. Two slightly different Riemann curvatures are then defined on the space-time lattice as the curvatures of the discrete spin connection. These two curvatures are closely related and one of them tends at the continuous limit towards the usual, continuous Riemann curvature. A simple example is also worked out in full.

DOI
10.3390/condmat4020040

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A particular family of Discrete Time Quantum Walks (DTQWs) simulating fermion propagation in 2D curved space-time is revisited. Usual continuous covariant derivatives and spin-connections are generalized into discrete covariant derivatives along the lattice coordinates and discrete connections. The concepts of metrics and 2-beins are also extended to the discrete realm. Two slightly different Riemann curvatures are then defined on the space-time lattice as the curvatures of the discrete spin connection. These two curvatures are closely related and one of them tends at the continuous limit towards the usual, continuous Riemann curvature. A simple example is also worked out in full.
A family of discrete-time quantum walks (DTQWs) on the line with an exact discrete U(N)\text{U}(N)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)\text{U}(N)U(N) gauge fields in two-dimensional spacetime. A discrete generalization of the usual U(N)\text{U}(N)U(N) curvature is also constructed. An alternate interpretation of these results in terms of superimposed U(1)\text{U}(1)U(1) Maxwell fields and SU(N)\text{SU}(N)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)\text{SU}(2)SU(2) fields. The results presented in this paper constitute a first step towards quantum simulations of generic Yang-Mills gauge theories through DTQWs.
A particular family of time- and space-dependent discrete-time quantum walks (QWs) is considered in one-dimensional physical space. The continuous limit of these walks is defined through a procedure discussed here and computed in full detail. In this limit, the walks coincide with the propagation of a massless Dirac fermion in an arbitrary gravitational field. A QW mimicking the radial propagation of a fermion outside and inside the event horizon of a Schwarzschild black hole is explicitly constructed and simulated numerically. Thus, the family of QWs considered in our manuscript provides an analog system to study experimentally coherent quantum propagation in curved spacetime.
A particular example is produced to prove that quantum walks can be used to simulate full-fledged discrete gauge theories. A new family of 2D2D2D walks is introduced and its continuous limit is shown to coincide with the dynamics of a Dirac fermion coupled to arbitrary electromagnetic fields. The electromagnetic interpretation is extended beyond the continuous limit by proving that these DTQWs exhibit an exact discrete local U(1)U(1)U(1) gauge invariance and possess a discrete gauge-invariant conserved current. A discrete gauge-invariant electromagnetic field is also constructed and that field is coupled to the conserved current by a discrete generalization of Maxwell equations. The dynamics of the DTQWs under crossed electric and magnetic fields is finally explored outside the continuous limit by numerical simulations. Bloch oscillations and the so-called E×B{\bf E} \times {\bf B}E×B drift are recovered in the weak-field limit. Localization is observed for some values of the gauge fields.
A particular example is produced to prove that quantum walks can be used to simulate full-fledged discrete gauge theories. A family of two-dimensional walks is introduced and its continuous limit is shown to coincide with the dynamics of a Dirac fermion coupled to arbitrary electromagnetic fields. The electromagnetic interpretation is extended beyond the continuous limit by proving that these discrete-time quantum walks (DTQWs) exhibit an exact discrete local U(1) gauge invariance and possess a discrete gauge-invariant conserved current. A discrete gauge-invariant electromagnetic field is also constructed and that field is coupled to the conserved current by a discrete generalization of Maxwell equations. The dynamics of the DTQWs under crossed electric and magnetic fields is finally explored outside the continuous limit by numerical simulations. Bloch oscillations and the so-called $\mathbf{E}\ifmmode\times\else\texttimes\fi{}\mathbf{B}$ drift are recovered in the weak-field limit. Localization is observed for some values of the gauge fields.
Nowadays, quantum simulation schemes come in two flavours. Either they are continuous-time discrete-space models (a.k.a Hamiltonian-based), pertaining to non-relativistic quantum mechanics. Or they are discrete-spacetime models (a.k.a Quantum Walks or Quantum Cellular Automata-based) enjoying a relativistic continuous spacetime limit. We provide a first example of a quantum simulation scheme that unifies both approaches. The proposed scheme supports both a continuous-time discrete-space limit, leading to lattice fermions, and a continuous-spacetime limit, leading to the Dirac equation. The transition between the two can be thought of as a general relativistic change of coordinates, pushed to an extreme. As an emergent by-product of this procedure, we obtain a Hamiltonian for lattice-fermions in curved spacetime with synchronous coordinates.
There are many problems that cannot be solved using current classical comput- ers. One manner to approach a solution of these systems is by using quantum computers. However, building a quantum computer is really challenging from the experimental side. Quantum simulators have been capable to solve some of these problems, as they are realizable experimentally.Discrete Time Quantum Walks (DTQWs) have been proved to be an useful tool to quantum simulate physical systems. In the continuous limit, a family of differential equations can be achieved, in particular, the Dirac equation can be recovered. In this thesis we study QWs as possible schemes for quantum simula- tion. Specifically, we can summarize our results in: i) We introduce a QW-based model in which a brane theory can be simulated in the continuum, opening the possibility to study more general theories with extra dimensions; ii) Elec- tromagnetic gauge invariance in QWs is discussed, presenting some similarities and differences to previous models. This QW model also makes a connection to gauge invariance in lattice gauge theories (LGT); iii) We introduce QWs over non- rectangular lattices, such a triangular or honeycomb structures, for the purpose of simulating the Dirac equation in the continuum. We also extent these models, by introducing local coin operators, that allow us to reproduce the dynamics of quantum particles under a curved space time.