I Introduction
We will describe two novel discretetime Quantum Walks (QW), one the honeycomb lattice, and other on the triangular lattice, whose continuum limit is the Dirac equation in –dimensions. Let us put this result in context.
Quantum walks. QWs are dynamics having the following characteristics: (i) the state space is restricted to the one particle sector (a.k.a. one ‘walker’); (ii) spacetime is discrete; (iii) the evolution is unitary; (iv) the evolution is homogeneous, that is translationinvariant and timeindependent, and (v) causal (a.k.a. ‘nonsignalling’), meaning that information propagates at a strictly bounded speed. Their study is blossoming, for two parallel reasons.
One reason is that a whole series of novel Quantum Computing algorithms, for the future Quantum Computers, have been discovered via QWs, e.g. BooleanEvalQW ; ConductivityQW and are better expressed using QWs. The Grover search has also been reformulated in this manner. In these QWbased algorithms, the walker usually explores a graph, which is encoding the instance of the problem. No continuum limit is taken.
The other reason is that a whole series of novel Quantum Simulation schemes, for the nearfuture Quantum simulation devices, have been discovered via QWs, and are better expressed as QWs BialynickiBirula ; MeyerQLGI . Recall that quantum simulation is what motivated Feynman to introduce the concept of Quantum Computing in the first place FeynmanQC . Whilst an universal Quantum Computer remains outofreach experimentally, more specialpurpose Quantum Simulation devices are seeing the light, whose architecture in fact often ressembles that of a QW WernerElectricQW ; Sciarrino . In these QWbased schemes, the walker propagates on the regular lattice, and a continuum limit is taken to show that this converges towards some wellknown physics equation that one wishes to simulate. As an added bonus, QWbased schemes provide: 1/ stable numerical schemes, even for classical computers—thereby guaranteeing convergence as soon as they are consistent ArrighiDirac ; 2/ simple discrete toy models of the physical phenomena, that conserve most symmetries (unitarity, homogeneity, causality, sometimes even Lorentzcovariance arrighi2014discrete ; DArianoLorentz , perhaps even general covariance MolfettaDebbasch2014Curved ; ArrighiGRDirac3D )—thereby providing playgrounds to discuss foundational questions in Physics LloydQG . It seems that QW are unravelling as a new language to express quantum physical phenomena.
Whilst the present work is clearly within the latter trend, technically it borrows from the former. Indeed, the QWbased schemes that we will describe depart from the regular lattice, to go to the honeycomb and triangular grid—which opens the way for QWbased simulation schemes on trivalent graphs.
Motivations. That quantum simulation schemes need not rely on the regular lattice grid is mathematically interesting—but there are numerous other motivations for this departure from the rectangular grid. One is the hot topic of simulating/modeling many quantum condensed matter systems dynamics, driven by the usual highbinding Hamiltonian or by the Diraclike Hamiltonian, for example in graphene, and within crystals in general neto2009electronic . This work would establish a connection between such physical phenomena and QWs. Another hot topic is related to topological phases. QW on triangulations should allow us to model all sorts of topologies as simplicial complexes, and hopefully help predict their transport properties Kitagawa2010 . The fact that our Triangular QW converges to the Dirac equation shows that we have have the right prediction at least in the flat case. Yet another motivation for exploring nonflat geometries is General Relativity. In fact, two of the authors have already developped QW models of the curved spacetime Dirac equation MolfettaDebbasch2014Curved ; ArrighiGRDirac3D ; DebbaschWaves . These were on the regular lattice, using a nonhomogeneous coin to code for the spacetimedependent metric. We wonder whether a QW on triangulations can also model the curved spacetime Dirac equation, using a homogeneous coin but a spacetimedependent triangulation. This problem is reminiscent of the question of matter propagation in triangulated spacetime, as arising, e.g., in Loop Quantum Gravity RovelliDirac . Here again, the fact that our Triangular QW converges towards the Dirac equation demonstrates that we have the right prediction at least in the triangulationofflatspace case. Finally, let us mention the work of two of the authors which models the massive Dirac equation as a Dirac QW on a cylinder MolfettaCylinder . QW on triangulations should allow us to vary the geometry of this cylinder, so as to model richer fields with just the massless Dirac QW.
Related works. The Grover quantum search algorithm has been expressed as a QW on the honeycomb lattice in Abal2010 (and also in Foulger2015 with continuous time). It has also been expressed as a QW on the triangular lattice Matsue2015 ; Abal2012 . Again for quantum algorithmic purposes, Karafyllidis2015 studies the possibility to use graphene nanoribbons to implement quantum gates. From the quantum simulation perspective, QWs on the triangular lattices have been used to explore transport in graphene structures Bougroura2016 ; chandrashekar2013two , and they have also been used to explore topological phases Kitagawa2010 —but no actual continuum limit is taken in these works. To the best of our knowledge, the only work that does take a continuum limit of a discretetime QW whilst departing from the regular lattice is sarkar2017effective , where a Diraclike hamiltonian is recovered. What we show is that the exact Dirac hamiltonian can be recovered, both in the honeycomb and the triangular lattices. That this can be done is somewhat surprising. Indeed, in Erba , the authors conducted a thorough investigation of isotropic QW of coin dimension
over arbitrary Caley graphs abelian groups, from which it follows that only the square lattice supports the Dirac equation. Our results circumvents this nogo theorem, whilst keeping things simple, by making use of twodimensional spinors which lie on the edges shared by adjacent triangles, instead of lying on the triangles themselves. Thus means that, per triangle, there are three thus including an additional degree of freedom associated to these edges.
Plan. In order to start gently, Sec. II, reexplains how the Dirac equation in –dimensions can be simulated by a QW on the regular lattice. In Sec. III, we reexpress the dimensional Dirac Hamiltonian in terms of derivatives along arbitrary three –rotated axes . We use this expression in order to simulate the Dirac equation with a QW on the honeycomb lattice. In Sec. IV, we introduce a QW on the triangular lattice, which will turn out to be equivalent to that on the honeycomb lattice. In V we provide a summary and some perspectives.
Ii On the regular lattice
In this section, we recall a now wellknown QW on the regular lattice with axis , and spacing , which has the Dirac Equation in the continuum limit. It arises by operatorsplitting FillionLorinBandrauk the original, onedimensional Dirac QW BenziSucci ; BialynickiBirula ; MeyerQLGI .
A possible representation of this equation is (in units such as ) :
(1) 
the Dirac Hamiltonian, () the Pauli matrices, the momentum operator components and the particle mass.
To simulate the above dynamics on the lattice, we define a Hilbert space , where represents the space degrees of freedom and is spanned by the basis states with , whereas describes the internal (spin) configuration. When acting on , the ’s are called quasimomentum operators (since they no longer satisfy the canonical commutation rules with the position operators). Still, the translation operators are given by and verify that
By analogy with these notations, we introduce the time evolution operator as . In this way, the time evolution of a state is given by
(2) 
After substitution of Eq. (1) into this definition, and making use of the LieTrotter product formula (assuming that is small) we arrive at:
since with the Hadamard gate, and with . Using the definition of , we get:
(3)  
where the matrices are partial shifts. This defines the Dirac QW, which is known to converge towards the Dirac equation in dimensions ArrighiDirac .
Iii On the honeycomb lattice
We now introduce a QW over the honeycomb lattice (Fig. 1) which we show has the Dirac equation as its continuum limit. The results of this Section will also help us in the next Section, when we introduce a QW over the triangular lattice. Our starting point is Eq. (2), with as defined in Eq. (1). The basic idea is to rewrite this Hamiltonian using partial derivatives (that will then turn into translations) along the three vectors that characterize nearestneighbors in the hexagonal lattice, instead of the and vectors that do so in the regular lattice. The vectors are given by
(4) 
with and the unit vectors along the and directions. In terms of momentum operators,
We then look for three matrices satisfying the following conditions:

We impose that , i.e. the Dirac Hamiltonian adopts the form
It was surprising to us that these conditions lead to unique matrices, up to a sign:
with . Let us choose , and notice that
(5) 
Thus
As before, we use the LieTrotter product formula and obtain:
(6) 
We now make use of condition (C1) to rewrite, for each ,
where now the partial shifts are defined through the operators, instead of and . Similarly, for all ,
Let . Wrapping it up, we have obtained a QW over the honeycomb lattice:
(7) 
which, by construction, has the Dirac Eq. (1) as its continuum limit as . By mere associativity the QW rewrites as
Thus, if the matrix products could be made independent of (with understood modulo ), the QW could be reformulated to have a constant coin operator. Surprisingly, this can be done thanks to a natural choice of the matrices, expressed in terms of wellchosen rotations in the Bloch sphere, understood as the set of possible spin operators. The natural choice for is , the rotation of angle around . Indeed , maps the Bloch vector of into the Bloch vector of :
(8) 
Next, we observe that the Bloch vectors are related by a rotation of angle around . For reasons that will become apparent, it matters to us that the cube of this rotation is the identity, which is obviously not the case for , since it represents a spin 1/2 rotation and will acquire a minus sign when applied three times. Hence we take instead. Then, the natural choice for the matrices and is:
Indeed, these again fulfill (C1): first the unitary brings to , and then the rotation brings to . Now, the fact that the matrices are related by a unitary which cubes to the identity entails that the products are independent of . We introduce
(9) 
Then, if we redefine the field up to an encoding, via
Then the Honeycomb QW rewrites as just:
(10) 
In other works, the Honeycomb QW just shifts the components along , applies the fixed matrix at each lattice point, shifts the components along , applies again, etc. For certain architectures it could well be that the time homogeneity of the coins makes the scheme easier to implement experimentally, compared to earlier alternate QW on the regular lattice ArrighiDirac .
Iv On the triangular lattice
Having understood how to obtain the Dirac Eq. over the honeycomb
lattice will make it much easier to tackle the triangular or related lattice such as the kagome lattice ye2018massive .
Let us first describe the lattice and its state space. Our triangles
are equilateral with sides , see Fig. 1.
Albeit the drawing shows white and gray triangles, these differ only
by the way in which they were laid—they have the same orientation
for instance. Our twodimensional spinors lie on the edges shared
by neighboring triangles. We label them , with a triangle and a side. But, since each spinor lies
on an edge, we can get to it from two triangles. For instance if triangle
(white) and (grey) are glued along their
side, then . In fact let us take
the convention that the upper (resp. lower) component of the spinor,
namely (resp. ), lies on the
white (resp. gray) triangle’s side. From this perspective each triangle
hosts a vector, e.g.
and .
The dynamics of the Triangular QW is the composition of two operators.
The first operator, , simply rotates every triangle anticlockwise.
Phrased in terms of the hosted vectors, the component
at side hops to side . For instance .
The second operator is just the application of the unitary
matrix given in (9), to every twodimensional spinor
of every edge shared by two neighboring triangles. Again we work on
preencoded spinors
(11) 
where the are those of Sec. III, but this time the chosen encoding depends on side . Altogether, the Triangular QW dynamics is given by:
(12) 
where is the neighbor of triangle alongside .
This Triangular QW is actually implementing the Honeycomb QW in a covert way. Indeed, whereas the Honeycomb QW propagates the walker along the three directions successively, the Triangular QW propagates the walker along the three translation simultaneously—depending on the edge at which it currently lies. Thus the walker will start moving along one of the three direction depending on its starting point, then another, etc. For instance, focusing on what happens to spinors on edges , we readily get
which is equivalent to a translation along (as is clear from Fig. 1), followed by the action of . But the result now lies on edges , and will undergo a translation along followed by the action of , etc.
As a sanity check we computed the continuum limit obtained by letting after three iterations of Eq. (12). The order is trivial. The is what defines the dynamics. Let us align the middle of the side side of triangle with the origin of the Euclidean space, so that in Cartesian coordinates. Expand the initial condition as:
where and are the coordinates in the lattice. As usual we also expand the inside the as . After three steps of the Triangular QW we obtain (with a the help of a computer algebra system):
Using that , and taking the limit , we arrive to the Dirac equation under the following form:
The factor comes from two reasons: the fact that continuous limit results from threetime steps and the fact that the distance between the middles of the sides of a triangle is . To get rid of this factor, it suffices to rescale the length of the spatial coordinates of the triangles by the same factor, or conversely to rescale time as .
V Summary and perspectives
Summary. We constructed a unitary , defined in Eq. (9), which serves as the ‘coin’ for both the Honeycomb QW and the Triangular QW. On the honeycomb lattice, each hexagon carries a spin. The Honeycomb QW, defined in Eq. (10), simply alternates a partial shift along along the direction of (4), followed by a on each hexagon, for . On the triangular lattice, each side of each triangle carries a , so that each edge shared by two neighbouring triangles carries a spin. The Triangular QW, defined in (12), simply alternates a rotation of each triangle, and the application of at each edge. The simplicity of these QWbased schemes, compared to those of the regular lattice (3), makes them not only elegant, but also easy to implement. Our main result states that, up to a simple, local unitary encoding given by (11), both the Honeycomb QW and the Triangular QW admit, as their continuum limit, the Dirac Eq. in –dimensions.
Perspectives. Thus we have shown that such quantum simulations results need not rely on the grid. We believe that this constitutes an important step towards : modelling propagation in crytalline materials; identifying substrates for QW implementations; studying topological phases; understanding propagation in discretized curved spacetime; coding fields in closed dimensions. In the near future, we wish run numerical simulations, and to understand what happens when deforming the triangles, and whether similar results can be achieved in –dimensions.
Update. We recently became aware that another, FrenchAustralian, team was tackling the same problem. We agreed to swap papers a few days before arXiv submission, so that the two works would be independent, and yet cite each other. Manuscript Manuscript is indeed very recommendable, as it goes further in terms of applications: electromagnetic field; gaugeinvariance; numerical simulations. Their triangular walk is, however, an alternation of three different steps, that use different coins—whereas the present paper just iterates the very same step. This is both mathematically more elegant, and easier to implement. Thus the two works have turned out nicely complementary.
Acknowledgements. We acknowledge an interesting discussion with M. C. Bañuls. This work has been funded by the ANR12BS0200701 TARMAC grant, the STICAmSud project 16STIC05 FoQCoSS and the Spanish Ministerio
de Economía, Industria y Competitividad , MINECOFEDER project FPA201784543P,
SEV20140398 and Generalitat Valenciana grant GVPROMETEOII2014087, the project INFINITI CNRS.
References
 [1] G. Abal, R. Donangelo, M. Forets, and R. Portugal. Spatial quantum search in a triangular network. Mathematical Structures in Computer Science, 22(3):521–531, 2012.
 [2] G. Abal, R. Donangelo, F. L. Marquezino, and R. Portugal. Spatial search on a honeycomb network. Mathematical Structures in Computer Science, 20(6):999–1009, 2010.
 [3] Andris Ambainis, Andrew M Childs, Ben W Reichardt, Robert Špalek, and Shengyu Zhang. Any andor formula of size n can be evaluated in time n^1/2+o(1) on a quantum computer. SIAM Journal on Computing, 39(6):2513–2530, 2010.
 [4] Pablo Arnault and Fabrice Debbasch. Quantum walks and gravitational waves. Annals of Physics, 383:645 – 661, 2017.
 [5] P. Arrighi and F. Facchini. Quantum walking in curved spacetime: (3+1) dimensions, and beyond. Quantum Information and Computation, 17(910):0810–0824, 2017. arXiv:1609.00305.
 [6] P. Arrighi, M. Forets, and V. Nesme. The Dirac equation as a Quantum Walk: higherdimensions, convergence. Preprint arXiv:1307.3524, 2013.
 [7] Pablo Arrighi, Stefano Facchini, and Marcelo Forets. Discrete lorentz covariance for quantum walks and quantum cellular automata. New Journal of Physics, 16(9):093007, 2014.
 [8] I. BialynickiBirula. Weyl, Dirac, and Maxwell equations on a lattice as unitary cellular automata. Phys. Rev. D., 49(12):6920–6927, 1994.
 [9] Eugenio Bianchi, Muxin Han, Carlo Rovelli, Wolfgang Wieland, Elena Magliaro, and Claudio Perini. Spinfoam fermions. Classical and Quantum Gravity, 30(23):235023, 2013.
 [10] Alessandro Bisio, Giacomo Mauro D Ariano, and Paolo Perinotti. Quantum walks, weyl equation and the lorentz group. Foundations of Physics, 47(8):1065–1076, 2017.
 [11] Hamza Bougroura, Habib Aissaoui, Nicholas Chancellor, and Viv Kendon. Quantumwalk transport properties on graphene structures. Physical Review A, 94(6):1–11, 2016.
 [12] Luis A Bru, German J De Valcarcel, Giuseppe Di Molfetta, Armando Pérez, Eugenio Roldán, and Fernando Silva. Quantum walk on a cylinder. Physical Review A, 94(3):032328, 2016.
 [13] CM Chandrashekar. Twocomponent diraclike hamiltonian for generating quantum walk on one, twoand threedimensional lattices. Scientific reports, 3:2829, 2013.
 [14] Giacomo Mauro D’Ariano, Marco Erba, and Paolo Perinotti. Isotropic quantum walks on lattices and the weyl equation. Physical Review A, 96(6):062101, 2017.
 [15] Giuseppe Di Molfetta, Marc Brachet, and Fabrice Debbasch. Quantum walks in artificial electric and gravitational fields. Physica A: Statistical Mechanics and its Applications, 397:157–168, 2014.
 [16] R. P. Feynman. Simulating physics with computers. International Journal of Theoretical Physics, 21(6):467–488, 1982.
 [17] François FillionGourdeau, Emmanuel Lorin, and André D Bandrauk. Numerical solution of the timedependent dirac equation in coordinate space without fermiondoubling. Computer Physics Communications, 183(7):1403–1415, 2012.
 [18] Iain Foulger, Sven Gnutzmann, and Gregor Tanner. Quantum walks and quantum search on graphene lattices. Physical Review A  Atomic, Molecular, and Optical Physics, 91(6):1–15, 2015.
 [19] Maximilian Genske, Wolfgang Alt, Andreas Steffen, Albert H Werner, Reinhard F Werner, Dieter Meschede, and Andrea Alberti. Electric quantum walks with individual atoms. Physical review letters, 110(19):190601, 2013.
 [20] Gareth Jay, Fabrice Debbasch, and Jingbo Wang. Dirac quantum walks on triangular and honeycomb lattices, 2018.
 [21] I G Karafyllidis. Quantum walks on graphene nanoribbons using quantum gates as coins. Journal of Computational Science, 11:326–330, 2015.
 [22] Takuya Kitagawa, Mark S. Rudner, Erez Berg, and Eugene Demler. Exploring topological phases with quantum walks. Physical Review A  Atomic, Molecular, and Optical Physics, 82(3), 2010.
 [23] S. Lloyd. A theory of quantum gravity based on quantum computation. ArXiv preprint: quantph/0501135, 2005.
 [24] Kaname Matsue, Osamu Ogurisu, and Etsuo Segawa. Quantum walks on simplicial complexes. arXiv, 7(5):1–36, 2015.
 [25] D. A. Meyer. From quantum cellular automata to quantum lattice gases. J. Stat. Phys, 85:551–574, 1996.
 [26] AH Castro Neto, Francisco Guinea, Nuno MR Peres, Kostya S Novoselov, and Andre K Geim. The electronic properties of graphene. Reviews of modern physics, 81(1):109, 2009.
 [27] Linda Sansoni, Fabio Sciarrino, Giuseppe Vallone, Paolo Mataloni, Andrea Crespi, Roberta Ramponi, and Roberto Osellame. Twoparticle bosonicfermionic quantum walk via integrated photonics. Phys. Rev. Lett., 108:010502, Jan 2012.
 [28] Debajyoti Sarkar, Niladri Paul, Kaushik Bhattacharya, and Tarun Kanti Ghosh. An effective hamiltonian approach to quantum random walk. Pramana, 88(3):45, 2017.
 [29] Sauro Succi and Roberto Benzi. Lattice boltzmann equation for quantum mechanics. Physica D: Nonlinear Phenomena, 69(3):327–332, 1993.
 [30] Guoming Wang. Efficient quantum algorithms for analyzing large sparse electrical networks. Quantum Info. Comput., 17(1112):987–1026, September 2017.
 [31] Linda Ye, Mingu Kang, Junwei Liu, Felix von Cube, Christina R Wicker, Takehito Suzuki, Chris Jozwiak, Aaron Bostwick, Eli Rotenberg, David C Bell, et al. Massive dirac fermions in a ferromagnetic kagome metal. Nature, 2018.
Comments
There are no comments yet.