Electromagnetic lattice gauge invariance in two-dimensional discrete-time quantum walks

Abstract

Gauge invariance is one of the more important concepts in physics. We discuss this concept in connection with the unitary evolution of discrete-time quantum walks in one and two spatial dimensions, when they include the interaction with synthetic, external electromagnetic fields. One introduces this interaction as additional phases that play the role of gauge fields. Here, we present a way to incorporate those phases, which differs from previous works. Our proposal allows the discrete derivatives, that appear under a gauge transformation, to treat time and space on the same footing, in a way which is similar to standard lattice gauge theories. By considering two steps of the evolution, we define a density current which is gauge invariant and conserved. In the continuum limit, the dynamics of the particle, under a suitable choice of the parameters, becomes the Dirac equation, and the conserved current satisfies the corresponding conservation equation.

Quantum walk hydrodynamics

“…The results presented in this article can be generalised to quantum fluids with non vanishing traditional pressure by working with non-linear QWs such as those already considered in 40–42 . The non-linearity of these walks can be reconciled with the linearity of Quantum Physics by viewing the non-linear terms as an effective description of (self-)interaction, generated for example by the coupling of QWs through gauge fields 21,43–45 . Let us note that non-linear QWs can in principle be realized experimentally, at least through non-linear optics experiments 46,47 .…”

Creating anomalous Floquet Chern insulators with magnetic quantum walks

“…We acknowledge financial support from the ERC grant DQSIM (Project Nr. 291401), and the collabo- We briefly summarize how a general 2D magnetic vector potential can be realized with a quantum walk [51,54,56]. To create an artificial vector potential A with both its x-and y-components nonvanishing, two magnetic-field operators are employed in the time-step operator:Ŵ…”

“…In this form, one can clearly recognize [51] that the quasimomentum operator is shifted by an amount proportional to the vector potential, in a similar fashion as minimal coupling in classical electromagnetism. Equivalently, the magnetic-field operator can be thought of as a way to ensure discrete local gauge invariance [52][53][54][55][56] of the discrete-time quantum-walk protocol. A further insight into the effect of the magnetic field operator is obtained by considering the dynamics of a magnetic quantum walk in the weak-field regime, φ 1, when the magnetic length scale is much larger than the lattice constant, B a.…”

Quantum machine learning for particle physics using a variational quantum classifier

“…Quantum annealers, for example, perform continuous time quantum computations and are therefore well-suited to study the dynamics of quantum systems, even quantum field theories [19,20], and in solving optimisation problems [21]. Quantum gate computers are in particular a popular choice to calculate multi-particle processes [22][23][24][25][26][27][28][29][30][31][32], often with field theories mapped onto a discrete quantum walk [33][34][35][36] or a combined hybrid classical/quantum approach [37][38][39][40].…”