Simulating Dirac Hamiltonian in Curved Space-time by Split-step Quantum Walk

Dirac particle represents a fundamental constituent of our nature. Simulation of Dirac particle dynamics by a controllable quantum system using quantum walks will allow us to investigate the non-classical nature of dynamics in its discrete form. In this work, starting from a modified version of one-spatial dimensional general inhomogeneous split-step discrete quantum walk we derive an effective Hamiltonian which mimics a single massive Dirac particle dynamics in curved $(1+1)$ space-time dimension coupled to $U(1)$ gauge potential---which is a forward step towards the simulation of the unification of electromagnetic and gravitational forces in lower dimension and at the single particle level. Implementation of this simulation scheme in simple qubit-system has been demonstrated. We show that the same Hamiltonian can represent $(2+1)$ space-time dimensional Dirac particle dynamics when one of the spatial momenta remains fixed. We also discuss how we can include $U(N)$ gauge potential in our scheme, in order to capture other fundamental force effects on the Dirac particle. The emergence of curvature in the two-particle split-step quantum walk has also been investigated while the particles are interacting through their entangled coin operations.


Introduction
Quantum walk, an effective algorithmic tool for simulating quantum physical phenomena where classical simulator fails or when the computational task is hard to realize via classical algorithm, has been shown to be very useful for realization of universal quantum computation [1][2][3]. The similarity between discrete quantum walk (DQW) and the dynamics of Dirac particles [4][5][6][7][8][9][10][11][12], at the continuum limit, elevates the DQW as a potential candidate to simulate various phenomena where the Dirac fermions play a crucial role [13][14][15]. With advancement in field of quantum simulations where many quantum phenomena are mimicked in table-top experiments, algorithmic schemes which can simulate Dirac particle dynamics in quantum field theory has garnered considerable interest in recent days. Simulation of Dirac particle dynamics in the presence of the external abelian and nonabelian gauge field by DQW has been recently reported [16,17]. Other recent works [18,19] investigated the inhomogeneous DQW that mimics the Dirac particle dynamics under the influence of external gauge-potential and curved space-time as a background. Two-step stroboscopic DQW with space-time dependent U(2) coin operator was used to produce gravitational and gauge potential effect in single Dirac fermion, but their approach was unable to capture mass, gravity and gauge potential in one Hamiltonian [18,19]. A generalized single particle Dirac equation in curved space-time was derived from a special DQWgrouped quantum walk (GQW)-which needs prior unitary encoding and decoding at last [20][21][22]. DQW with SU(2) coin operator parameters which are spatially independent but depend randomly on time-steps, has also been studied in the context of random artificial gauge fields [23]. The randomized coin parameters which mimic random gravitational and gauge field act as transition knobs from non-classical probability distribution to classical probability distribution. A DQW with a single evolution step which contains four spatial shift

Effective Hamiltonian corresponding to the standard Dirac equation in curved space-time
The general curved space-time Dirac equation [46,47] is written as where the covariant derivative ∇ μ = ∂ μ +Γ μ ; while in presence of U(1) gauge potential, ∇ μ = ∂ μ +Γ μ − iA μ , γ ( a) are local γ matrices and satisfy the conditions: g g h ={ } will be obeyed, where  n and n are mutually orthonormal. The identity matrix in this case will be expressed as σ 0 . The torsion-free and metric compatible connection is defined as  ) m x y t m , , will be independent of position and time, but for emergent particles which appear in condensed matter systems, the mass can in general be a function of position and time. For + ( ) 1 1 dimension the variable y will not be present in equation (5). However, for notational convenience we will express the Hamiltonian given in equation (5) as follows,

General split-step DQW
As the conventional DQW is a discrete quantum version of the classical random walk [48][49][50], the SS-DQW is a generalized version of the conventional DQW, first introduced in [26]. Multiple evolution parameters give more control over the evolution of the walk to engineer the dynamics at our desire. The general single-particle SS-DQW in + ( ) 1 1 dimensional space-time can be defined as a unitary evolution operator that evolves a state y ñ | ( ) t at time t to a state y t : or x is the position Hilbert space. The general state Here the unitary coin operation is defined as represent the elements of SU(2) group operation and after including x t ( ) x t , , j , we have a general U(2) group operation on coin space [25]. The coin state dependent position shift operators are defined as å å = Ä + ñá + Ä ñá = Ä ñá + Ä -ñá These position shift operators act homogeneously on all positions, at all time steps. The usual implementation method of the unitary operator U which implement one complete step of SS-DQW is in the following orderthe coin operation t ( ) C t, 1 is followed by the shift operation -S which is further followed by the coin operation t ( ) C t, 2 and then the shift operation + S . Here  S are by definition unitary operators. The coin operations are generalized U(2) operations on the coin space while they keep the position state of the particle intact, but the parameters of this U(2) operation depend on the position of the coin. + S shifts the particle one-step further in position points along the direction of increasing x if the coin state of the particle is in the up-state or ( ) 1 0 T and does nothing if the coin state of the particle is in the down-state or ( ) 0 1 T . -S does nothing if the coin state of the particle is in the up-state or ( ) 1 0 T and shifts the particle one step further in position points along the direction of decreasing x if the coin state of the particle is in the down-state or ( ) 0 1 T . Using the expressions given in equations (8)- (9), the whole evolution operator in equation (7) can be written in the form: 10 11 where * * * * * Because of the inhomogeneity of the evolution operator in space-time in equation (7), it is difficult to diagonalize the whole operator and derive the effective Hamiltonian as done in [25]. Instead of that, we will derive the Hamiltonian by Taylor series expansion with respect to the variables t a , assuming that t t = a c , have the same order of magnitude and = t a lim 0 0 (as we are taking c as a finite constant).
We are assuming that the are analytic functions of τ, so that we can do Taylor series expansion of them as well as the overall SS-DQW evolution operator.
Imposing the condition that , , 1 for all t x t , , ;as the coefficient of t n should be zero separately for each n, where  Î n ; we get From the condition we have a difference equation which, after expansion upto the first order in a gives , 0  lim  1  , , 0  , , 0  lim  1 , , 0 , , 0 .
x j a j j a j j 0 0 The similar definition will be used for the functions ( We further assume that all the higher order difference equation in a of all these functions are well defined, so that, at the limit  a 0, we can neglect higher order terms compared to the first order term in a in the Taylor series expansion with respect to the variable a.

Modified evolution operator and effective hamiltonian
Our conventional single-particle SS-DQW evolution operator does not directly satisfy the condition in equation (15), unless we impose some extra conditions on the functions: . But there may be a possibility that finding the valid conditions is not simple or may be the limit itself does not exist. Moreover, the condition will reduce the number of independent parameters. So, instead of using t ( ) U t, as our evolution operator we will use Note that this modification won't affect the relation of Dirac cellular automaton (DCA) and SS-DQW established in [25], because in that case . We can write the matrix form of the modified evolution operator U t ( ) t, in the coin basis as: 10 11 where , 0  ,  , 0  , ,  ,  , 0  ,  , 0  , ,   ,  , 0  ,  , 0  , ,  ,  , 0  ,  , 0  , .  20   00  00  00  10  10  01  00  01  10  11   10  01  00  11  10  11  01  01 11 11 The detailed forms of these operators are calculated in appendix B. Expanding these operators upto first order in τ and a, we can calculate the effective Hamiltonian using the similar form of the equation (13) defined for the conventional SS-DQW evolution operator, i.e. now we use the definition: For the detailed derivation of this Hamiltonian see appendix C.1. The derived effective Hamiltonian is of the form: The terms X ( ) in Hamiltonian operator are explicit functions of The explicit expressions of these terms are given in appendix C.1.
Note that, in the standard Hamiltonian in equation (6) in + ( ) 1 1 dimension, the total possible number of independent coefficients of the momentum operatorp 1 is two and they are a a Ä Ä

When the coin operation of modified SS-DQW is restricted to span
The Hamiltonian, derived in the preceding section corresponds to the general U(2) coin operation. Now we will consider a special case where the coin operations are only rotations about the spin-x-axis, such that , , 0 , , , 0 j j j . In this case: x t x t x t 0, cos , , 0 sin 2 , , 0 , , 0 , 1 2 cos 2 , , 0 1 2 cos 2 , , 0 2 , , 0 , 23 , 2 , , 0 , , , 2 , , 0 , 24  Note that, for this choice , the effective Hamiltonian in equation (22) will reduce to the case of the flat space-time: for all x t , .   After omitting all the zero-valued terms, the Hamiltonian in equation (27) becomes where we identify   In case we want to study the fundamental particle, the mass m should be taken position-time independent, we can choose ( ) , , ,0 1 . In condensed matter, many kinds of emergent particles are possible whose masses may depend on both the time and position steps, so, we can set = ( ) , , ,0 , , 0 1 1 , g 11 term of any metric can be captured by this through some constant value scaling.

Numerical simulations
The main purpose of this work is to unify all the possible background potential effects in the single particle massive Dirac Hamiltonian in its first quantized version and simulate it in an operational form using quantum walks. For proper depiction one should do numerical analysis for all possible common mathematical forms of the metric and gauge potentials. So that one can predict the mathematical forms of metric and gauge potentials corresponding to the experimentally observed phenomena where the metric and gauge potential functions are unknown. Here in the numerical section we have given examples of few common mathematical forms of metrics and external gauge potentials. Our numerical results are obtained by considering  = 1 unit, c=1 unit, t ≔ L 1 unit and ≔ a L 1 unit. Here, L should not be confused with the system size, we have used it merely to parameterize τ and a. We choose to work with the mass=m=0.04 unit. We have plotted the probability as a function of time (SS-DQW steps) and position for two different cases. This probability is irrespective of the coin state of the particle, i.e. we have traced over whole coin states at every time-step.

A static metric
For a static case we will run our simulation considering L=250.
1. Figure 1 is for flat space time without U(1) potential:   The coin parameter functions are: In figure 2, the probability which spreads only to the right side of the origin is seen. The gauge potential is captured by the parameters: The other coin parameter functions are:

A non-static metric
Here we will show the numerical simulation of a non-static case. We will take L=150.
1. Figure  , the coin parameter functions are:  , the coin parameter functions are: In this work we should note that the initial state of the quantum walk system is taken to be a pure state ä  Ä c x , and hence under the modified SS-DQW evolution which is also an unitary, the state will remain pure. As we have dealt with a quantum walk particle which is always in a pure state , ensemble is the collection of the identically prepared quantum walk systems, all of them are in the state y ñ | ( ) t at time-step t. But during measurement of position irrespective of the coin state of the particle, we actually measure on the partial state of the system which is traced out over coin Hilbert space= y y ñá . For example, the ensemble contains total + n n 0 1 number of systems, at a particular time-step all are described by the state y ñ | ( ) t , n 0 among them are in the ñ |0 coin state and other n 1 are in the ñ |1 coin state (after coin measurement). Among the n 0 systems r 0 systems are in position x, among the n 1 systems r 1 systems are in position x. So the probability to be in the position x is å frequency of coin state×positional probability of that coin state=å= + 2 with mass=0.04 unit and the initial state used for the evolution is in presence of gauge potential. in absence of gauge potential with mass=0.04 unit. The initial state used for the evolution is , has been shown to get a comparable idea about the other plots.
The parameters considered for the figure 2 will give Hamiltonian at the continuum limit. This Hamiltonian is the same as the Rindler Hamiltonian: , except an additional potential term. In the figure 2 from the probability profile it may seem that after long times the particle has a probability to exist outside the light-cone described by the figure 1. But in that case the light-cone should be described by equation: , where ds is usually taken as the infinitesimal distance in world space-time and x 0 is the position of the particle at time t=0. The trajectories should not cross the light-cone described by equation (37), as the coordinate system is not flat now. So it will not violate the causality principle even if it crosses the Minkowski light-cone. Although, in the unit system c=1,  = 1, = a L 1 that we have used while plotting the figures this light-cone in equation (37) will always remain within the Minkowski light-cone, and the particle trajectory never crosses the light-cone described by the equation (37). Because in the figures the axes labels are actually dimensionless quantities.   In this case:

Simulating
, , 2 , , 0 , 42 , , 2 , , 0 , 43 The total number of variables   Therefore, the metric . We should note here that this kind of choice implies that the effect of the momentum k y of the hidden coordinate express itself as a part of the gauge potential A 2 . Other choices are possible which may give rise to different metrics.

Implementation of our scheme in Qubit-system
The shift operations  S in equation (9), and the coin operations t ( ) C t, j in equation (8) are kinds of controlledunitary operations. The shift operations  S change the position distribution while the coin state acts as the controller, and the coin operations t ( ) C t, j change the coin state while positions act as controllers. Coin state is already represented by a qubit (a 2-dimensional quantum-state), but the position space is  dimensional if the total number of lattice sites is  , so, in general it can be any dimensional. Here, we will represent the position states by n-qubit system such that the total number of position will now be 2 n and each position is indexed by the decimal value-of the corresponding binary bits expression. Although  = 2 n represents only a particular kind of numbers, any general number of lattice sites can be constructed by neglecting some extra degrees of freedom. Below we demonstrate this scheme by a simple example. Suppose our working system is a periodic lattice with 4 lattice sites, i.e.lattice system is ñ {|x such that  Î } x 4 . We can build it by 2-qubit only-representing each qubit in the computational basis x a x i i 00 01 01 10 10 11 11 00 For the periodic lattice case with total  number of lattice sites we use the relation x a x k a n k n a 1 1 .
. Therefore the minimum possible gap between two kʼs = Here ñ |k is the eigenvectors of the generator of the positional translation operator of the quantum walk: . For even  , we can choose Thus the whole evolution operator U t t t = + - ,0 is implementable in a simple qubit system.
There is opinion of unnecessity of a quantum simulator for the simulation of the dynamics of a single particle quantum system-properly classical simulator can do the whole job. But the following two aspects can be used to counter this opinion. (i) A single quantum particle can be in a superposition of wave and particle state according to the [51]. But classical particle and wave are two independent entities and they never mix. (ii) Entanglement between two different degrees of freedom (coin and position in our case) in a single quantum particle have contextual origin, but classical physics shows non-contextual behavior. For detailed discussion please look into the [52]. So unless one explicitly proves that, in case of quantum simulation these two aspects are not important or can be captured by classical means after some kinds of encoding, it is better to work with quantum simulators.

Inclusion of U(N) potential in our SS-DQW scheme
In the above cases we are able to include the effect of the U(1) gauge potential. But we can define the coin operations in such a way that influence of general U(N) potential on single Dirac particle in + ( ) 1 1 dimension can also be derived. Here we will follow a similar kind of procedure as in the [16].
In this case the coin Hilbert space  where  N is the N×N identity matrix defined on the coin Hilbert space. The coin operations are now defined as where the matrices L q are the generators of U(N) group. We will define our evolution operator as ,0 ,0 , , which is similar to the case having U(1) potential only. Using the definition of the effective Hamiltonian as in equation (21), we get The term for all q. For a proper choice of N the U(N) can be the composition of all possible abelian and non-abelian gauge potential effects, and hence, the derived Hamiltonian can capture all the possible fundamental force effects on a single Dirac particle. For example we can include SU(3) and SU(2) interactions by choosing =´= N 2 3 6. One important point is that the dynamical characters of these potentials have not been considered, they act as background potentials on the single Dirac particle.
Sometimes the fermion doubling problem [53,54] appears, when the fermion particle dynamics is discussed in lattice position framework. The corresponding no-go theorem-Nielsen-Ninomiya theorem describes the impossibility of lattice simulation of local fermion field theory consistently without avoiding the fermion doubling problem. In [55,56], it is discussed that the Nielsen-Ninomiya theorem may not be applicable for discrete time evolution.
In our case all the evolution operator is discrete in both time and position. In the homogeneous SS-DQW case when we allow only rotation about the spin-x axis in the coin operations, according to the [25] the positive energy eigenvalue ka cos cos cos cos sin sin 59 is a monotonic function of the modulus of momentum: In case of position, time-step dependent coin parameters, the overall effect can be thought as a introduction of space-time dependent potential effects on the homogeneous SS-DQW case. It is expected that for the scalar potential, i.e.while the potential does not depend on the chirality of the particle, it does not change the monotonic nature of the energy as a function of the modulus of momentum. So, in those cases, fermion doubling does not occur. But for chirality dependent potentials, it is not so obvious that the fermion doubling problem does not appear, so these cases need further investigations.
8. Extending our ( + 1 1) dimensional SS-DQW scheme to two-particle case Here we will apply our SS-DQW framework into a two-particle system. In order to extend to two-particle case we will use entangled coin operations and the separable shift operations. We extend the conventional evolution operator that evolves a two-particle state at time t to a state at time t + t , such that the modified or actual evolution operator will now be ,0 acting on the Hilbert space correspond to the position Hilbert spaces of the first and second particles, correspond to the coin Hilbert spaces of the first and second particles, respectively. Note that, we have synchronized the time-steps of both the particles to the time-step t same for both, which is a special case. The shift operators for the individual particle are now defined as where r=1 and r=2 are for the first and the second particles, respectively. Therefore,  The coin operators are now defined as where , , , has to be real for all , , , , , , in order to make the coin operations unitary. . Hence, this kind of evolution operator can capture distinguishable as well as indistinguishable two-particle evolution depending on the functional form of C t ab ( ) . In the separable coin operation case while there is no interaction among the particles we must have 1, 2, 3 and any t x t , , . Thus for the nontrivial case-when these coefficients are nonzero, the particles can be in general entangled in their coin space by the whole SS-DQW evolution. Here our main purpose is to study the emergence of the curvature effects from the coin-coin entanglement, so, we will choose to work in a special entangled coin operation: , , , and all other , , , 0. Our choice of the whole coin operator is already symmetric in two-particle coin states, so it may describe indistinguishable particles if q . We will consider the case when q t ( ) are analytic in all of their arguments for all = j 1, 2. So, we can consider the Taylor series expansion in variable τ as q where the higher order terms in τ are chosen to be zero. Following the same procedure as in the case of the single particle, we get the effective two-  where X å X ñ á mn mn  . For detailed derivation and the explicit expression of the coefficient functions of the two-particle Hamiltonian see appendix E.
The two-particle effective Hamiltonian can be split into three parts as looks like local Hamiltonian part for the first particle whose effective mass=0 and the curved nature of spacetime which is influenced by the presence of the second particle, is captured by the term Q ( ) , and looks like local Hamiltonian part for the second particle whose effective mass=0 and the curved nature of space-time which is influenced by the presence of the first particle, is captured by the term Q ( ) . The part H eff inter of the Hamiltonian has no proper local analogy. This appears as a purely two-particle interaction term originated from the entangled coin operation.
Note: The coin operation is global, which can entangle two separable particles, and this entanglement has in general nonlocal features. Thus implementation by local operation is in general impossible. But the coefficients (or strength) of the interaction term controlled by C t ab ( ) x x t , , , outside the light-cone, the interaction can be made local. In case of quantum simulation these particles are usually very near to each other, i.e. the distance between the particles is hardly space-like. Almost of all the simulation cases they remain within time-like distance, so that information transfer from one to another is possible during any bipartite local operation. Once local and twoparticle controlled local simulators implement s s Ä a b operations, an entanglement between these particles can be created. After that it is possible that they possess nonlocal nature in Bell inequality violation sense, when get separated beyond light-like distance. At the current stage we do not have any explanation of this nonlocal interaction in terms of any gauge boson exchange. This is very interesting point, but need further investigation.  ,  2  sin  , , , 0  , , , 0  ,   , ,  cos  , , , 0  ,   , ,  2  , , , 0  , , , 0  , ,  , , , ,

Conclusion
In this work we are able to show that single-step SS-DQW with slight modification, can simulate massive Dirac particle dynamics under the influence of external abelian gauge potential and curved space-time. The modification of evolution operator is just an extra coin operation after applying the conventional SS-DQW. We have shown that the same Hamiltonian can capture pseudo + ( ) 1 1 dimensional or + ( ) 2 1 dimensional Dirac particle dynamics when the momentum of the hidden dimension remains fixed. We provided an implementation scheme by qubit systems which is realizable in current experimental set-up. By increasing the dimension of the coin-space, the influence of general U(N) gauge potential has been included in our scheme which paves a way towards simulation of four fundamental force effects on a single Dirac particle. We extended our study to the case of two-particle SS-DQW where the interaction of the particles is solely comes from the entangled coin operations and showed that the parameters of this entanglement can be included in the curvature effect. Our study shows a way to investigate nonclassical properties as well as the curvature effects which are difficult to observe in real situation.

Acknowledgments
. Generalization to the curved space-time is given by , A μ is the U(1) potential. Now in view of the following relations,  g g g g h g h g g g g g g g g g To derive the current density we need to derive also the dual equation satisfied by y y b = † , where b g = ( ) 0 and it is given by the following equation, with the assumption that all the vielbeins are real, e  x  iA  e  i  mc  2  ,  2  ,  A 3   a  a  a  a  a  5  2 From equation (A2) and equation (A3) it is possible to derive the four vector current m j , and they are given as yg y y y yg y = - where H is the Hermitian Hamiltonian operator. So the probability density is given by, After we multiply equation (A2) by β, we get a similar equation like equation (A5)  Similarly, and, x g x g x g g x g x g x g g x 1 2 1 2 We can evaluate this easily by using the following relation for any arbitrary matrix M, . Finally using all the relations described above, we can write, (which will not make any lose of generalization as the number of independent vielbeins in the metric is less than the total number of vielbeins-see [46] for details) and the properties in equations (A11), (A12) we can show that second, third, and eighth terms of the above equation will cancel with each other. Finally we can write,   We now expand the unitary evolution operators in equation (B1) upto first order in variables τ and a. We use here the definition of generator of translation as: From the calculation in equation (B1) we get the matrix elements of t ( ) U t, in coin basis as follows • The first-row first-column term of SS-DQW Evolution operator in coin-basis  • The first-row second-column term of SS-DQW Evolution operator in coin-basis • The second-row first-column term of SS-DQW Evolution operator in coin-basis x • The second-row second-column term of SS-DQW Evolution operator in coin-basis x , , ,  00  00  00  10  10  , ,0  , ,0  2  1   2  1 , , ,   , 0  , , 0  , , 0  , , 0  , , 0  , , 0  , , 0  , , 0   , , 0  , , 0  , , 0  , , 0  , , 0  , , 0   , , 0  , , 0  , , 0  , , 0  , , 0  , , 0  , , 0  , , 0  , , 0  , , 0  , , 1  2  2  1  2  1  2  2  2  1   2  2  1  1  1  2  2   1  2  1  2  1  2  2   1  2   1  2  2  1  2  2  2  1  2   2  1  2  1   2  2  2  1  2  2  2  1  2   2  1  2  1  2 where we have used the definition B.2. The first-row second-column term of our modified Evolution operator in coin-basis 01  00  01  10  11  , ,0  , ,0  2  1  2  1 , , , , , ,    B.3. The second-row first-column term of our modified Evolution operator in coin-basis x a t  x x e   ,  ,0  ,  ,0  ,   , , 0  , , 0  , , 0  , , 0   , ,  , , , , x 10  01  00  11  10  , ,0  , ,0  2  1  2  1   , ,  , ,  2  1   , ,  , ,  2  1  , ,0  , ,0  2  1  2  1 , , ,   where we have used the definition , , ,   where we have used the definition Appendix C. Calculating the operator terms of the effective Hamiltonian for the single particle         Here for notational convenience we will omit the arguement ( ) x t , , 0 from all the functions ( ) j and will be represented as F j , G j , x j , l j , f j , g j respectively. Then the operator terms of this effective Hamiltonian can be written as follows      Appendix D. Calculating the modified evolution operator for the two-particle case Here we will calculate the modified evolution operator U t ( ) t, two in section 8 parts by parts.           are zeros.