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Coin-Embedded U(1) Gauge Coupling in Discrete-Time Quantum Walks and the Emergent Dirac Equation: From the Temporal Gauge to a General Potential

Parth Verma

Published Oct 25, 2026• 0 citations
Publisher

Abstract

A discrete-time quantum walk (DTQW) is a lattice model of a single particle whose two-component internal state is repeatedly mixed by a unitary "coin" and shifted in space, chirality by chirality. In this work, we present a framework for coupling such a walk to a background electromagnetic (U(1)) gauge field. Instead of introducing traditional link variables common to the existing literature, we embed the gauge phase directly within the coin operator already required by the free walk, thereby eliminating the need for extra local degrees of freedom. We demonstrate this construction in two stages: first, using only coin phases to generate a purely spatial potential (A_1) while structurally forcing the temporal potential (A_0) to vanish (the temporal, or Weyl, gauge); and second, incorporating a single post-shift phase to recover a fully general potential (A_mu = (A_0, A_1). In both regimes, we show that the gauge transformation law for the coin phases is rigorously forced by the consistency requirement between the left- and right-moving sectors rather than assumed. We verify the construction non-perturbatively through the exact lattice dispersion relation, the walk’s convergence rate to the continuum Dirac equation, and its intrinsic avoidance of the fermion doubling problem.

DOI
10.5281/zenodo.23630299

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A discrete-time quantum walk (DTQW) is a lattice model of a single particle whose two-component internal state is repeatedly mixed by a unitary "coin" and shifted in space, chirality by chirality. In this work, we present a framework for coupling such a walk to a background electromagnetic (U(1)) gauge field. Instead of introducing traditional link variables common to the existing literature, we embed the gauge phase directly within the coin operator already required by the free walk, thereby eliminating the need for extra local degrees of freedom. We demonstrate this construction in two stages: first, using only coin phases to generate a purely spatial potential (A_1) while structurally forcing the temporal potential (A_0) to vanish (the temporal, or Weyl, gauge); and second, incorporating a single post-shift phase to recover a fully general potential (A_mu = (A_0, A_1). In both regimes, we show that the gauge transformation law for the coin phases is rigorously forced by the consistency requirement between the left- and right-moving sectors rather than assumed. We verify the construction non-perturbatively through the exact lattice dispersion relation, the walk’s convergence rate to the continuum Dirac equation, and its intrinsic avoidance of the fermion doubling problem.
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.
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.
Quantum simulation is an important way to study the Dirac particles in a general situation. Discrete quantum walk (DQW), is a powerful quantum simulation scheme, and implementable in well controllable table-top set-ups. We first identify that the conventional DQW can't exactly simulate Dirac Cellular Automaton (DCA), a discretized theory of free Dirac Hamiltonian (DH). We found some choice of coin parameters of the split-step (SS) DQW, a generalization of DQW can fully simulate single-particle DCA. Next we question whether the same SS-DQW can simulate dynamics of free Dirac particle with extra degrees of freedom like colors, flavors besides the spin or chirality. One such example is Neutrino oscillation. By moving from the U(2) coined SS-DQW to the U(6) coined SS-DQW we have simulated the exact probability profile of Neutrino flavor transitions. We further probe towards simulating single particle massive DH in presence of background potentials and space-time curvature. By using a SS-DQW with position-time dependent coin parameters, and we realize that it will give us an unbounded effective Hamiltonian, at the continuum limit of position-time. So we have introduced a modified version of SS-DQW which will produce a bounded effective Hamiltonian. This modified SS-DQW with U(2) coin operations produces single-particle massive DH in presence of abelian gauge potentials and space-time curvature. Introducing higher dimensional---U(N) coin operations in the modified SS-DQW we can include non-abelian potentials in the same DH. In order to simulate two-particle DH in presence of curved space-time and external potentials, we have used two particle modified SS-DQW, where the shift operations act separately on each particle, the coin operations which act simultaneously on both particles contain all kinds of interactions.
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.
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.
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.