From curved spacetime to spacetime-dependent local unitaries over the honeycomb and triangular Quantum Walks

A discrete-time Quantum Walk (QW) is an operator driving the evolution of a single particle on the lattice, through local unitaries. In a previous paper, we showed that QWs over the honeycomb and triangular lattices can be used to simulate the Dirac equation. We apply a spacetime coordinate transformation upon the lattice of this QW, and show that it is equivalent to introducing spacetime-dependent local unitaries —whilst keeping the lattice fixed. By exploiting this duality between changes in geometry, and changes in local unitaries, we show that the spacetime-dependent QW simulates the Dirac equation in (2 + 1)–dimensional curved spacetime. Interestingly, the duality crucially relies on the non linear-independence of the three preferred directions of the honeycomb and triangular lattices: The same construction would fail for the square lattice. At the practical level, this result opens the possibility to simulate field theories on curved manifolds, via the quantum walk on different kinds of lattices.

Examples of this class are the simulation of condensed matter systems modeled by a tight-binding Hamiltonian, such as graphene 11 or the Kagome lattices 12 -where the dynamics of electrons can be effectively recast as a Dirac-like equation. In fact the QW introduced in this paper may be useful as a simple point of departure to predict electronic transport properties in the graphene like-materials 13 and exploring how varying their geometry may influence the dispersion relations, and lead to topological phases 14 , with interesting consequences on the conducting properties.
Another motivation for this work is to understand how fermions would propagate if spacetime were a triangulated manifold, at the fundamental level. Indeed, triangulated manifolds are being used to describe curved spacetime since 15 -when Regge introduced his simplicial, discrete formulation of General Relativity. This discrete formulation then motivated a number of quantum gravity theories, such as Loop Quantum Gravity 16 and Causal Dynamical Triangulation 17 -which seek to recover Regge calculus in the classical limit. Most often quantum gravity research focuses on the core issue of the quantum dynamics of discrete spacetime itself-overlooking the question of how matter would propagate within the discrete spacetime structure it prescribes. The present ideas may help address the question.
Duality. In a previous work, we showed how a QW can be defined on the honeycomb and the triangular lattice 18 (see also 19 ), whose continuous spacetime limit is the Dirac equation in (2 + 1)-dimensional spacetime. Here, we extend these definitions to allow for spacetime dependent local unitaries, and introduce a dynamics that, in the continuum limit, corresponds to the Dirac equation in a curved (2 + 1)-dimensional spacetime.
The construction, we feel, is interesting. Indeed, given a lattice made of equilateral triangles, we begin by distorting the metric just via a coordinate transformation, following the initial step of the derivation of the Dirac equation in ordinary curved spacetime. But then we realize that the coordinate transformation can be absorbed by a suitable choice of the three gamma matrices that are associated to the three directions provided by the triangles-a possibility offered by the fact that these three directions are, of course, linearly-dependent in the plane. Recall that the role of the gamma matrices is to prescribe a basis of the spin, in which spin up goes one way, and spin down goes the opposite way. In the QW, the local unitaries implement precisely the corresponding changes of base. Thus, the gamma matrices determine the local unitaries in the QW. This, therefore, unravels an equivalence, in the continuum limit, between changing the actual geometry of the lattice, or keeping it fixed but changing the local unitaries in a suitable manner. The final step is to allow the local unitaries to be spacetime dependent and take the continuum limit, thereby recovering the Dirac equation in curved spacetime.
Notice that having three directions in two-dimensional space, as in the honeycomb or triangular lattices, is what provides that extra degree of freedom allowing for the transfer of the geometric distortions into the local unitaries-the square lattice is too rigid in this respect.

Related works.
It is already well known that QW can simulate the Dirac equation 3,4,8,[20][21][22][23] , the Klein-Gordon equation [24][25][26] and the Schrödinger equation 27,28 and that they are a minimal setting in which to simulate particles in some inhomogeneous background field [29][30][31][32][33] , with the difficult topic of interactions initiated in 34,35 . Eventually, the systematic study of the impact inhomogeneous local unitaries also gave rise to QW models of particles propagating in curved spacetime. This line of research was initiated by a QW simulations of the curved Dirac equation in (1 + 1)-dimensions, for synchronous coordinates 30,36 , and later extended by 37 to any spacetime metrics, and generalized to further spatial and spin dimensions in 38,39 . A related work, from a slightly different perspective, can be found in 40 . All of these models were on the square lattice: to the best our knowledge no one had modeled fermionic transport over non-square lattices. The present paper shows that over the honeycomb and triangular lattices the problem becomes considerably simpler, and the solution elegant.
In a recent work 41 , quantum transport over curved spacetime has been compared to electronic transport in deformed graphene, where a pseudo-magnetic field emulates an effective curvature in the tight-binding Hamiltonian (see also 42 ). Back to the quantum computing side, the Grover search has been expressed as a QW over the honeycomb lattice 43 (see also 44 for continuous time approach). Reference 45 evaluates the use graphene nanoribbons as a substrate to build quantum gates.
plan. The paper is organized as follows. First, we remind the reader of the basic concepts and notations surrounding the Dirac equation in a curved spacetime, in (3 + 1) and (2 + 1)-dimensions. In Methods we revisit our earlier Dirac QW on a honeycomb and on a triangular lattice, and why it worked. Also we show how a simple, homogeneous coordinate transformation impacts the continuum limit of the Dirac QW. In the end of this section, it is shown the duality, i.e. how the coordinate transformation can be absorbed into a choice of local unitaries. Finally, we present the main results: a QW that reproduces the Dirac equation with curvature in the continuum limit, both for the honeycomb and for the triangular lattices. We use = = c 1  units.

Dirac equation in curved Spacetime: a Recap
In this Section we recall the basic properties of the Dirac equation in curved spacetime. We refer the reader to [46][47][48] for a review. We start by describing the case of a (3 + 1)-dimensional spacetime with coordinates x μ , μ = 0, …4, where x 0 is the time coordinate, and metric tensor g μν (x) in these coordinates. At each point x, it is possible to introduce a set of four vectors μ = … μ e x a { ( )/ , 0, 4} a , referred to as the tetrad or vierbein, that locally diagonalizes the metric tensor i.e., (here and thereafter, summation over repeated indices is assumed), where η ab = Diag(1, −1, −1, −1). Notice that, given a vierbein, one can obtain a new one, which would also satisfy Eq. (1), by performing an arbitrary Lorentz transformation. The inverse of the vierbein is denoted μ e a (interchanged indices), satisfying Using (1) and (2), one has Thus, tetrads can be understood as normalized tangent vectors that relate the original coordinates to a local inertial frame. We use the common convention that inertial coordinates are designated by latin indices, and original coordinates by greek indices. Latin indices are lowered and raised by η ab , greek indices by g μν . In the local inertial frame, one is legitimated to use the Dirac γmatrices, i.e. matrices satisfying the Clifford algebra Given a Dirac field ψ(x), the action of a local Lorentz transformation Λ can be written as and θ ab (x) are the parameters of the transformation, defined by . One can prove that this operator acts on Dirac gamma matrices as follows: With the above notations, the Dirac equation in curved space a a  where m is the particle mass, is invariant under a local Lorentz transformation provided the generalized derivative that we use is where Γ μ transforms according to The correction Γ μ to the derivative can then be obtained as 47 ab ab where ω abμ (x) is the so-called spin connection, and can be expressed in terms of the tetrads and the affine connection as From Eq. (7) one can define a four-vector current where g is the (absolute value of) the determinant of the metric, so that it is conserved: This justifies the normalization condition 0 0 0 with dv the volume element in space.
(2 + 1)-dimensions. When the space dimension is lower than 3, the γ-matrices become 2 × 2. Then, the Dirac Eq. (7) can be simplified to give www.nature.com/scientificreports www.nature.com/scientificreports/ We will now express this equation in Hamiltonian form. We name the greek indices μ = t, x, y, and the latin indices a = 0, 1, 2. By performing a local Lorentz transformation, it is possible to arrive to a form of the tetrad such that = e 0 t a for a = 1, 2. Then, by introducing the change of wavefunction given by 49 :  , with the usual Dirac α-matrices α a ≡ βγ a . In particular, one can make the choice γ 0 = σ z , γ 1 = iσ y and γ 2 = −iσ x . Then α 0 becomes the identity matrix, α 1 = σ x and α 2 = σ y , with σ i (i = 1, 2, 3) the Pauli matrices.
According to Eqs (14) and (16), the normalization condition becomes simply

Methods
It is now very well-known that one can define a QW on the lattice that converges, in the limit of both the lattice spacing and the time step going to zero, towards the solutions of (19). This is done by defining a Hilbert space  stands for the space degrees of freedom, as spanned by the basis states x y , the p i will now denote the quasimomentum operators defined by ε ε ε ε x y The Dirac QW will evolve a state ψ(t) into ψ ε εσ ε σ ε σ ε ψ z x x y y D using the Trotter-Kato formula. It follows that one recovers the Dirac Eq. (19) in the continuum limit when ε goes to zero, where the p i become the true momentum operators p i = −i∂ i . Recently 18 we showed that Dirac dynamics can be implemented by a QW, not only over square lattices, but also over the honeycomb and triangular lattices (see also 19 ). The honeycomb lattice QW is easier to introduce. It defines three directions u i , i = 0, 1, 2 having relative angles of 120°, let u i j denote their coordinates. The idea is to introduce three unitary 2 × 2-matrices τ i with eigenvalues ±1 such that H D can be written as Next we start from a QW that reproduces the flat equation, and introduce a deformation (described by the transformation Γ) that will end up with a more generic metric g′. We can make a simple choice, given by the canonical tetrads δ = μ μ e a a for the initial coordinates, and then transform them according to Eq. (23). Since we are considering a deformation of the spatial sites of the lattice, the time components will be left unchanged, and the matrix Γ will take the form 1 0 0 0 0 (25) 11 12 21 22 where each λ ij are position independent, although they are allowed to depend on time.
Under this restriction, we can reduce the problem to a transformation on a bidimensional space, where = e 1 t 0 , which implies that Eq. (17) adopts the simpler form Let us consider how this transformation will affect the QW defined on a triangular lattice, as introduced in Sect. III (see 18 ). Such transformation will imply modifying the vectors u i , yielding the new vectors 21 22 Introducing these vectors in our algorithms and calculating the continuum limit, we arrive at the following equation This procedure can be used for a homogeneous transformation, such as the one defined above. In the next section, we introduce an alternative, which consists in redefining the τ i matrices. As we shall see, this redefinition also allows for an inhomogeneous (i.e., space-time dependent) Λ(t, x, y) transformation, thereby resulting in a Dirac equation in curved space. curved dirac equation from a non-homogeneous QW. We now generalize the ideas developed in the previous Sect. with the purpose to obtain, in the continuum limit, the Dirac equation on a curved spacetime, for a given metrics with a triangular tetrad, as discussed in Sect. II. We start by looking at the set of matrices α = B s a e e s a t 0 , as a linear transformation over the set of usual Pauli matrices, in the same spirit as Eqs (29)  We now make use of the property that relates the τ i matrices, defined in Eq. (22), with the Pauli matrices: 18 ). In this way, we arrive to The above equation can be understood as a transformation performed on the u i vectors, c.f. Eq. (27), as the origin of the curved spacetime equation.
Instead of introducing a distortion Λ(t, x, y) on the lattice via the modification of the u i vectors, the unitary matrices τ i can be transformed to produce the same effect. In other words, we seek for a set of matrices β i (t, x, y) that fulfill the following conditions: (2019) 9:10904 | https://doi.org/10.1038/s41598-019-47535-4 www.nature.com/scientificreports www.nature.com/scientificreports/ Notice that condition (C1) implies that the coordinate transformation dictated by Λ t x y ( , , ) k j is transferred to the unitary operations, which become new spacetime dependent β i (t, x, y), instead of the original τ i . Additionally, condition (C2) will allow us to rewrite the QW evolution in terms of the usual state-dependent translation operators. Let us apply these ideas to the honeycomb and the triangular lattice.
To alleviate the notations, in what follows we will omit the spacetime dependence both in these matrices and in the U i (t, x, y), and write simply β i and U i . The above conditions allow to calculate the β i matrices, which can be written as a combination of Pauli matrices, i.e.
Before we proceed to examine the induced QW on the honeycomb and triangular lattices together with their limits, let us discuss what the situation would have been in the square lattice, had we implement the above procedure. In this case, the original Dirac matrices can be chosen to be the Pauli matrices, and the two unit vectors u i can be taken to be the canonical ones, so that the requirement of Eq. (34) simply becomes k j k j But then, since condition (C2) implies that det(β j ) = −1 for each j, we need that ∑ Λ = .
( ) 1 (39) k k j 2 Thus the square lattice only allows for a limited form of "duality", i.e. only those transformations satisfying condition (39) can be absorbed into the unitaries, whereas the honeycomb and triangular lattices allow for arbitrary transformations.

Results
Honeycomb QW. In this section we define the QW over the honeycomb, following a similar procedure as in 18 . After the ideas developed in Methods, we define the following Hamiltonian to be used in the QW: Expanding the Hamiltonian, we arrive to: After substitution of Eq. (37), one obtains  (17).
Numerical simulations. In order to illustrate how the above scheme can be used to describe the dynamics of a particular system, we have computed the behaviour of a massless fermion in a (2 + 1)-dimensional spacetime black hole, whose metric in Lemaître coordinates is given by: 3 , and r s is the Schwarzschild radius. To simplify the simulations and the plots, we have not considered the angular motion, so that the variation in θ is zero. This allows us to describe the QW probability density in the plane (t, x), where x plays the role of ρ. The deformation Λ(t, x) to induce the former metric reads: www.nature.com/scientificreports www.nature.com/scientificreports/ s In Fig. 1 we can observe the dynamics of the walker in the projected plane (t, x). Depending on the initial position of the walker, the trajectories in the spacetime vary. The event horizon is given by . Therefore, when the particle is initialized inside the horizon with = .  30 , in which they study a QW with the same metric in (1 + 1)-dimensional spacetime.

Discussion
We introduced a Quantum Walk (QW) over the honeycomb and the triangular lattice. In both cases, our starting point was the possibility to rewrite the targeted Hamiltonian as a sum of momentum operators along the three relevant directions of the lattice, each weighted by a suitably chosen gamma matrix. This procedure has been introduced in 18 -our targeted Hamiltonian was then that of the Dirac equation, which we recovered in the continuum limit. In the present work, we realized that due to the linear dependence of the three preferred directions of the honeycomb and the triangular lattices, one could also obtain the Hamiltonian of the Dirac equation under an arbitrary change of coordinates. We emphasized that applying the same procedure, but for the square lattice, only allows for a very limited set of changes of coordinates.
Then, by making the gamma matrices to be spacetime dependent, we obtained the Curved Dirac equation in an arbitrary background metric. Overall, the QW hereby constructed over the honeycomb and the triangular lattices thus recovers, in the continuum limit, the Dirac equation in curved (2 + 1)-dimensional spacetime. We believe that the duality between changes of metric, and changes of gamma matrices weighting non linearly-independent momentum operators, is profound and may lead to further developments. , where the factor 3 6 is a necessary rescaling of the time coordinate 18 , and X = εx with ε = 1 3 . The number of time steps is t = 300. The initial condition is ψ(0, v, 1) = g(v − v 0 )(1, 1) T where g(v) is a Gaussian function with σ = 3. See the text for an explanation of the different panels.