Continuum Limits of the 1D Discrete Time Quantum Walk

The discrete time quantum walk (DTQW) is a universal quantum computational model. Significant relationships between discrete and corresponding continuous quantum systems have been studied since the work of Pauli and Feynman. This work continues the study of relationships between discrete quantum models and their ostensive continuum counterparts by developing a formal transition between discrete and continuous quantum systems through a formal framework for continuum limits of the DTQW. Under this framework, we prove two constructive theorems concerning which internal discrete transitions ("coins") admit nontrivial continuum limits. We additionally prove that the continuous space limit of the continuous time limit of the DTQW can only yield massless states which obey the Dirac equation. Finally, we demonstrate that the continuous time limit of the DTQW can be identified with the canonical continuous time quantum walk (CTQW) when the coin is allowed to transition through the continuous limit process.


I. INTRODUCTION
The discrete time quantum walk (DTQW) has been the subject of much attention since its applications to quantum computing were discovered in the analysis of Hadamard Walks [2]. The DTQW has since been used in a variety of quantum computing algorithms, including the Oracular Search [3] and Element distinctness [4] algorithms (for a full list, see Ref. 5).
As noted in Ref. 6, a now well-studied limit of the DTQW was introduced by Feynman and Hibbs in Ref. 7 in constructing a path integral formulation for the propagator of the Dirac equation. According to Feynman, a particle zig-zags at the speed of light across a space-time lattice, flipping its chirality from left to right with an infinitesimal probability at each time step [6]. The Dirac equation results when the continuous space-time limit is taken, with the mass of the particle determined by the flipping rate. More recent works have produced notions of discrete space-times (see Refs. 8 and 9) and consequent questions regarding how they produce our apparent continuum.
Recently, Mlodinow and Brun in Ref. 10 demonstrated how to constrain a 3D DTQW to obtain a resulting fully Lorenz invariant continuum limit. They showed that their symmetry requirement necessitates the inclusion of antimatter, and, in Ref. 11, discuss experimental methods to distinguish between the DTQW and its continuum limiting Dirac equation as a description of fermion dynamics. These limits were also central to Refs. 12 and 13. Their continuum limits for DTQWs transformed discrete time evolution equations to partial differential equations (PDEs), as the PDEs analyzed were much simpler than the discrete recursion relations of the DTQW.
In Ref. 6, Strauch also used the continuum limit to connect the DTQW and CTQW, and Refs. 14 and 15 demonstrate that the free particle Dirac evolution could be obtained by taking continuum limits of the DTQW. Strauch also demonstrated, in Ref. 14, the DTQW's connections with zitterbewegung, which is an interference effect among free relativistic Dirac particles between their positive and negative energy parts that produces a quivering motion [16]. Strauch shows that zitterbewegung in the DTQW can be tuned based on the value of its coin rotation parameter, and shows that the CTQW contains zitterbewegung-like oscillations (which Strauch denotes as anomalous zitterbewegung) even though there is only one energy for the CTQW [14]. The purpose of this work is to formulate a general framework within which continuum limits of the DTQW can be taken, and to analyze the corresponding dynamics in the various limits. From our analysis, we have concluded that it is only possible to keep space discrete while continuizing time for particular coins (section IV); that taking time and space limits simultaneously with a fixed coin is possible when steps in the walk are allowed and yields a massless dirac equation (section V); that the ensuing time evolution derived from taking a continuous space limit of the continuous time limit of the DTQW is a massless dirac equation as well (section VI); and that the solutions of the continuous time limit of the DTQW can always be related to the solutions of the CTQW for any choice of coin allowed to undergo the continuous time limit of the DTQW (section VII).

A. DTQW Definition
The one dimensional DTQW assumes a time dependent probability amplitude where the operations S and C (defined below) represent external and internal unitary operations, respectively, S being an external translation operation and C being an internal rebalancing of the two spin amplitudes ψ R and ψ L .
For example, if the coin operation C is implemented by the Hadamard matrix, then: With ∆t and ∆x the time and space intervals for the quantum walk, the full change SC acting in one time iteration ∆t is then: The unitary time evolution is then: letting m = t ∆t . We also express this in discrete differential form for the purpose of forming subsequent continuum limits: We will also often represent the walk in Fourier space and define our discrete Fourier transform convention here. Let #» Ψ(k,t) be the Fourier transform of #» Ψ(x,t) and x = n∆x for n ∈ Z. We use the following conventions for the forward and inverse Fourier transforms, for Fourier variable A standard procedure here will be to represent operators in Fourier space as follows: given an oper- with f (x) ∈ Y , so thatÕ is the Fourier representation of O. The operator we will be most commonly representing in Fourier space is the shift operator S, defined by S: where σ z is a Pauli matrix.

II. DEFINING CONTINUUM LIMITS
Skipping Steps. Before formulating a universal definition of continuum limits for the quantum walk, we want to establish the important notion of so-called alternating limits, in which only steps of a certain parity (e.g. even or odd) are considered observed. We first provide an informal example demonstrating that trivial divergences occur in the ∆t → 0 limit arising from multiple parity-dependent limits in the discrete walk. Such limits were considered in Ref. 6.
Consider the DTQW with coin C = ie iθ σ x , with σ x a standard Pauli matrix and θ ≡ θ (∆t) a real number (modulo 2π) depending on the time discretization parameter ∆t. For the example we construct an informal continuous time limit, to be formalized in Definition II.1. Essentially we will take the ∆t → 0 limit in Equation (5). The continuous time limit then amounts to identifying the limiting operator assuming a fixed space of functions X on which it acts; here I is the identity. This is defined more carefully later in this section within Formal Definitions.
For this analysis of a continuous time limit for the discrete space and time quantum walk, we will seek the most general scaling of walk parameters that admit nontrivial limits as ∆t → 0. In this case we will admit all scalings for the coin parameter of the form θ = π/2 + γ∆t, with γ > 0, which were introduced by Strauch in Ref. 6. Thus from the operator standpoint we seek a limit of the form lim ∆t→0 ie ik∆xσ z iσ x e iγ∆tσ x −I ∆t , which in fact does not exist generically. We show here however, that if we consider only even parity steps (i.e. even numbers of steps, effectively considering only every other step), then non-trivial limits exist. Thus we will be considering only iterations of the even parity operator SCSC rather than the fundamental step SC, and we will identify a limit lim ∆t→0

S(∆x)C(∆t) S(∆x)C(∆t)−I ∆t
, which structurally is: As might be expected, it will be clear below that replacing the above even power (SC) n with n = 2 by n = 3, the above limiting process will no longer exist; existence of the limit will hold only for even powers n. In general, restricting to fixed even step sizes n will lead to continuous limiting processes as above (with scaling of the coin based on ∆t), while non-even step sizes will never admit such limits (see Theorem IV.1, proved in Appendix A).
Formal Definitions. Definitions of our operator limits require common spaces for their domains. We will redefine all operators on such a common space, given as where Σ is the space spanned by |L = x ∈ R for fixed t).
We will consider general quantum walks that have ∆t → 0 limits when step numbers n = km are restricted to whole multiples of an integer n, i.e. generalizing the above parity restriction for step numbers (n = 2) to accommodate more general step number restrictions. Thus let #» Ψ(x,t) ∈ X, n be the number of skipped steps, ∂ t be the time derivative operator, and define the discrete derivative We denote the level of discretization of our space and time operations by η = (∆x, ∆t). We will consider a discrete space and time quantum walk governed by on X, with H η the above family of operators parametrized by η = (∆x, ∆t). The continuous time limit of the walk in Equation (10) exists if the right hand side of the equation has a limit (for #» Ψ ∈ X) as η → (0, 0) along a given prescribed path, for which both the DTQW functions and continuum limit of the DTQW functions are in X. Continuum space limits in the absence of any change in ∆t will not be considered here because S → I as ∆x → 0, so the walk reduces simply to a coin acting on the spin portion of the wave function at each time step. With no traversal of the lattice there results a trivial walk. Formally, we state the definition of continuous time limit and continuous space-time limit as: Definition II.1. Let the operators H η ≡ H ∆x,∆t and H ∆t act on functions #» Ψ(x,t) ∈ X. Then we have the following definitions: • The continuous time limit of the DTQW governed by coin C skipping n steps is the time where H∆t is defined (when the limit exists) by , with the limit taken in the space X.
• The continuous space-time limit of the DTQW governed by coin C skipping n steps is We call the operators H ∆t and H ∆x,∆t the generators of time evolution, or Hamiltonians, in their respective continuum limits. Note that the second limit above may depend on the ratio ν = ∆x ∆t , and can also be generalized to allow any manner of approach of η → (0, 0).
Our goal is to explore the most general possibilities for these two cases. We remark that our inclusion of n expands the number of continuum limits that exist; in particular this possibility was not considered in Ref. 17 Additionally, we need the following definition to allow parameterized coin variations: Definition II.2. Consider a continuous space-time limit where ∆x and ∆t have the same scaling, so ∆x = v∆t = vε for some non-zero v ∈ R. A coin varies in this continuum limit if the coin depends on ε = ∆t.

III. GENERAL CONDITIONS FOR CONTINUUM LIMITS
The discussion here is based on terminology and results in Ref. 17. We will study an important aspect of coins that change in the process of continuous time and space-time limits; this will help to interpret the theorems in Sections IV and V. We will follow the DTQW wave function through n time steps of length ∆t. Note here that all limits in this section will be in the topology of the space X. We begin with the basic equation with S = S(∆x) and C = C(∆t) both dependent on the increment η = (∆x, ∆t). If a continuous space-time limit is taken with (∆t, ∆x) → (0, 0), then because S → I when ∆x → 0, we must have lim ∆t,∆x→0 (C(∆t)) n = I, as otherwise the limit could not exist. In particular, unless C(∆t) is constantly the identity, it must vary (as in Definition II.2) in the continuous time limit.
If only a continuous time limit is taken (i.e. ∆t → 0), then (by continuity of the left side in t) for the left and right sides of Equation (11) to be equal, we must have lim ∆t→0 (S(∆x)C(∆t)) n = I where we include ∆t dependence in C for generality. Note that the constraint in the continuous time limit involves both the coin and the shift operator, not just the coin as in the continuous space-time limit. Now for the following definition:

. Consider a matrix A(t) which depends on some continuous parameter t. A(t)
homotopically approaches a root of unity if A(t) depends continuously on t and there exists some non-zero integer m and some real number t ′ such that lim t→t ′ A(t) m = I. By the previous definition and the above analysis of Equation (11), we have the following theorem: Theorem III.1. A coin for which a continuous space and time limit exists must homotopically approach a root of unity. The product of the shift and coin operator for which a continuous time limit exists must homotopically approach a root of unity as well.
Proof. Recall from definition II.1 that we define the space-time limit H ∆x,∆t with ∆x = v∆t = vε as: for Now we see that for the left hand side to equal the right hand side, C must be of the form C n = . Thus, by definition III, C must homotopically approach a root of unity in the continuous space-time limit. The proof for the continuous time limit is similar, except now S does not converge to identity, so instead SC must be of the form thereby satisfying definition once again.
From this analysis, we have obtained a general property of coins which undergo continuum limit transformations, and we will refer to this property in the future.

IV. GENERAL CONTINUOUS TIME LIMIT
In this section, we will find the set of DTQWs for which a continuous time limit exists, as according to definition II.1. We will then analyze the properties of the resulting time evolution in the continuous time limit.
We consider a general unitary coin, which can be written the following way: We wish to know for which 2 × 2 matrices, as parameterized by Equation (14), does the continuum limit exist, according to definition II.1. Before introducing a theorem answering such a problem, a few notes. First, a constraint on δ is necessary to satisfy the finiteness condition of the existence of the limit in definition II.1. The value of δ is arbitrary as it amounts to an overall energy shift in the Hamiltonian, which does not matter. This point is explained further in the proof of lemma A.4.
Second, assuming that the elements of C cannot depend on the elements of S in any way, observe that the limit in definition II.1 cannot not be finite unless C depended on ∆t in some way. Thus, we must assume that the coin varies in the continuum limit, where variation is defined in II.2. A proof of the following theorem is presented in Appendix A.
Theorem IV.1. Let C(δ , ψ, θ , φ ) be a 2×2 unitary matrix as defined in Equation (14) such that the set of angles ψ, θ , φ parameterizing C depends on ∆t the following way: When using the conditions for the coin from Theorem IV.1, we see that the class of coins which can undergo continuous time limits, as determined in the theorem, are the following (parameterized by ψ 0 , θ 1 , and φ 0 ): An interesting observation is that the final hamiltonian after the continuous time limit is taken does not depend on any parameters that are coefficients of terms O(∆t) except θ 1 (so they really do not need to be included in Equation (16). θ 1 can be interpreted as a driving factor for the final hamiltonian's time evolution. Its value completely determines how much mixing between the left and right states there will be due to the time evolution operator SC, as when θ 1 = 0 all operators in the coin commute with the shift operator and no mixing occurs, which corresponds to the wave function coming back to itself every other step in the DTQW. Another interesting point is the reason for the skipping of steps (i.e. n must be even). An explicit proof of why this must be can be seen in Appendix A, but for a more intuitive explanation, consider the following. For the continuous time limit of the DTQW to exist, the following Fourier space hamiltonian must be finite:H The operator e ik∆xσ z C does not homotope to identity as ∆t → 0 for any C, so no continuous time limit can exist for n = 1. However, for the coin in Equation (16), the operator e ik∆xσ z Ce ik∆xσ z C does homotope to the identity because of the following property of the coins in Equation (16): The repercussions of not setting δ = pπ for odd integer p leads to a final H having constant infinite energy contributions, which are then ignored, as energy differences are the physical quantities.
Another interesting property of the coins derived in IV.1 is that the coins themselves homotopically approach a root of unity as well, so lim ∆t→0 C n = I, as one can check. This was not implied by our analysis in section III for the continuous time limit case, and a full derivation of all the possible coins which can undergo continuous time limits was needed to obtain this property. Also, the resulting hamiltonian in Theorem IV.1 will be used in section VI to determine how the continuous time limit followed by the continuous space limit compares to the simultaneous continuous Then the following is the analytical form of the time evolution for #» Ψ(x,t) for all t in terms of its initial state, where x = m∆x for m ∈ Z and J m (t) is the m th order Bessel function of the first kind, α = φ 0 +ψ 0 2 , and β = φ 0 −ψ 0 2 : This solution does reduce to that found in Ref. [6] when the corresponding parameters in equation 18 are used, except for some minor sign differences stemming from their shift operator being defined as the inverse of our shift operator. We see that the locations for which The ensuing Hamiltonian for such a walk will be the following massless Dirac hamiltonian: We see that the massless Dirac hamiltonian is the limiting hamiltonian of this continuum limit.
In the continuous space-time limit the mass term is generated by the coin's variation with time step in the continuum limit, as can be seen in Appendix F. Because our coin does not vary in the continuum limit, the ensuing continuous space-time hamiltonian will have no mass.
Another interesting note is that the above theorem states that a coin with no variation will have a continuum limit if it is itself a root of unity. This makes sense in light of Theorem III.1, as the continuous parameter in our coin is no longer there, so the coin itself must be a root of unity. Also, this theorem might seem at odds with the discussion at the start of Ref. 17 (which we repeat in section III) but they did not consider skipping steps in the walk when taking the continuum limit, which is how we were able to get a limit for this walk even when our coin did not vary at all in the continuum limit. The reason why we do not allow for φ 0 and ψ 0 to not depend on ∆x is given by the following conjecture: Conjecture 1. There is no dependence φ 0 and/or ψ 0 can have on ∆x that would allow for spatial derivative(s) in the continuum limit The reason why we are searching for time evolution equations with spatial derivatives is because without them, no spatial translation will occur for our wave function in the continuous space limit, thus resulting in a trivial stationary walk. An interesting note of Theorem VI.1 is that a different time evolution equation occurs when a simultaneous continuous space-time limit is taken.
As can be seen in Appendix F, when a simultaneous space-time continuum limit is taken, a massive Dirac equation results.

VII. GENERAL DTQW RELATIONSHIP TO CTQW
In the following section, we will be building on a result of Strauch's from Ref. 6, in which a connection was found between the DTQW and CTQW by taking a continuous time limit of the DTQW to relate it to the CTQW. Strauch used a specific coin e −iθ σ x , and let θ = π 2 − γ∆t when the continuous time limit was taken. Now that a general parameterization of all the possible coins which can undergo a continuous time quantum walk has been obtained from Theorem IV.1, we have the opportunity to investigate if, for a general coin, whether or not a relationship between the CTQW and DTQW exists. We begin by reviewing Strauch's specific results in Ref. 6.

A. Review of Strauch
In this section, we will be reviewing the connection between the DTQW and CTQW made by Strauch in Ref. 6. To start, consider a DTQW dictated by the shift operator (in Fourier space) S = e ik∆ x σ z and coin operator C = e −iθ σ x such that the time evolution of a Fourier space wave function #» Ψ(k,t) is given by #» Ψ(k,t + ∆t) = SC #» Ψ(k,t). When a continous time limit (∆t → 0) is taken on #» Ψ(x,t), letting θ = π 2 − γ∆t and skipping every other step, the following time evolution is recovered for #» Ψ(x,t): Now for the profound relation discovered by Strauch. If we define two wave functions (which is the CTQW time evolution equation). In other words, Strauch found that the continuous time limit of his DTQW can be written as a superposition of two copies of the CTQW. This relation helped clarify the then longstanding mystery about the exact relationship between the two ways of quantizing the quantum walk, the DTQW and CTQW. Next we show that this relationship holds for a general coin, and we will use the relation to see how the DTQW coin parameters effect the final probability distribution in the discussion following the theorem VII.1.

B. General Coin CTQW-DTQW relation
We summarize our findings in the following theorem, the proof of which is in Appendix E: Theorem VII.1. Let #» Ψ(x,t) be the following 2 component wave function resulting from the continuous time limit of the DTQW with a general coin as found in Theorem IV.1: where θ 1 , φ 0 , and ψ 0 are real numbers which cannot depend on x or t. Also let #» Ψ ± (x,t) be wave functions which satisfy the following CTQW time evolution equations: Then #» Ψ(x,t) can be written as a superposition of #» Ψ + (x,t) and #» Ψ − (x,t) in the following way Now that we have a general relationship between the continuous time limit of the DTQW and the CTQW, we can analyze exactly how the coin parameters α and β effect the probability distribution of the continuous time limit. First of all, #» Ψ ± (x,t) are fixed momentum traveling wave states with time evolution which does not depend on α or β because #» Ψ ± (x,t) satisfy the CTQW (which does not depend on α or β ), so the time evolution of just these wave functions would be to just spread their initial distribution across the sites. Now we write equation 22 more suggestively: has the effect of boosting #» Ψ + (x,t) to a frame traveling right (if α > 0) at speed | αθ 1 2 | or left (if α < 0), and e i(α x ∆x + t) to a frame moving at speed | αθ 1 2 | in the opposite direction as #» Ψ + (x,t). The last effect these parameters will have will be on the initial condition 0). We see the only effect β has is on the initial conditions, while α effects both the initial condition and the frames #» Ψ + (x,t) and #» Ψ − (x,t) are boosted to.

A. Conclusions
Given our definitions of continuum limit from section II, we have concluded by theorem IV.1 that keeping space discrete while continuizing time is only possible for particular coins, which must be of the form C = e −i(ψ 0 )σ z /2 σ y e −iθ 1 ∆tσ y /2 e −i(φ 0 )σ z /2 , granted the limit is taken 2 steps at a time. We have also concluded, from theorem VII.1 in section VII B, that the continuous time limit of the DTQW can be always be related to the CTQW if the coin is of the form exp − iθ 2 (σ y cos ψ 0 − σ x sin ψ 0 ), where θ → pπ + θ 1 ∆t in the continuum limit for some odd integer p and θ 1 ∈ R.
The implications of these two theorems are that there exists unitary matrices used as coins in the DTQW that do not have continuous time limit, and thus cannot be related to the CTQW.
We also found from theorem V.1 that certain coins do not need to vary in the continuum limit, as long as they are roots of unity. Finally, we concluded from theorem VI.1 that various types of Dirac equations can be obtained depending on how the continuum limit of the DTQW is taken. Space and time limits taken simultaneously yield different answers than when time is taken followed by space.

B. Open Questions
There are many open questions pertaining to these ideas. Which types of coins have continuum  Before we begin, let's reiterate the theorem we wish to prove: Theorem IV.1. Let C(δ , ψ, θ , φ ) be a 2×2 unitary matrix as defined in Equation (14) such that the set of angles ψ, θ , φ parameterizing C depends on ∆t the following way: φ = φ 0 + φ 1 ∆t + O(∆t 2 ), ψ = ψ 0 +ψ 1 ∆t +O(∆t 2 ), and θ = θ 0 +θ 1 ∆t +O(∆t 2 ), where φ 0 , ψ 0 , θ 0 , φ 1 , ψ 1 , θ 1 ∈ R do not have any space or time dependence. The continuous time limit will exist, as defined in II.1, for such a class of coins if and only if, θ 0 = pπ, δ = − pπ 2 (for odd integer p), and n is even. The Hamiltonian obtained in such a limit is the following, where S is the shift operator defined in equation 8: We begin by stating the continuous time limit of the DTQW for coin C and skipping n steps, from definition II.1: Now we construct lemmas to prove Theorem IV.1. Our first lemma will be an algebraic expansion of ( SC) n that will help make manifest later lemmas, where S is the Fourier transform of S, which is defined by S = e ik∆xσ z .

and S be the Fourier transform of S. Then the following is true up to O(∆t):
Proof. After substituting ψ, φ , and θ in terms of φ 0 , ψ 0 , θ 0 , φ 1 , ψ 1 , and θ 1 , the rotation matrices in Equation (14) become R z (ψ) = R z (ψ 0 )(1 − iψ 1 ∆t 2 σ z + (∆t 2 )) and so on for R z (φ ) and R y (θ ). After doing this substitution and going to Fourier space (so S → S = R z (−2k∆x)), we get the following for ( SC) n : We now make the substitution ψ ′ 0 = ψ 0 − 2k and expand Equation (A3) further: Now we make a statement concerning the O(∆t 2 ) terms: Lemma A.2. is the identity operator itself, but e iδ A cannot even equal identity, as A has k dependence from containing S, and the angles are not permitted to depend on k, so there is no possible way to cancel out the k dependence. Thus, there is no continuous time limit defined in Equation (A1) for n = 1.
Next we use the reasoning from lemma A.3 to further constrain the values θ 0 and δ can take.
Lemma A.4. For the limit defined in Equation (A1) to be finite, θ 0 and δ must be constrained such that θ 0 = pπ and δ = 2πl n − pπ for odd integer p and any integer l.
Proof. Following up on the constraint that (e iδ A) n = I from lemma A.3, let U be the diagonalization matrix of A, and let D be the matrix of eigenvalues of A. Then we have the following: so if we set the eigenvalues of e iδ A equal to an n th root of unity e 2πl/n where l = 0, 1, 2, ... (which is equivalent to the constraint (e iδ A) n = I), we recover the following constraint for θ 0 : Because none of the angles have k dependence, the only way this condition can hold true is if cos θ 0 /2 = 0 or θ 0 = pπ where p = 1, 3, 5, .... This also gives a constraint on δ , being δ = 2πl n − pπ 2 . As a remark, the reason why the choice of overall phase is important here is that it shifts the zero point energy of the Hamiltonian in question and will make the dependence on other variables more manifest in the continuum limit (physical quantities are the differences in energies/eigenvalues of a Hamiltonian, not the eigenvalues themselves).
The next lemma uses the constraints on θ 0 and δ from lemma A.4 to impose a constraint on n.
Lemma A.5. For the limit defined in Equation (A1) to be finite, n must be even.
Proof. Substituting our constraint for θ 0 from lemma A.4 into A, we get the following: satisfy (e iδ A) n = I. As for odd n, we can write n = 2m + 1 for some integer m to obtain the following: This cannot equate to identity, as we showed in lemma A.3 that for n = 1 no parameterization of A can make e iδ A = I. Thus, n must be even to have a finite continuum limit as defined in Equation Because the constraints on n and θ 0 hold true for all l from the last two lemmas, we will choose l = 0 for the remainder of the proof without loss of generality.
Proof. To find SC, we take the n th root of both sides of the third line of Equation (A5). This will yield the following: Plugging in for the constrained versions of A and B and reducing, we obtain the following: To find C, we simply multiply Equation (A13) by S −1 = R z (2k), obtaining the following: Next we will find ( SC) n by evaluating the sum in the last line of Equation (A5) using the constrained form of A in Equation (A9). One can show that A 2 = −1 and A −1 = −A, so the sum in Equation (A5) becomes the following: Now we split up the sum into even and odd terms: We used Equation (A9) in the last line, so Equation (A5) becomes the following: Now we can find H by evaluating the limit in Equation (A1) using Equation (A17) to obtain the following Hamiltonian in Fourier space: Fourier transforming back and resubstituting for ψ ′ 0 , we obtain the following H: where S is the shift operator defined in Equation (8).
Then the following is the analytical form of the time evolution for #» Ψ(x,t) for all t in terms of its initial state, where x = m∆x for m ∈ Z and J m (t) is the m th order Bessel function of the first kind, α = φ 0 +ψ 0 2 , and β = φ 0 −ψ 0 2 : with corresponding eigenvectors Proof. The hamiltonian written in fourier space is the following: Proof. If U is the diagonalization matrix of eigenvectors of H(k), we can write U † H(k)U = λ σ z .
Therefore, because UU † = I by unitarity, we have the following: Our next lemma recovers the explicit expression for U e −iλ (k)σ z t U † #» Ψ(k, 0): We obtain lemma B.3 through straightforward matrix multiplication. Our next lemmas will introduce some integrals and convolutions that we will need when computing the inverse fourier F −1 (e ± iθ 1 t 2 cos (k−α) Ψ L,R (k, 0)) = F −1 (e ± iθ 1 t 2 cos α cos k e ± iθ 1 t 2 sin α sin k Ψ L,R (k, 0)) (B6) Proof. The first equality of the equation in lemma B.4 is true by elementary trigonometric identities, and the second line is true by the convolution theorem. For the third line, we need the following inverse fourier transforms dke ikn∆x e ± iθ 1 t 2 cos α cos k = (±i) n J n ( θ 1 t 2 cos α) (B9) dke ikn∆x e ± iθ 1 t 2 sin α sin k = (∓1) n J n ( θ 1 t 2 sin α) (B10) Now we find F −1 (e ± iθ 1 t 2 cos α cos k ) * F −1 (e ± iθ 1 t 2 sin α sin k ): Now we will be taking a continuous space-time limit of the DTQW with this coin, and we will see what resultant PDE we obtain.
Lemma C.2. The Hamiltonian for the continuous space-time limit of the DTQW with coin of the form in Equation (19) will be the following: Proof. We have the following continuous space-time Next, we can ignore the O(∆t 2 ) terms, as they will be zero in the end. So we obtain the following: Again, we can ignore the O(∆t 2 ) terms, and we used the fact that C m = 1. We can reduce the series in the last expression of C7 in the following way, setting α = π j m : = (cos α + in · #» σ sin α)σ z (cos α − in · #» σ sin α) = 2 sin α(−n y cos α + n x n z sin α)σ x + (2n y n z sin 2 α + n x sin 2α)σ y + (cos 2α + 2n 2 z sin 2 α)σ z (C8) The only terms to survive the sum will be those proportional to sin 2 α and cos 2 α. Thus, we recover the sum: Proof. We rewrite H C as above, but now we factor out e −iψ 0 σ z : e −iψ 0 σ z (e 2ik∆xσ z + e i(φ 0 +ψ 0 )σ z )σ y (D4) In order for the term e 2ik∆xσ z + e i(φ 0 +ψ 0 )σ z to look like a spatial derivative, e i(φ 0 +ψ 0 )σ z must be proportional to −identity, which equates to φ 0 + ψ 0 = π.
For the sake of completeness, these conditions on the parameters of the coin correspond to the following pre-continuous time limit coin: − exp(iφ 0 σ z /2)σ x exp −i α∆t 2∆x σ y exp(−iφ 0 σ z /2) (D5) (from Equation (16)). Putting these two lemmas together, we get that the only finite continuous space limit H C can have which contains spatial derivatives is the following: This equates to the following time evolution equation in position space for wave function #» ψ (x,t) which is in the form of a dirac hamiltonian for a massless particle in the σ x basis (and reduces to the familiar form when ψ 0 = 0).