Bounds on the dynamics of periodic quantum walks and emergence of the gapless
and gapped Dirac equation
Abstract
We study the dynamics of discretetime quantum walk using quantum coin operations, and in timedependent periodic sequence. For the twoperiod quantum walk with the parameters and in the coin operations we show that the standard deviation [] is the same as the minimum of standard deviation obtained from one of the oneperiod quantum walks with coin operations or , . Our numerical result is analytically corroborated using the dispersion relation obtained from the continuum limit of the dynamics. Using the dispersion relation for one and twoperiod quantum walks, we present the bounds on the dynamics of three and higher period quantum walks. We also show that the bounds for the twoperiod quantum walk will hold good for the splitstep quantum walk which is also defined using two coin operators using and . Unlike the previous known connection of discretetime quantum walks with the massless Dirac equation where coin parameter , here we show the recovery of the massless Dirac equation with nonzero parameters contributing to the intriguing interference in the dynamics in a totally nonrelativistic situation. We also present the effect of periodic sequence on the entanglement between coin and position space.
I Introduction
The quantum walk is a generalization of the classical random walk equivalent in a quantum mechanical framework Ria1958 (); Fey86 (); Par88 (); ADZ93 (); Mey96 (). By exploiting the quantum interference in the dynamics, quantum walks outperform the classical random walk by spreading quadratically faster in position space Kem03 (); Ven12 (). At certain computational tasks, quantum walks provide exponential speedup CDF03 (); CG04 () over classical computation and are used as a powerful tool in most of the efficient quantum algorithms Amb07 (); MSS07 (); BS06 (); FGG08 (). Both the variants, continuoustime and discretetime quantum walks have been shown to be universally quantum computation primitive, that is, they can be used to efficiently realize any quantum computation tasks Chi09 (); LCE10 (). With the ability to engineer and control the dynamics of the discretetime quantum walk by controlling various parameters in the evolution operators, quantum simulations of localization Joy12 (); Cha12 (); CB15 (), topological bound states KRB10 (); OK11 (), relativistic quantum dynamics where the speed of light is mimicked by the parameter of the evolution operator Mey96 (); Fre06 (); CBS10 (); Cha13 (); MBD13 (); MBD14 (); AFF15 (); Per16 (), and neutrino oscillations MMC17 (); MP16 () have been shown. The quantum walk has also played an important role in modeling the energy transfer in the artificial photosynthetic material Eng07 (); MRL08 (). Faster transport HM09 (), graph isomorphism DW08 (), and quantum percolation CB14 (); KKN14 () are few other application where the quantum walk has found application.
Experimentally, controlled evolution of quantum walks has also been demonstrated in various physical systems such as NMR RLB05 (), trapped ions SMS09 (); ZKG10 (), cold atoms KFC09 () and photonic systems PLP08 (); PLM10 (); SCP10 (); BFL10 () making it a most suitable dynamic process which can be engineered for quantum simulations.
Among the two variants of quantum walks, the dynamics of the continuoustime variant are described directly on the position Hilbert space using an Hamiltonian. The dynamics of each step of the discretetime variant are defined on a Hilbert space composed of both, the position and particle Hilbert space using a combination of unitary quantum coin operation acting only on the particle space followed by a position shift operation acting on both, particle and position space. By exploring different forms of quantum coin and position shift operators in homogeneous NV00 (); CSL08 (), periodic MK09 (), quasiperiodic XQT14 (); GAB17 () and random AVM11 (); Cha12 () sequence, ballistic spreading to the localization of the wavepacket of the particle has been studied. One of the mathematically rigorous approaches to understand the asymptotic behavior of the dynamics is to compute the limit distribution function KLS13 (); Kon10 (). In Ref. MK09 (), limit distribution function for the twoperiod quantum walk using two orthogonal matrices as alternate quantum coin operations has been computed. Inspite of the important role of quantum interference in the dynamics of the quantum walk it has been shown that the limit distribution of the twoperiod quantum walk is determined by one of the two quantum coin operations (orthogonal matrix).
This is an important observation which needs to be explored in more detail to understand the intricacy involved in the dynamics of the periodic quantum walks. Particularly, when twoperiod quantum walks is shown to produce the dynamics identical to the splitstep quantum walk ZGS17 () which has been used to simulate topological quantum walks, Diraccellular automata MC16 () and Majorana modes and edge states ZGS17 () where both the coin operations play an important role.
Obtaining the limit density function for the nonorthogonal unitary matrix as the quantum coin operation for the twoperiod and for other period quantum walks has been a hard task. Even if one succeeds in meticulously obtaining a limit theorem, it will give us an asymptotic behavior and fails to lay out the way evolutions modulate during each sequence of periodic operations.
In this paper we revisit the dynamics of the twoperiod discretetime quantum walk using nonorthogonal unitary quantum coin operations and . For the twoperiod quantum walk with the parameters and in the coin operations we show that the standard deviation () is the same as the minimum of the standard deviation obtained from the oneperiod quantum walk with coin operations or , . Our numerical result is analytically corroborated using the dispersion relation obtained from the continuum limit of the dynamics. Though the standard deviations are identical, the spread in position space after steps is bounded by the . And the interference pattern is also clearly distinct. This shows up with the prominent presence of both the parameters and in the differential form of the dynamics expression. We also show that the bounds we obtained for the twoperiod quantum walk will hold good for the splitstep quantum walk which is defined using two coin operators using and . Our dispersion relationship approach can be extended to study bounds on the dynamics of three and higher period quantum walks. Unlike the previous known connection of discretetime quantum walks with the massless Dirac equation where coin parameter , here we show the recovery of the gapless (massless) and gapped (massive) Dirac equation with nonzero parameters contributing to the intriguing interference in the dynamics in a totally nonrelativistic situation. We also study the effect of periodic sequence on the entanglement between coin and position space.
In Sec. II we will give a basic introduction to the operators that define the evolution of the discretetime quantum walk. Using that as a basis we will define the periodic quantum walk and present the numerical results for the twoperiod quantum walk. In Sec. II.1, we obtain the dispersion relation for the one and twoperiod quantum walk and use it to arrive at the bounds on the dynamics of two and three and higher period quantum walks. In Sec. III, we present the emergence of the Dirac equation from the twoperiod quantum walk and present the enhancement of entanglement for periodic quantum walks in Sec. IV. We conclude with our remarks in Sec. V.
Ii Periodic Quantum Walk
Dynamics of the onedimensional discretetime quantum walk on a particle with two internal degrees of freedom is defined on an Hilbert space where the coin Hilbert space and position Hilbert space , representing the number of position states available to the walker. The generic initial state of the particle, , can be written using a two parameters , in the form,
(1) 
Each step of the walk evolution is defined by the action of the unitary quantum coin operation followed by the position shift operator. The single parameter quantum coin operator which is a nonorthogonal unitary and acts only on the particle space can be written in the form,
(2) 
The position shift operator that translates the particle to the left and/or right conditioned on the internal state of the particle is of the form,
(3) 
The state of the particle in extended position space after steps of the homogeneous (oneperiod) quantum walk is given by applying the operator on the initial state of the particle and the position,
(4) 
Probability of finding particle at position and time will be
(5) 
Using we can compute the standard deviation () of the probability distribution after steps of the walk.
Twoperiod quantum walk : To describe the periodic quantum walk we will use two quantum coin operations and . The evolution operator for the step of the twoperiod quantum walk will be of the form,
(6) 
For the period quantum walk the evolution is described using operation for every multiple of steps and for all other steps. We should note that the twoperiod quantum walk we have defined is a timedependent periodic evolution but for the localized initial state and evolution operators we have defined it is equivalent to the positiondependent twoperiod quantum walk. This equivalence should be attributed to the probability distribution which will be zero at the odd (even) position when is even (odd). But this equivalence will not hold good to any period quantum walk in general.
From earlier results we know that the spread of the oneperiod quantum walk probability distribution using evolution operation is bounded between and () and NV00 (); CSL08 (). For a twoperiod walk it looks natural to expect the spread to be bounded somewhere between positions and . But in reality the spread is bounded between .
In Fig. 1, the probability distribution and standard deviation () after 200 steps of the quantum walk using different values of and , separately (oneperiod) and together in twoperiod sequence is presented. We can see that the spread of the probability distribution of the twoperiod quantum walk is always bounded within the spread of the probability distribution and . But the interference pattern is not identical. In Fig. 2, after 100 steps as a function of when is fixed is presented. In Fig. 3, as a function of and after 25 steps of the quantum walk is shown. Analyzing the dependence of on the two coin parameters we can note that the .
In Ref. MK09 (), for a combination of orthogonal matrices in the twoperiod quantum walk, the limit distribution () was computed for a specific combination of parameters and shown to be identical to the limit distribution of the quantum walk using the single coin operation, . However, from the probability distribution shown in Fig. 1, the interference pattern within the bound is different and the limit distribution function fails to capture that. To get more insight into the dynamics of the twoperiod quantum walk and explore the physical significance we will study the dynamic expression at time and obtain the dispersion relation for it in the continuum limit.
ii.1 Dispersion relation and bounds on spread of wave packet
Oneperiod quantum walk. The state of the particle after number of steps of a oneperiod discretetime quantum walk can be written as,
(7) 
where the left and right propagating components of the particle is given by,
(8a)  
(8b) 
This can be written in the matrix form,
(9) 
By adding and subtracting the lefthand side of Eq. (9) by and the righthand side by we get a difference operator which can be converted to a differential operator which will result in the differential equation of the form,
(10) 
By reorganizing the preceding expression we get a simultaneous equation of the form,
(11a)  
(11b) 
For the above expression governing the dynamics of each step of the oneperiod quantum walk in the continuum limit, we can seek a Fouriermode wave like solution of the form
(12) 
where is the wave frequency and is the wave number. Upon substitution into the real part of Eq. (11) we get,
(13) 
and the group velocity will be
(14) 
From this we can say that the wavepacket spreads at a rate of during each step of the quantum walk and after steps the spread will be between . Though we have used only one form of the quantum coin operation with complex elements in it, the group velocity will be even when a most generic unitary operator is used as a quantum coin operation Cha12 ().
Twoperiod quantum walk : For the twoperiod quantum walk the evolution is driven by two quantum coin operations and . First, we will write the state at position and time , as a component of at time ,
(15a)  
(15b) 
In the preceding expression, dependency of the state on the coin parameter can be obtained by writing the state as component of at time ,
(16a)  
(16b)  
(16c)  
(16d) 
Now, substituting Eq. (16) into Eq. (15) we obtain,
(17a)  
(17b) 
Without loosing any generic feature in the preceding evolution expression we can replace with . After that we can effectively reduce the twostep evolution expression using coins with parameters and to a combined singlestep evolution expression by replacing in the righthand side by and in the lefthand side by . This will result in
(18a)  
(18b) 
In the matrix form this can be written as,
(19) 
By adding and subtracting the lefthand side of Eq. (19) by and the righthand side by we get a difference operator which can be converted to a differential operator which will result in the differential equation of the form,
(20) 
The preceding matrix representation can be reorganized and written as a simultaneous equations,
(21a)  
(21b) 
For the above expression effectively governing the dynamics of the twoperiod quantum walk in the continuum limit, we can seek a Fouriermode wave like solution of the form . Upon substitution into the real part of Eq. (21) we get,
(22) 
and the group velocity will be
(23) 
In Fig. 4 we have plotted group velocity for the twoperiod quantum walk, . This gives an effective displacement of the wave packet for each step of the twoperiod quantum walk when the twostep evolution using and is combined to one effective step evolution. Comparing Figs. 4 and 3, and as a function of and we can see an identical pattern and only when the dominance of one over the other happens, the transition is smooth for . This is due to the continuum approximation we made in the analytics.
From the expression for group velocity, Eq. (23), we can infer that
(24) 
Therefore, the bound on the group velocity sets the bound on the standard deviation, . This bound on the group velocity and standard deviation corroborates with the bounds we obtained from the numerical analysis.
Three and period quantum walk : First three step of the threeperiod quantum walk using two quantum coin operations and is implemented with the evolution operator in sequence,
(25) 
In Fig. 5, the probability distribution after 200 steps of the threeperiod quantum walk is presented and the spread of the probability after steps is always lower than but not bounded by the minimum of the two as it was for twoperiod quantum walk. The standard deviation plot in Figs. 5(b), 5(d), and 5(f) shows that will be around and does not match explicitly. In Fig. 6, the standard deviation as a function of and after 45 steps of the threeperiod quantum walk is shown. Except for the evolution parameter where and close to unity, the standard deviation is very low. This can be attributed to multiple peaks in the distribution where peaks with higher probability are closer to origin.
Unlike the twoperiod case where only two peaks were seen in the probability distribution, multiple peaks can emerge in the three and period quantum walks (see Fig. 8). This can result in a mismatch between the linear scaling of group velocity with the standard deviation. However, group velocity can give us a definite bound on the maximum spread of the probability distribution in position space for three and period quantum walks.
The evolution operator for the first three steps of the threeperiod walk can be rewritten as,
(26) 
where represent the twoperiod operator sequence for which we already know the dispersion relation and [Eq. (23)] when it is treated as an effective one step evolution. Extrapolating and from oneperiod and twoperiod quantum walks we can write the group velocity for the threeperiod walk in the form,
(27) 
For any given values of ’s, we can get multiple valid value for . This can be interpreted as the wave packet simultaneously evolving with different resulting in multiple peaks in the probability distribution. Among the possible values for the contribution for a maximum spread in position space will be from,
(28) 
From the preceding expression we can conclude that the bound on the spread of the wave packet in position space after the step of the threeperiod walk will be,
(29) 
In Fig. 7, bounds on the spread of the probability distribution in position space after 100 steps of the threeperiod quantum walk as a function of and is shown. This bound on the spread is obtained from the maximum of group velocity . By substituting finite values for and into Eq. (29) we can confirm that the bounds we get from a maximum of group velocity matches with the maximum range of spread of probability distribution obtained from numerical evolution [Fig. 5(a), Fig. 5(c), Fig. 5(e) and Fig. 8].
For the period quantum walk, the spread of the probability distribution will be bounded by,
(30) 
In Fig. 8, the probability distribution and standard deviation after 200 steps of period quantum walks is shown. We can verify that the spread in probability distribution is bounded by a maximum of group velocity for all period quantum walks.
Iii Twoperiod quantum walk, splitstep quantum and Dirac equation
The splitstep quantum walk was first introduced to define the topological quantum walk KRB10 () and was shown to simulate Dirac cellular automata MC16 (). Recently, the decomposed form of the splitstep quantum walk was shown to be equivalent to the twoperiod quantum walk and simulate Majorana modes and edge states ZGS17 (). In this section, staring from the splitstep quantum walk we arrive at the differential equation form of the evolution equation which is equivalent to the twoperiod quantum walk evolution equation. From this we can establish that all bounds applicable to the twoperiod quantum walk will hold good for the splitstep quantum walk and equivalent form of Dirac equations.
Each step of the splitstep quantum walk is a composition of two half step evolutions,
(31) 
where and are the quantum coin operation and we will define it in the same form as Eq. (2). The position shift operators are defined as,
(32a)  
(32b) 
The state at any position and time after the operation of at time will be , where
(33a)  
(33b) 
The preceding expression is identical to Eq. (18) which we have obtained for the twoperiod quantum walk. Therefore, the differential equation form of the evolution will be the same as Eq. (20). By controlling the parameters and we can arrive at the Dirac equations for massless and massive particles.

Multiplying Eq. (20) by and setting and to a small value (mass of subatomic particles) we recover Dirac equation in the form,
(34) 
By choosing and such that in Eq. (20), and multiplying by we get an expression identical to Dirac equation of massless particle,
(35) Here the coefficient of the position derivative is a more general Hermitian matrix which depicts the oscillation of the spin (eigen state) during the dynamics.

By choosing to be extremely small and corresponding such that in Eq. (20), and multiplying by we get the Dirac equation in the form,
(36)
In Ref. MC16 (), it was shown that and small value of is required to recover Dirac cellular automata from splitstep quantum walk and both to recover massless Dirac equation. Here, we have shown the other possible configurations of nonzero values where we can recover the massless Dirac equation. From bounds on the twoperiod quantum walk (equivalently the splitstep walk) we can imply that the spread of the wave packet for the massive and massless, that is, the gapped and gapless Dirac equation of the form, Eq. (34) and Eq. (36), respectively is bounded by the parameter . The spread will be very wide for the former and small for the latter (remaining around the origin). For the massless Dirac equation with general the Hermitian matrix, Eq. (35), the spread will be bounded by .
Iv Entanglement in periodic quantum walks
Entanglement of the particle with position during quantum walk evolution has been reported in many earlier studies. Entanglement during the temporal disordered (spatial disorder) quantum walk is reported to be higher (lower) than the homogenous (oneperiod) quantum walk Cha12 (). In the homogenous quantum walk, mean value of entanglement generated is independent of the initial state of the particle. But in the splitstep quantum walk, the dependence of mean value of entanglement is prominently visible MC16 (). Therefore, for the twoperiod quantum walk, entanglement behavior will be identical to the one reported in Ref. MC16 (). In this section we will see how the entanglement manifests and reaches maximum value for the period quantum walk.
As we have considered only a pure quantum state evolution in this study, we will use the partial entropy as a measure of entanglement, which is enough to give correct measure of entanglement for the pure state evolution with unitary operators. We will first take the partial trace with respect to space (position space) of the time evolved state = Then according to our measure the entanglement at time is given by,
(37) 
where the suffix c represents the coin space.
In Fig. 9, we present the entanglement between the particle and position space as a function of time for one, two, and period quantum walks. For the twoperiod quantum walk, in contrast to standard deviation, the mean value of entanglement is bounded around the maximum of the two oneperiod quantum walks. For the threeperiod quantum walk, entanglement reaches a maximum value, higher than the entanglement due to both, oneperiod quantum walks is seen. This is also in contrast to the way the spread in position space and standard deviation decreases for periodic quantum walks. For the higher period quantum walk we can see that the change of coin induces the increase in entanglement.
V Conclusion
In this work we have presented the dynamics of the timedependent periodic quantum walk. In particular, we have shown the way the probability distribution spreads, standard deviation increases and entanglement varies for the periodic quantum walk; and we have shown the way they are bounded when compared with the dynamics properties of the homogeneous (single coin driven) quantum walk. For the twoperiod quantum walk with the parameters and in the coin operations we show that . Our numerical results were corroborated with analytical analysis from the dispersion relation of the twoperiod quantum walk. Revisiting the splitstep quantum walk dynamics we have also shown that all the bounds we have presented for the twoperiod quantum walk will be identical to the splitstep quantum walk. Unlike the computing limit density function which is meticulously hard, we have used the dispersion relation from oneperiod and twoperiod quantum walk to understand the bounds on the spread of the wave packet for the period quantum walk, . By revisiting the connection of quantum walks with the Dirac equation, we have shown the configuration of periodic quantum walk evolution which can recover the Dirac equation for both, massive and massless particles with the nonzero coin parameter . Thus, the evolution configuration that results is the emergence of the gapless and gapped Dirac equations. This can contribute to quantum simulation of dynamics in Dirac materials. We also showed that the periodic sequence will enhance the entanglement between the coin and position space in the quantum walk dynamics.
Depending on the convenience of the experimental system, either the, twoperiod or splitstep quantum walk can be used for quantum simulations of various lowenergy and higher energy particle dynamics defined by Dirac equations. The bounds we have presented will further help to understand the transition from the diffusive to the localized state.
Acknowledgment:
CMC and NPK would like to thank Department of Science and Technology, Government of India for the Ramanujan Fellowship grant No.:SB/S2/RJN192/2014. CMC and RL would also acknowledge the support from US Army Research Laboratory.
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