Nonreversal and nonrepeating quantum walks
T. J. Proctor, K. E. Barr, B. Hanson, S. Martiel, V. Pavlovic, A. Bullivant, V. M. Kendon
aa r X i v : . [ qu a n t - ph ] J un Nonreversal and nonrepeating quantum walks
T. J. Proctor, ∗ K. E. Barr, ∗ B. Hanson, S. Martiel,
1, 2
V. Pavlovi´c,
A. Bullivant, and V. M. Kendon School of Physics and Astronomy, E. C. Stoner Building, University of Leeds, Leeds LS2 9 JT Universit`e Nice Sophia Antipolis, Laboratoire I3S, UMR 7271,2000 Route des Colles, 06903 Sophia Antipolis, France Faculty of Science and Mathematics, University of Nis, Serbia (Dated: September 21, 2018)We introduce a variation of the discrete time quantum walk, the nonreversal quantum walk,which does not step back onto a position which it has just occupied. This allows us to simulatea dimer and we achieve it by introducing a new type of coin operator. The nonrepeating walk,which never moves in the same direction in consecutive time steps, arises by a permutation of thiscoin operator. We describe the basic properties of both walks and prove that the even-order jointmoments of the nonrepeating walker are independent of the initial condition, being determined byfive parameters derived from the coin instead. Numerical evidence suggests that the same is thecase for the nonreversal walk. This contrasts strongly with previously studied coins, such as theGrover operator, where the initial condition can be used to control the standard deviation of thewalker.
PACS numbers: 03.67.-a
I. INTRODUCTION
Quantum walks have been extensively studied sincetheir introduction [1–3]. Initial interest in cellular au-tomata [3] led to more general algorithmic applications[4–6] precipitating a rapid development of the theory ofcomputation by quantum walks. Quantum walks canprovide quadratically enhanced searching [7, 8] with gen-eralizations to related computational tasks such as ele-ment distinctness [9] and subset finding [10, 11]. Quan-tum walks have been shown to have interesting transportproperties in a variety of scenarios. On the line theyachieve ballistic transport [6] and they were first shownto have an exponential speedup over the classical randomwalk on the hypercube by Kempe [12, 13] for the discretetime walk and Childs et al. for the continuous time walk[14], followed by an algorithm with a proven exponentialspeed up [15].The quantum walks introduced thus far model ideal-ized walkers with no spatial extension. Whilst these havemany uses in modeling physical and biological processes,e.g., [16], we may also want to consider walkers whichdo have spatial extension, and hence can only move intopositions which they are not already occupying. In aclassical setting, self-avoiding random walks were devel-oped to model precisely such processes, initially the fold-ing of polymers. The simplest case of self-avoidance is adimer occupying two adjacent lattice sites. For a dimerwith distinguishable halves, a “head” followed by a “tail”,self-avoidance means the head cannot step back onto thepreviously occupied position, since that is now occupiedby the tail, see Fig. 1. This is thus known as the non-reversal walk. In this paper, we introduce a quantumversion of such a walk. The motivation for studying the ∗ The first two authors contributed equally to this work nonreversal quantum walk is much the same as that forstudying the classical version: more realistic simulationof physical systems.
FIG. 1. The head and tail of a dimer on a square lattice, thediamond marking the head, and the triangle marking the tail.As the tail prevents the head moving left, the head can moveup, down, or right, resulting in a nonreversal walk
In both the classical and quantum cases, the self-avoiding or nonreversal walk on the line is trivial. Thisis because there are only two degrees of freedom in themovement, so if one of those is prohibited by the model,then unidirectional ballistic transport is obtained. Thewalks studied in this paper take place over a square lat-tice, in which case the dynamics are highly non-trivial.The paper proceeds as follows: in Section I A classicalself-avoiding and nonreversal walks are described in moredetail, to provide background and context. Then forthe sake of comparison, the properties of the quantumwalk on the square lattice are briefly outlined in SectionI B. The nonreversal and the closely related nonrepeatingquantum walks are then defined in Section II. The prop-erties of the nonrepeating walk are explored analyticallyin Section III. With the aid of numerical simulations, thenonreversal walk is investigated in Section IV. We finishin Section V with some concluding remarks.
A. Classical self-avoiding random walks
The classical self-avoiding walk has proven difficult totreat analytically, hence the results concerning it have allso far been numerical [17] and there remain many openquestions. Even enumerating the number of self-avoidingwalks has proven very difficult, despite them being sorare that coming upon one by mistake when examininga random walk is highly improbable. If we denote by c n the number of self-avoiding walks of precisely n steps,then the total number of self-avoiding walks up to length n is P n ≥ c n . Some facts are clear, for example that c n + m ≤ c n + c m . The set of self-avoiding walks of length n concatenated with those of length m contains not onlythe self-avoiding walks of length n + m but some whichoverlap, hence the inequality. While determining the pre-cise number of walks is difficult, some bounds have beenestablished. On a square lattice, the number of nonre-versal walks of length exactly n steps is 3 n , since thereare three choices of direction at each step. The numberof self-avoiding walks must be less than 3 n , as the nonre-versal walks include the self-avoiding walks as a subset.Additionally, it is possible to construct subsets of self-avoiding walks which grow as 2 n , hence we know thatthere are between 2 n and 3 n self-avoiding random walks.The best evidence so far suggests that the number of self-avoiding walks of length n is proportional to 2 . n , andthis is provided as a non-rigorous estimate in [18]. Theevidence for this value was obtained by enumerating eachsuch walk of length up to 51 and required a 1024 proces-sor supercomputer [19]. Without new algorithms it isunlikely that we will be able to enumerate much furtherthan this.As even counting the walks has proved difficult, it isunsurprising that little is known regarding other proper-ties. Interesting quantities with which to compare dif-ferent walks include the average distance from the ori-gin, denoted h r i , the average of the square of this dis-tance h r i , and the standard deviation of r . For the self-avoiding walk, h r i is conjectured to be proportional to n / though so far, even a proof that the exponent mustbe between 1 and 2 is elusive [17]. Another interestingproperty of self-avoiding walks demonstrates a key differ-ence between the self-avoiding walk and its standard andnonreversal counterparts. The self-avoiding walk doesnot necessarily continue to evolve indefinitely. This isbecause it is possible to reach a lattice site whose onlyadjacent lattice sites have previously been visited, hencethe walker becomes stuck.The nonreversal walk is in some ways more tractable.As already noted, on the square lattice there are 3 n suchwalks of exactly n steps. Its mean squared displacementis h r i = 2 n , so it spreads twice as fast as the standardrandom walk. There is very little literature on the nonre-versal walk, and what there is tends to examine specificcharacteristics of the walk relevant to the study of poly-mer chains [20], rather than its general features. B. Quantum walks on the square lattice
We first define the formalism for the discrete timequantum walk on a square lattice before discussing previ-ous results for such quantum walks. The walk is definedon Z = { ( x, y ) : x, y ∈ Z } where Z denotes the set of in-tegers. The state of the system, Ψ, is then described by afour-dimensional vector at each lattice site, correspond-ing to four possible coin states that are internal degreesof freedom of the walker. We denote this as:Ψ( x, y, t ) = ψ x + ( x, y, t ) ψ y + ( x, y, t ) ψ y − ( x, y, t ) ψ x − ( x, y, t ) , (1)where each component is a complex function of the dis-crete position of the walker, ( x, y ), and discrete time t and where P x,y,j | ψ j ( x, y, t ) | = 1 with j taking the sym-bols x + , y + , y − , x − . These four coin states are associ-ated with the walker moving in the positive x , positive y , negative y and negative x directions respectively.The evolution is then defined by a coin operator, whichacts only on the coin subspace of the walker, and a shiftoperator which acts on the entire Hilbert space. The coinoperator at a particular site is therefore an operator in SU (4). Different coin operators can in general be chosenfor different lattice sites and they may vary in (discrete)time. In what follows the same coin operator is chosen atall lattice sites and it does not vary in time. We denotethe coin operator that acts on the state of the walker(and so is constructed from the individual coin operatorsat each site) as C c , where c labels a particular choice ofcoin operator. The shift operator, S , is defined by: S Ψ( x, y, t ) = ψ x + ( x − , y, t ) ψ y + ( x, y − , t ) ψ y − ( x, y + 1 , t ) ψ x − ( x + 1 , y, t ) . (2)Therefore, the action of the shift operator is to movethe ψ x + coin state at a particular vertex ( x, y ) one stepin the positive x direction to the vertex ( x + 1 , y ), andanalogously for the three other coin states. This can beseen from the definition of the shift operator as the ψ x + coin state at ( x, y ) depends on the pre-shift ψ x + coinstate at ( x − , y ). We then define the operator thatevolves the walk by one time step U c , by the action ofthe coin operator C c , followed by the shift operator, S .That is:Ψ( x, y, t + 1) = U c Ψ( x, y, t ) = S · C c Ψ( x, y, t ) . (3)The choice of coin operator and the initial state of thewalker then completely define the walk as we may writeΨ( x, y, t ) = U tc Ψ( x, y,
0) = ( S · C c ) t Ψ( x, y, . (4)The properties of the discrete time quantum walk onthe square lattice were extensively explored in [21] fol-lowing initial investigations in [22]. In particular they (a) x ( l a tt i c e un i t s ) −40 −30 −20 −10 0 10 20 30 40 y ( l a tt i c e u n i t s ) −40−30−20−100 10203040 p r o b a b ili t y (b) x ( l a tt i c e un i t s ) −40 −30 −20 −10 0 10 20 30 40 y ( l a tt i c e u n i t s ) −40−30−20−100 10203040 p r o b a b ili t y (c) x ( l a tt i c e un i t s ) −40 −30 −20 −10 0 10 20 30 40 y ( l a tt i c e u n i t s ) −40−30−20−100 10203040 p r o b a b ili t y FIG. 2. (Color online) Probability distributions arising from(a) the Hadamard coin and b) the DFT coin for the initialstate of Eq. (8), and c) the Grover coin for initial state ofEq. (11). examined the mean distance from the origin at time t : h r i t = X x,y p ( x, y, t ) p x + y , (5)where r is the radial distance from the origin, and p ( x, y, t ) is the probability of finding the walker at po-sition ( x, y ) at time t . They also characterise the walksin terms of the standard deviation of r , σ = p h r i − h r i , (6) which characterises how spread out the walker is overthe lattice. Larger σ indicates greater spreading or vari-ation in r , while small σ indicates the walker is mov-ing out radially in a well-defined ring. The authors of[21, 22] carried out a comparison between three differentcoin operators. Their first choice is a tensor product oftwo Hadamard operators for the walk on a line: H ⊗ H = 12 − −
11 1 − − − − . (7)This creates a separable unitary evolution in the x + y and x − y directions [23] and so a two dimensional version ofthe distribution of the walk on the line is obtained. Thisis shown in Fig. 2(a), where the initial state is taken asthe walker at the origin with the separable coin stateΨ(0 , ,
0) = 12 ii − . (8)More interestingly, they consider the Grover coin: G = 12 − − − − , (9)and the discrete Fourier transform (DFT) coin: D = 12 i − − i − − − i − i . (10)These operators were tested for a number of initial con-ditions. These coins are both unbiased, in that they dis-tribute amplitude equally between each coin state. As forthe walk on the line, they find the dynamics for a spe-cific coin depend strongly on the choice of initial state.However, the dynamics differ markedly depending on thecoin used. The lowest and highest standard deviationsobtained for the position of the walker were found usingthe Grover operator. It was observed that the reason forthis is that regardless of the initial state, the distributionforms a central spike, with a ring around it which prop-agates outwards. The choice of initial condition controlsthe amount of amplitude that is situated in the centralspike, and the amount of amplitude that is situated inthe ring. The distribution for the DFT coin, given theinitial state of Eq. (8), is shown in Fig. 2(b). The dis-tribution for the Grover coin, where the initial state istaken to be the walker at the origin, with the coin stateΨ(0 , ,
0) = 12 − − , (11)is shown in Fig. 2(c). Additionally, the authors of [21]studied the set of unbiased four dimensional unitary op-erators with entries equal to either ± / ± i/ / II. DEFINITION
We now define the nonrepeating and nonreversal walksin terms of particular choices for the coin operator. The first coin we will consider will generate the nonrepeatingquantum walk and hence will be called the nonrepeatingcoin, denoted C ! rep . This coin is defined by: C ! rep = λe iα γe iβ f ( λ, γ ) e iθ λe − i ( φ + δ + α ) − f ( λ, γ ) e i ( ψ − θ + β ) γe iψ − γe − i ( δ + α + ψ ) − f ( λ, γ ) e i ( φ − θ + α ) λe iφ f ( λ, γ ) e i ( θ − α − ψ − φ − β ) − γe i ( δ + α − β ) λe iδ , (12)where all of the variables are real, 0 ≤ γ + λ ≤ , and f ( λ, γ ) = p − ( λ + γ ). This is the most general SU (4) operator with zeros on the diagonal. It is clear,with reference to the shift operator defined in Eq. (2),that the coin never permits amplitude to move in thesame direction in two consecutive steps, and so it is nat-ural to refer to this as a nonrepeating walk. We nowdefine the nonreversal coin operator, in terms of a per-mutation of the nonrepeating coin operator, by: C ! rev = C ! rep · . (13)In analogy to the nonrepeating walk, the walk definedby this coin never permits amplitude to move back tothe vertex where it was at the previous time step, andso hence is a nonreversal quantum walk. It is importantto note that the interpretations of each of these walks isstrongly linked to the definition of the shift operator. Ifan alternative definition is used, such as that in [24], thenthe interpretations of the walks created by these coins ischanged. Whilst the nonreversal walker is a single parti-cle, when interpreted as a dimer it is presumed that thetwo parts of the nonreversal walker are distinguishable,so one leads the other, as shown in Fig. 1.As the walk dynamics that are obtained from the non-reversal and the nonrepeating coins have many proper-ties in common we discuss both together. A simple ex-ample of a nonreversal coin, used to produce the prob-ability distribution shown in Fig. 3, takes θ = φ = π , α = β = δ = ψ = − π and λ = γ = f ( λ, γ ) = √ in C ! rev . This leads to the following coin: C = e − i π √ − − − − − , (14)where the global phase factor can be dropped. Fig. 3shows examples of typical probability distribution arisingfrom a nonreversal and a nonrepeating quantum walk.The nonreversal walk displays a greater average radialdistance from the origin than the nonrepeating walk.This is quantified in section IV. In both cases, the dy-namics are similar for all initial conditions, tracing outroughly a diamond shape, larger for the nonreversal walk,with peaks at each corner. The initial condition deter-mines the height and number of distinctive peaks. In thecase of the nonreversal walk, it is possible to see a smallersquare insider the larger outer diamond that is character-istic of this walk. In the case of the nonrepeating walk,the outline of the possible sites that the walk can havereached after t steps is given by a square with sides oflength t ( t + 1) if t is even (odd) centered on the originwith the sides parallel to the x and y axes. However, itcan be seen from Fig. 3(b) that the characteristic shapeof the peaks of the probability amplitude for this walkis also a diamond, as in the nonreversal case, and withdimensions much smaller than t . III. FOURIER ANALYSIS
In order to analytically study the large t behaviour ofthe quantum walks defined above, we use Fourier anal-ysis. The Fourier transform from position space to mo- (a) x ( l a tt i c e un i t s ) −40 −30 −20 −10 0 10 20 30 40 y ( l a tt i c e u n i t s ) −40−30−20−100 10203040 p r o b a b ili t y (b) x ( l a tt i c e un i t s ) −40 −30 −20 −10 0 10 20 30 40 y ( l a tt i c e u n i t s ) −40−30−20−100 10203040 p r o b a b ili t y FIG. 3. (Color online) Probability distributions arising after100 steps of a) a typical nonreversal quantum walk and b)a typical nonrepeating quantum walk, with the initial condi-tions given by Eq. (8) and Eq. (11) respectively. mentum space isˆΨ( k x , k y , t ) = X x,y Ψ( x, y, t ) e i ( k x x + k y y ) , (15)with the inverse transform given byΨ( x, y, t ) = Z π − π Z π − π dk x dk y (2 π ) ˆΨ( k x , k y , t ) e − i ( k x x + k y y ) . (16)In momentum space the shift operator is given by S ( k x , k y ) = e ik x e ik y e − ik y
00 0 0 e − ik x . (17) The walk then evolves by the recurrence relationˆΨ( k x , k y , t + 1) = U c ( k x , k y ) ˆΨ( k x , k y , t ) , (18)where U c ( k x , k y ) = S ( k x , k y ) C c and C c is the chosencoin operator. Using the notation ˆΨ t = ˆΨ( k x , k y , t ) andΨ t = Ψ( x, y, t ), we can rewrite this asˆΨ t = U c ( k x , k y ) t ˆΨ (19)As the walker is initially at the origin ˆΨ is constantin both k x and k y . For analytical purposes, instead ofconsidering moments in terms of the radial distance fromthe origin we consider the joint moments of the positionoperators in the x and y directions, denoted X and Y respectively. For a two dimensional quantum walk, theseare given by D X ξt Y χt E Ψ = X x,y ∈ Z Ψ † t x ξ y χ Ψ t = Z π − π Z π − π dk x dk y (2 π ) ˆΨ † t (cid:18) i ∂∂k x (cid:19) ξ (cid:18) i ∂∂k y (cid:19) χ ˆΨ t , (20)where i ∂∂k x and i ∂∂k y are the momentum space represen-tations of the position operators X and Y . In order tocalculate the state of the walker at time t for a particular C c we need to calculate the eigensystem of U c . We willfirst of all take U ! rep and show that in this case the evenmoments, i.e., when ξ + χ is even, are independent of theinitial state of the walker for large t . In Appendix A it isshown that the eigenvalues of U ! rep ( k x , k y ), denoted by p j , can be expressed as p = − p = p ∗ = − p ∗ = e iω ( k x ,k y ) , (21)where ω is a function of k x , k y and all 8 coin param-eters. This is because the characteristic equation of U ! rep ( k x , k y ) is of the form p + Ap + B = 0 , (22)where A and B are real. We label a corresponding setof orthonormal eigenvectors by | v j ( k x , k y ) i , j = 1 , , , k x and k y dependence from the notation.We may represent the state in terms of the eigensystemby: ˆΨ t = U ( k x , k y ) t ˆΨ = X j =1 p tj h v j | Ψ i | v j i . (23)We will now show that the joint moments, as defined inEq. (20), are asymptotically independent of Ψ using themethod of Grimmett et al. [25, 26] to calculate the large t expression for the moments. First consider (cid:18) i ∂∂k x (cid:19) ξ (cid:18) i ∂∂k y (cid:19) χ ˆΨ t = (cid:18) i ∂∂k x (cid:19) ξ (cid:18) i ∂∂k y (cid:19) χ X j =1 ( − ( j − t (cid:0) e iωt h v j | Ψ i | v j i + e − iωt h v j +2 | Ψ i | v j +2 i (cid:1) = (cid:16) ( − ξ + χ e iωt X j =1 ( − ( j − t h v j | Ψ i | v j i + e − iωt X j =3 ( − ( j − t h v j | Ψ i | v j i (cid:17) t ξ + χ (cid:18) ∂ω∂k x (cid:19) ξ (cid:18) ∂ω∂k y (cid:19) χ + O ( t ξ + χ − ) . (24)Considering the whole of the integrand in Eq. (20), we then have thatˆΨ † t (cid:18) i ∂∂k x (cid:19) ξ (cid:18) i ∂∂k y (cid:19) χ ˆΨ t = n ( − ξ + χ X j =1 |h v j | Ψ i| + X j =3 |h v j | Ψ i| o t ξ + χ (cid:18) ∂ω∂k x (cid:19) ξ (cid:18) ∂ω∂k y (cid:19) χ + O ( t ξ + χ − ) . (25)Now as P j =1 |h v j | Ψ i| = 1 then if ξ + χ = 2 n , n ∈ N we haveˆΨ † t X ξ Y χ ˆΨ t = t ξ + χ (cid:18) ∂ω∂k x (cid:19) ξ (cid:18) ∂ω∂k y (cid:19) χ + O ( t ξ + χ − ) . (26)Therefore, by substituting Eq. (26) into Eq. (20) wesee that under the condition ξ + χ = 2 n , and in theasymptotic limit of large t , the moments are independentof the initial state of the walker and are a function ofthe coin parameters only. From appendix A it can beseen that the moments (asymptotically) depend only onthe five parameters m = α − β + δ + ψ , m = φ + δ , m = φ + α − θ + ψ + β , λ and γ . This is because ω ( k x , k y )can be written as a function of k x , k y , m , m , m , λ and γ . Although this is an asymptotic proof, numericalresults show that this is true for any t , suggesting thatthe dependence on the initial states cancels in a similarway for all orders, not just the leading order. A similarresult also holds for the Hadamard walk on a lattice, asdefined by the coin of Eq. (7) and the shift of Eq. (2).For this separable coin [23], in the limit of large t , h ( X t + Y t ) ξ , ( X t − Y t ) χ i is independent of the initial state when both ξ and χ are even. This follows directly from theproperties of a Hadamard walk on a line and is shownbriefly in appendix C. IV. THE NONREVERSAL WALK
The result derived above applies only to walks usingthe operator U ! rep . We conjecture that the same resultholds for the nonreversal walk. As its eigensystem is nottractable using the same methods as for the nonrepeatingwalk, as shown in Appendix B, the nonreversal walk istreated numerically rather than analytically. These walkswere investigated by varying the parameters in the coinas well as the initial condition. Independent uniformlyrandom choices for each variable were used to generate500 coins, and the walks arising from these operators wereinvestigated using roughly 1000 initial conditions for the (a) time step (time units) a v e r a g e r a d i a l d i s t a n c e ( l a tt i c e un i t s ) (b) time step (time units) a v e r a g e r a d i a l s p r e a d ( l a tt i c e un i t s ) FIG. 4. (Color online) Comparison of the nonreversal(blue right triangles), nonrepeating (green left triangles),Hadamard (red squares), Grover (open diamonds and circles)and DFT (black diamonds and circles) coin operators in termsof a) the average radial distance from the origin, defined inEq. (5) and b) the standard deviation of the radial distance,defined in Eq. (6). The diamonds are the largest and thecircles the smallest average radial distances for different ini-tial states for those two coins; the values are the same for allinitial states for the other three coins. coin state of a walker at the origin, parameterised usingΨ(0 , ,
0) = cos θ e iφ sin θ cos θ e iφ sin θ sin θ cos θ e iφ sin θ sin θ sin θ , (27)i.e., uniformly according to the Haar measure [27, 28].This means that we sample 0 < φ i ≤ < cos θ i ≤
1, for i = 1 , , t . Further investigations suggestthat all the joint moments of the distribution where the x and y exponents sum to an even number are independentof the initial condition.To compare with the nonrepeating coin, we testedwhether the moments were constant if the five parameters m = α − β + δ + ψ , m = φ + δ , m = φ + α − θ + ψ + β , λ and γ were held constant whilst varying their con-stituents. As in this case only twenty coins were tested,the results are not conclusive, but it appears that thesesame five parameters determine the moments in the caseof the nonreversal coin.The properties of both walks contrast strongly withthose arising from previously studied non-separablecoins. The mean and standard deviations as a func-tion of time are shown for a variety of coins in Fig. 4.For the nonreversal, nonrepeating and Hadamard walksthese are independent of the initial condition. For theGrover and DFT walks this is not the case and so theinitial conditions which give the largest and smallest val-ues for h r i are plotted for both coins. Fig. 4 shows thatboth the average radial distance and the average radialspread are greater for the nonreversal than for the non-repeating walk. The average radial distance for both ofthese walks lies within the range of the possible valuesachievable with the Grover coin with the use of specificinitial coin states. V. CONCLUSION
We have introduced two previously unstudied typesof coin operator for the discrete time quantum walk, thenonrepeating and nonreversal coins. We have shown thatthey have some notable properties, namely that the meanand standard deviation of the radial distance from the origin of the walker is independent of the choice of ini-tial condition, in contrast to all the commonly used non-separable coin operators. The standard deviation stillgrows linearly with t for much the same reason as it doesfor the walk on the line, as the coin operator always en-sures that some amplitude moves away from the startingpoint with each step. We have shown, analytically forthe nonrepeating operator and numerically for the non-reversal operator, that the even joint moments of the x and y positions are independent of the initial conditionof the walker; the odd moments do depend on the initialcondition. The even moments of the nonrepeating walkdepend on five parameters derived from the nonrepeat-ing coin. We have also provided numerical evidence thatthe moments of the nonreversal walk depend on the samefive parameters.For future work, it would be interesting to investigatethe properties of the nonrepeating and nonreversal walkson other lattices besides the square lattice. The self-avoiding random walk has been shown to have macro-scopic properties which are independent of the choice oflattice. In order to see if this property carries over intothe quantum case, analogous coins of varying dimensionare required, which can be constructed by parameteriz-ing a unitary matrix with the appropriate pattern of zeroentries. Further coins that exhibit the same analyticalproperties as the nonrepeating coin, i.e., the even mo-ments are independent of the initial state, could be con-structed by creating coin operators with eigenvalues thatare the solutions to characteristic equations that have theform of Eq. (22). ACKNOWLEDGMENTS
T.J.P. was funded by a University of Leeds ResearchScholarship. K.E.B. was funded by the UK Engineer-ing and Physical Sciences Research Council. V.M.K.was funded by a UK Royal Society Research Fellowship.T.J.P., B.H., and A.B. received support through a schol-arship from the UK Royal Society and the UK Engineer-ing and Physical Sciences Research Council. V.P. wassupported by the International Association for the Ex-change of Students for Technical Experience and Ministryof Education, Science and Technological Development ofthe Republic of Serbia (project ON171025). S.M. wasfunded by JFT Grants No. 15619 and No. ANR-10-JCJC-0208 CausaQ. [1] S. Gudder,
Quantum Probability (Academic Press Inc.,CA, USA, 1988).[2] Y. Aharonov, L. Davidovich, and N. Zagury, Phys. Rev.A , 1687 (1993).[3] G. Grossing and A. Zeilinger, Complex Systems , 197(1988). [4] E. Farhi and S. Gutmann, Phys. Rev. A , 915 (1998).[5] D. Aharonov, A. Ambainis, J. Kempe, and U. Vazirani,in Proceedings of the 33rd Annual ACM Symposium onTheory of Computing (STOC) (2001) pp. 50–59.[6] A. Ambainis, E. Bach, A. Nayak, A. Vishwanath, andJ. Vishwanath, in
Proceedings of the 33rd Annual ACM
Symposium on Theory of Computing (STOC) (2001) pp.37–49.[7] N. Shenvi, J. Kempe, and K. B. Whaley, Phys. Rev. A , 052307 (2003).[8] F. Magniez, A. Nayak, P. C. Richter, and M. Santha, in Proceedings of the twentieth Annual ACM-SIAM Sym-posium on Discrete Algorithms , SODA ’09 (Society forIndustrial and Applied Mathematics, Philadelphia, PA,USA, 2009) pp. 86–95.[9] A. Ambainis, in (IEEEComputer Society Press, Los Alamitos, CA, 2004) pp.22–31.[10] A. M. Childs and J. M Eisenberg, Quantum Informationand Computation , 593 (2005).[11] F. Magniez, M. Santha, and M. Szegedy, in Proceed-ings of 16th ACM-SIAM Symposium on Discrete Algo-rithms (Society for Industrial and Applied Mathematics,Philadelphia, PA, USA, 2005) pp. 1109–1117.[12] J. Kempe, in
Approximation, Randomization, and Com-binatorial Optimization. Algorithms and Techniques , Lec-ture Notes in Computer Science, Vol. 2764 (Springer,Berlin/Heidelberg, 2003) pp. 354–369.[13] J. Kempe, Probability Theory and Related Fields , 215 (2005).[14] A. Childs, E. Farhi, and S. Gutmann, Quantum Inf.Process. (2002).[15] A. Childs, R. Cleve, E. Deotto, E. Farhi, S. Gutmann,and D. Spielman, in Proceedings of the 35th Annual ACMSymposium on Theory of Computing (STOC) (2003) pp.59–68.[16] M. Mohseni, P. Rebentrost, S. Lloyd, and A. Aspuru-Guzik, J. Chem. Phys. , 174106 (2008).[17] B. Hayes, AmSci , 314 (1998).[18] I. Jensen, J. Phys. A: Math. Gen , 5731 (2003).[19] A. J. Guttmann and A. R. Conway, Ann. Comb. , 319(2001).[20] A. Skliros, W. Park, and G. S. Chirikjian, J. Algebr.Stat. , 27 (2010).[21] B. Tregenna, W. Flanagan, R. Maile, and V. Kendon,New J. Phys. (2003).[22] T. D. Mackay, S. D. Bartlett, L. T. Stephenson, and B.C. Sanders, J. Phys. A: Math. Gen. (2002).[23] In [21, 22] the quantum walk takes place on the diag-onals with the shift operator defined in the orthogonal x + y and x − y directions (i.e the shifts are by one latticepoint in both x and y ), with the coin basis given by thetensor product (+ , − ) y ⊗ (+ , − ) x = (++ , + − , − + , −− ).This is a 45 ◦ rotation from the coin basis used here of( x + , y + , y − , x − ). Equivalent results to [21, 22] hold forthe shift operator as defined here which can be seen us-ing the change of variables in the (momentum space) shiftoperator of k ± = ( k x ± k y ), as is demonstrated in ap-pendix C.[24] V. Kendon, Int. J. of Quant Inf. , 791 (2006).[25] G. Grimmett, S. Janson, and P. F. Scudo, Phys. Rev. E , 026119 (2004).[26] K. Watabe, N. Kobayashi, M. Katori, and N. Konno,Phys. Rev. A , 062331 (2008).[27] K. Nemoto, J. Phys. A: Math. Gen (2000).[28] D. J. Rowe, B. C. Sanders, and H. de Guise, J. Math.Phys. , 3604 (1999). Appendix A
Here we calculate the eigenvalues of U ! rep ( k x , k y ). Us-ing algebraic manipulation software it can be shown thatthe characteristic equation for this matrix is given by p +2 p (cid:0) γ cos Θ − λ cos Θ − f ( λ, γ ) cos Θ (cid:1) +1 = 0 , where Θ = m − k x + k y , Θ = m − k x − k y andΘ = m with m = α − β + δ + ψ , m = φ + δ and m = φ + α − θ + ψ + β . If we solve this equation for p we will obtain the 4 eigenvalues. As there are no firstor third order terms in the characteristic equation, if p isa solution then so to is − p . As the coefficients are realthen if p is a solution then so is p ∗ . As the coin is unitarywe therefore have that the eigenvalues can be writtenas p = e iω ( k x ,k y ) , p = − e iω ( k x ,k y ) , p = e − iω ( k x ,k y ) , p = − e − iω ( k x ,k y ) where ω is a function of λ , γ , Θ , Θ ,and Θ and so hence is a function of k x , k y and the coinparameters λ , γ , m , m and m . By solving the effectivequadratic it can be shown that the solutions are given by p i = ± s √ a r ± t ia i , where the subscripts denote that ± s and ± t are independent and a r = − b , a i = (cid:12)(cid:12) √ b − (cid:12)(cid:12) , b = γ cos Θ − λ cos Θ − f ( λ, γ ) cos Θ . Appendix B
The characteristic equation for U ! rev can be shown tobe p + ∆ p + Ξ p + ∆ ∗ p + 1 = 0 , where ∆ = f ( λ, γ )( e i ( b − k y ) + e i ( b + k y ) − e − i ( b + b + θ + k x ) − e i ( θ + k x ) ) and Ξ = 2( f ( λ, γ ) cos( b + b )+( λ −
1) cos( k x + k y + b + θ )+( γ −
1) cos( k x − k y + b + θ )) with b = α + φ − θ and b = β + ψ − θ . This quartic does not in general havethe properties of that in appendix A. However if b = − b then ∆ = ∆ ∗ and so the characteristic equation is quasi-symmetric and has real parameters. As the parametersare real, if p is a solution then so is p ∗ . We can then writethe solutions in terms of two parameters ω and ω suchthat p = p ∗ = e iω and p = p ∗ = e iω . It can be shownthat the solutions are of the form p = − ∆2 ± s p ∆ − − ± t q − ∓ s ∆ p ∆ − − . We can therefore show that we cannot write the solutionsin the form derived in Appendix A, that is we cannotwrite p = p ∗ = − p = − p ∗ = e iω and so the proofmethod for U ! rep does not follow for U ! rev . Appendix C