The Lackadaisical Quantum Walker is NOT Lazy at all
TThe Lackadaisical Quantum Walker is NOT Lazy at all
Kun Wang, Nan Wu, ∗ Ping Xu, and Fangmin Song State Key Laboratory for Novel Software Technology,Department of Computer Science and Technology, Nanjing University, Jiangsu 210093, China National Laboratory of Solid State Microstructures,School of Physics, Nanjing University, Jiangsu 210093, China (Dated: October 11, 2018)In this paper, we study the properties of lackadaisical quantum walks on a line. This model isfirst proposed in [1] as a quantum analogue of lazy random walks where each vertex is attached τ self-loops. We derive an analytic expression for the localization probability of the walker at theorigin after infinite steps, and obtain the peak velocities of the walker. We also calculate rigorouslythe wave function of the walker starting from the origin and obtain a long time approximation forthe entire probability density function. As an application of the density function, we prove thatlackadaisical quantum walks spread ballistically for arbitrary τ , and give an analytic solution forthe variance of the walker’s probability distribution. PACS numbers:
I. INTRODUCTION
Since the seminal works by [2–4], quantum walks havebeen the subject of research in two decades. They wereoriginally proposed as a quantum generalization of ran-dom walks [5]. Asymptotic properties such as mixingtime, mixing rate and hitting time of quantum walks ona line and on general graphs have been studied exten-sively [6–10]. Applications of quantum walks in quan-tum information processing have also been investigated.Especially, quantum walks can solve the element distinct-ness problem [11, 12] and perform the quantum search-ing [13]. In some applications, quantum walks basedalgorithms can even gain exponential speedup over allpossible classical algorithms [14]. The discovery of theircapability for universal quantum computations [15, 16]indicates that understanding quantum walks is helpfulfor better understanding quantum computing itself. Fora more comprehensive review, we refer the readers to[17, 18] and the references within.Lackadaisical quantum walks (LQWs), first consideredby Wong et al. [1], are quantum analogous of lazy randomwalks. This model also generalizes three-state quantumwalks on a line [19–22], which only have one self-loopat each vertex. In [1], the authors mainly investigatethe effect of extra self-loops on Grover’s algorithm whenformulated as search for a marked vertex on completegraphs. They find that adding self-loops can either slowdown or boost the success probability by choosing dif-ferent coin operators. On the other hand, three-statequantum walks on a line have been investigated exhaus-tively. Most notably, if the walker of a three-state quan-tum walk is initialized at one site, it will be trapped withlarge probability near the origin after walking enoughsteps [19, 20]. This phenomenon is previously found in ∗ Correspondence to: [email protected] quantum walks on square lattices [23] and is called lo-calization. Researches show that the localization effecthappens with a broad family of coin operators in three-state quantum walks [21, 24, 25]. Moreover, a weak limittheorem is recently derived in [20, 26] for arbitrary coininitial state and coin operator. However, the propertiesof LQWs, such as localization and spread behavior, arestill open. In this paper, we give a in-depth study theLQWs on a line. Since the lackadaisical model is morecomplicated than the standard one, we could expect moreintrinsic characteristics.The rest of this paper is organized as follows. InSec. II, we give formal definitions of LQWs and describethe Fourier transformation method which is often usedin analyzing quantum walks. In Sec. III, we providea mathematical framework for the walker’s localizationprobability on the time limit. In Sec. IV, we find theexplicit forms to compute the velocities of the left- andright-travelling peaks appeared in the walker’s probabil-ity distribution. And in Sec. V, we obtain a long timeapproximation for the entire probability density functionand prove that all LQWs spread ballistically. Finally, weconclude in Sec. VI.
II. DEFINITIONSA. Lackadaisical quantum walks
In this paper, a LQW is defined to be a quantum walkon an infinite line with τ self-loops attached to each ver-tex. An illustrative example is given in Fig. 1, in whicheach vertex has 2 additional self-nodes. We term thenumber of self-loops τ as the laziness factor . If τ = 0, itis the standard quantum walk (also called the Hadamardwalk). In this paper we consider τ >
0. It’s obvious thatin lazy random walks, the greater the τ is, the more thewalker prefers to stay. The total system of a LQW withlaziness factor τ is given by H = H P ⊗ H C , where H P is a r X i v : . [ qu a n t - ph ] D ec FIG. 1: An illustrative example of an infinite line with 2 self-loops attached to each vertex. the position space defined as H P = Span {| n (cid:105) , n ∈ Z } , and H C is the coin space. In each step, the walker has∆ (∆ = τ + 2) choices - it can move to the left, or theright, or just stay in current position via a self-loop. Foreach of these options, we assign a standard basis of thecoin space H C . Thus H C is defined as H C = C ∆ = Span {| (cid:105) , | (cid:105) , · · · , | ∆ (cid:105)} . A single step of quantum walk is given by U = S · ( I P ⊗ C )where S is the position shift operator, I P is the identityof H P and C is the coin flip operator. For LQWs, theposition shift operator S is S = (cid:88) n ∈ Z (cid:110) | n − (cid:105)(cid:104) n | ⊗ | (cid:105)(cid:104) | + | n + 1 (cid:105)(cid:104) n | ⊗ | (cid:105)(cid:104) | + ∆ (cid:88) j =3 | n (cid:105)(cid:104) n | ⊗ | j (cid:105)(cid:104) j | (cid:111) . For the coin operator C , a common choice is the Groveroperator G , which is defined asG = 1∆ − τ · · · − τ · · ·
22 2 − τ · · · · · · − τ . (1)Let | Ψ( t, n ) (cid:105) = [ ψ ( t, n ) , ψ ( t, n ) , · · · , ψ ∆ ( t, n )] † ∈ H C bethe probability amplitude of the walker at position n attime t , then the system state can be expressed by | Ψ( t ) (cid:105) = (cid:88) n ∈ Z | n (cid:105) ⊗ | Ψ( t, n ) (cid:105) . | Ψ( t ) (cid:105) can be obtained by applying U to the initial state | Ψ(0) (cid:105) for t times, i.e. | Ψ( t ) (cid:105) = U t | Ψ(0) (cid:105) . The walker X t can be found at position n at time t with probability P ( X t = n ) = (cid:104) Ψ( t, n ) | Ψ( t, n ) (cid:105) = ∆ (cid:88) j =1 | ψ j ( t, n ) | . (2)Expanding | Ψ( t + 1) (cid:105) = U | Ψ( t ) (cid:105) in terms of | Ψ( t, n ) (cid:105) , weobtain the master equation for the walker at position n | Ψ( t + 1 , n ) (cid:105) = G | Ψ( t, n + 1) (cid:105) + G | Ψ( t, n − (cid:105) + ∆ (cid:88) j =3 G j | Ψ( t, n ) (cid:105) , (3)where G j = (cid:80) ∆ k =1 G j,k | j (cid:105)(cid:104) k | , j = 1 , , · · · , ∆, and G j,k are the elements of G defined in Eq. 1. B. Fourier analysis
Eq. 3 can be solved by Fourier transformation on thesystem state | ˜Ψ( t, k ) (cid:105) = (cid:88) n ∈ Z e − ikn | Ψ( t, n ) (cid:105) , k ∈ ( − π, π ] . (4)From now on, a tilde indicates quantities with a k depen-dence. The inverse Fourier transform is | Ψ( t, n ) (cid:105) = (cid:90) π − π dk π e ink | ˜Ψ( t, k ) (cid:105) . (5)Substituting Eq. 4 to Eq. 3 yields the master equation inthe Fourier space˜Ψ( t + 1 , k ) = (cid:2) G e ik + G e − ik + ∆ (cid:88) j =3 G j (cid:3)(cid:124) (cid:123)(cid:122) (cid:125) ˜ U k ˜Ψ( t, k ) . (6)Let κ = e ik , ˜ U k has the form of˜ U k = 1∆ − τ κ κ κ · · · κ /κ − τ /κ /κ · · · /κ − τ · · · · · · · · · · · · · · · · · · · · · − τ . Since ˜ U k is unitary, its eigenvalues have the forms of λ j = e iω j . We denote | λ j (cid:105) as the corresponding eigen-vectors. After some calculation, we get explicit forms ofthe eigenvalues ω j = θ, j = 1 , − θ, j = 2 , , j = 3 ,π, j ≥ , where θ satisfiescos θ = − τ cos k + 2 τ + 2 , sin θ = (cid:112) τ (1 − cos k )( τ + 4 + τ cos k ) τ + 2 . The corresponding eigenvectors are | λ j (cid:105) = (cid:112) N j e i ( ωj − k ) e i ( ωj + k ) e iωj ...... e iωj , for j = 1 , , | λ j (cid:105) = 1 √ − ← j -th row , ∀ j ≥ , (7)where N j is the corresponding normalization factor.Putting ˜ U k in its eigenbasis, we can rewrite Eq. 6 as˜Ψ( t, k ) = ˜ U k ˜Ψ( t, k ) = ˜ U tk ˜Ψ(0 , k )= ∆ (cid:88) j =1 λ tj | λ j (cid:105)(cid:104) λ j | ˜Ψ(0 , k ) , (8)where ˜Ψ(0 , k ) is the Fourier transformed initial state.In this paper, we assume the walker always starts atposition 0 and the initial coin state satisfies | Ψ(0 , (cid:105) = α | (cid:105) + β | (cid:105) , where α, β ∈ C , α + β = 1. This assumptionis quite reasonable if we want to have a same initial statefor walks with different τ . We guarantee the quantumwalker starts with its coin state in superposition of onlyleft and right bases. Therefore, the system’s initial statecan be formulated asΨ(0) = (cid:88) n ∈ Z δ n, | (cid:105) ⊗ (cid:2) α | (cid:105) + β | (cid:105) (cid:3) , (9)where δ n, is the Kronecker function. By Eq. 4 theFourier transformed system’s initial state becomes˜Ψ(0 , k ) = [ α, β, , · · · , † , ∀ k ∈ ( − π, π ] . (10) III. PROBABILITY AT ORIGIN
In this section, we focus on the localization phe-nomenon on LQWs. To determine whether the local-ization will occur at the origin, we need to calculatelim t →∞ P ( X t = 0). Let the probability be ˆ P , wherewe use a caret (ˆ) to indicate the asymptotic limit of t . THEOREM 1.
For a LQW with laziness factor τ , if thewalker starts with the state given by Eq. 9, the asymptoticlimit of the probability of the walker at origin is ˆ P = 2 · τ + 4 − √ τ + 4 τ . (11)It’s obvious from the theorem that, if the walker startson a superposition of only left and right directions, thelocalization probability of the walker is independent onthe coin initial state, and is totally dominated by thelaziness factor τ . When τ = 1, we get ˆ P = 2(5 − √ FIG. 2: Numerical probabilities of finding the walker at theorigin as a function of walking steps for LQWs with variouslaziness factors τ = 1 (blue dots), τ = 6 (red dots), and τ = 20 (green dots). These factors are carefully chosen toshow different oscillating behaviors. The initial coin state is α = , β = i for all walks. The horizontal lines are thecorresponding theoretical localization probabilities obtainedby Eq. 11. β = 0. We perform numerical simulations and the con-clusions are summarized in Fig. 2. The figure manifeststhe probabilities P ( X t = 0) oscillate around their corre-sponding theoretical limiting values ˆ P for τ = 1, τ = 6,and τ = 20. It is clearly that, for different laziness fac-tors, the probability at the origin oscillates periodicallywith different patterns. These oscillations clearly exhibittendencies to converge, indicating that the walker doeshave a non-zero probability to be localized. Furthermore,we observe that the larger the τ is, the faster the proba-bilities converge. Proof.
By Eq. 2, we haveˆ P ≡ lim t →∞ P ( X t = n ) = lim t →∞ (cid:104) Ψ( t, | Ψ( t, (cid:105) . (12)To obtain ˆ P , we have to calculate lim t →∞ | Ψ( t, (cid:105) . Sub-stitute Eq. 8 into Eq. 5 and let n = 0, we derive theexplicit form for | Ψ( t, (cid:105)| Ψ( t, (cid:105) = (cid:90) π − π dk π ˜Ψ( t, k ) = (cid:90) π − π dk π ∆ (cid:88) j =1 λ tj | λ j (cid:105)(cid:104) λ j | ˜Ψ(0 , k ) (cid:105) . From
Lemma 1 in [27], we know that the contributionsto | Ψ( t, (cid:105) from items with j = 1 , t approaches infinity. As a re-sult, | Ψ( t, (cid:105) is totally determined by the integrals with j ≥
3. Since λ = 1, and ∀ j ≥ , λ j = −
1, we canfurther simplify the equation above by substituting intothese constant eigenvalues. The final expression is shownin Eq. 13. In this equation, only | λ (cid:105) is a function of k ,while for all j ≥ , | λ j (cid:105) and ˜Ψ(0 , k ) are independent on k according to Eq. 7 and 10 respectively. Actually, Eq. 13can be understood as a series of linear maps from theinitial state ˜Ψ(0 , k ) to Ψ( t, F j definedaslim t →∞ | Ψ( t, (cid:105) ∼ (cid:90) π − π dk π (cid:104) | λ (cid:105)(cid:104) λ | + ∆ (cid:88) j =4 ( − t | λ j (cid:105)(cid:104) λ j | (cid:105) | ˜Ψ(0 , k ) (cid:105) = (cid:34) (cid:90) π − π dk π | λ (cid:105)(cid:104) λ | + ∆ (cid:88) j =4 ( − t (cid:90) π − π dk π | λ j (cid:105)(cid:104) λ j | (cid:35) | ˜Ψ(0 , k ) (cid:105) , (13) F = (cid:90) π − π dk π | λ (cid:105)(cid:104) λ | , (14) F j = (cid:90) π − π dk π | λ j (cid:105)(cid:104) λ j | = | λ j (cid:105)(cid:104) λ j | , ∀ j ≥ . (15)The matrices F j capture all information about thewalker’s behavior at its original position when t → ∞ .The existence of localization is directly related to the sys-tem’s initial state via the matrices F j . For all j ≥ F j is independent on k , so it is a constant matrix and can beeasily calculated. The exact form of F can be obtainedby exploiting the eigenvector | λ (cid:105) . Let κ = e − ik , κ = e ik , it’s obvious that κ † = κ and κ † = κ . Sub-stitute ω = 0 into Eq. 7 for j = 3, we get the explicitform of | λ (cid:105)| λ (cid:105) = (cid:112) N (cid:2) κ , κ , , · · · , (cid:3) † , where N = κ κ † + κ κ † + τ = τ +4+ τ cos k k is the normaliza-tion factor. Then F = (cid:90) π − π dk π N κ κ κ κ κ · · · κ κ κ κ κ κ · · · κ κ κ · · · κ κ · · · . Define Θ = τ − √ τ +4 τ ( τ +2) , Θ = √ τ +42 τ +4 , and Θ = τ − ( τ +4) √ τ +42 τ ( τ +2) , we can show after some tedious calculations (cid:90) π − π dk π N = (cid:90) π − π dk π N κ = (cid:90) π − π dk π N κ = Θ , (cid:90) π − π dk π N κ κ = (cid:90) π − π dk π N κ κ = Θ , (cid:90) π − π dk π N κ κ = (cid:90) π − π dk π N κ κ = Θ . Thus the explicit form of F is F = Θ Θ Θ · · · Θ Θ Θ Θ · · · Θ Θ Θ Θ · · · Θ ... ... ... ... ...Θ Θ Θ · · · Θ . Substitute F j into Eq. 13, we getlim t →∞ | Ψ( t, (cid:105) = (cid:104) F + ∆ (cid:88) j =4 ( − t F j (cid:105) | ˜Ψ(0 , k ) (cid:105) , (16)where | ˜Ψ(0 , k ) (cid:105) is given in Eq. 10. Let lim t →∞ | Ψ( t, (cid:105) =[ φ , · · · , φ ∆ ] † , as F j has no impact on the first two com-ponents of | ˜Ψ(0 , k ) (cid:105) for all j ≥
4, Eq. 16 can be easilysolved: φ = Θ α + Θ β,φ = Θ α + Θ β,φ j = Θ ( α + β ) , ∀ j ≥ . The asymptotic limit of the probability at origin ˆ P canbe obtained nowˆ P = | φ | + | φ | + τ | φ | = 2 · τ + 4 − √ τ + 4 τ . Though Theorem 1 only considers localization proba-bilities for a special class of initial states (given in Eq. 9),we should point out that we are able to calculate thelocalization probability by Eq. 12 and 16 for arbitrarysystem’s initial state that satisfiesΨ(0) = (cid:88) n ∈ Z δ n, | (cid:105) ⊗ ∆ (cid:88) j =1 α j | j (cid:105) , where α j ∈ C , and (cid:80) ∆ j =1 | α j | = 1. IV. PEAK VELOCITY
In this section, we determine the peak velocity at whichLQWs spread on the line. The analytical method we usehere is first described in [24]. From their arguments, weknow that the peak velocity is given by the first order ofthe stationary points of the phase˜ ω j ≡ ω j − nt k. The stationary point of the second order of ˜ ω j corre-sponds to the solution of k . We should notice that boththe first and second derivatives of ˜ ω j with respect to k vanish. Therefore in order to obtain the peak velocity,we need to solve equations d ˜ ω j dk = dω j dk − nt = 0 ,d ˜ ω j dk = d ω j dk = 0 . (17)Assume k is the solution of the second equation inEq. 17, then by the first equation we obtain the posi-tion of the peak after t steps n = dω j dk (cid:12)(cid:12)(cid:12) k t. The peak propagates with a constant velocity dω j dk (cid:12)(cid:12) k .Now we show the peak velocities for LQWs on a line.As the phases ω j are constant for all j ≥
3, we immedi-ately know that their corresponding peak velocities are v S = 0. This is easy to understand as the constant phasesresult in the central peak of the probability distributionstaying . Thus the velocities of left and right travellingpeaks are dominated by ω , . We find the equations inEq. 17 can be solved by investigating ω and ω dω , dk = ± τ sin k (cid:112) τ (1 − cos k )( τ cos k + τ + 4) , (18) d ω , dk = ± (cid:115) τ (1 − cos k )( τ cos k + τ + 4) . In k ∈ ( − π, π ], d ω , /dk = 0 has a solution when k =0. Evaluating ω , /dk at k , we get the peak velocitiesof the left and right traveling probabilities v R = lim k → + dω dk = (cid:114) ττ + 2 , (19) v L = lim k → + dω dk = − (cid:114) ττ + 2 . (20)When the laziness factor satisfies τ = 1, we recover theresults presented in [24]. As an illustrative example, weplot the walker’s probability distribution of the LQWwhose laziness factor is 10 in Fig. 3. The probabilitydistribution contains three dominant peaks, the left and FIG. 3: Probability distributions of LQWs after T = 50steps, for various laziness factors τ . The coin initial state is α = 1 / √ β = i/ √
2, This state will give a symmetric walk.We can easily identify three dominant peaks in each probabil-ity distribution. We visualize the right peaks (cid:112) / T ≈ (cid:112) / T ≈
46 in grid lines for τ = 1 and τ = 10. These the-oretical peaks coincide with the positions of peaks obtainedfrom numerical simulations. right travelling peaks are given by the peak velocities v L and v R respectively.From Eq. 19 (Eq. 20) we can see that as the lazinessfactor increases, the right (left) peak velocity also be-comes larger. In this sense, we can control the spreadbehavior the quantum walker and achieve faster spread-ing than the standard quantum walks. In [24], the au-thors offer a different way to control the spread behaviorof the walker by tuning the parameter ρ of the general-ized Grover coin operator (see Eq. 14 in their paper), theunderlying quantum walks are still three-state quantumwalks. While in our paper we actually propose a multi- state quantum walk scheme by introducing different num-ber of self-loops to each vertex, the spread behavior of thewalker can be controlled by tuning the laziness factor τ ,the underlying coin operator is always Grover operator.In the extreme case we have τ → ∞ ⇒ | v R,L | = 1 . Thisindicates that if there is infinite self-loops in each ver-tex, the quantum walker will propagate on the line withconstant speed 1. This can be explained by investigatingthe coin operator G defined in Eq. 1. When τ → ∞ , G satisfieslim τ →∞ G ≡ lim τ →∞ | ψ (cid:105)(cid:104) ψ | − I C = lim τ →∞ τ + 2 (cid:88) j =1 (cid:88) k =1 | j (cid:105)(cid:104) k | − I C ∼ − I C , where I C is the identity of H C . That is, when τ → ∞ ,the coin operator G approximates to − I C , which resultsin a trivial quantum walk. This fast spread behavior ofLQWs is striking different from lazy random walks, inwhich the additional self-loops will slow down the spreadspeed. In the extreme case where τ → ∞ , the classicalwalker will localize in the origin and never spread. V. WEAK LIMIT
In this section, we present a weak limit distribution forthe rescaled LQW X t /t as t → ∞ . It expresses an asymp-totic behavior of the walk after long enough time. Thelimit distribution is composed of a Dirac δ -function re-lated to the localization probability calculated in Sec. IIIand a continuous function with a compact support whosedomain is given by the peak velocities given in Sec. IV.We also prove that LQWs spread ballistically. The an-alytical method we use in this section is first proposedin [28] and we mainly follow the proof procedure out-lined in [26]. What’s more, one should keep in mind thatin this paper we only consider a special class of system’sinitial states defined in Eq. 9. THEOREM 2.
For any real number x , we have lim t →∞ P ( X t t ≥ x ) = (cid:90) x −∞ dy (cid:110) δ ,y ˆ P + f ( y ) I ( − Ω , Ω) ( y ) (cid:111) , where • δ ,y is the Dirac δ -function at the origin, • ˆ P is the sum of localization probabilities in all po-sitions and satisfies ˆ P = √ τ + 42 τ + 4 + 2 (cid:110) τ − ( τ + 4) √ τ + 42 τ ( τ + 2) (cid:111) R ( α † β ) , • f ( y ) is the weak limit density function defined inEq. 24, • Ω = (cid:113) ττ +2 is the bound of the compact supportdomain, and • I Γ ( y ) is the compact support function whose do-main is Γ and defined as I Γ ( y ) = (cid:26) , y ∈ Γ , , y / ∈ Γ . From the theorem we can see that the limit densityfunction rescaled by time has a compact support andits domain ( − Ω , Ω) is totally determined by the walker’stravelling peak velocities. A weak limit theorem of three-state walks is presented in [26]. Our results are the sameas theirs when we let τ = 1 and set the parameters c = − / s = 2 √ / β = 0 in Theorem 2 of their paper.One should note the difference between ˆ P and ˆ P (the localization probability at the origin) studied in Sec. III.Actually, ˆ P is the sum of localization probabilities in allpositions, i.e., ˆ P = (cid:80) n ∈ Z ˆ P n . We are unable to derivean analytic expression for ˆ P n for n (cid:54) = 0, but luckily wecan still calculate ˆ P . Proof.
The r -th moment of the quantum walker’s proba-bility distribution can be calculated as E ( X rt ) = (cid:88) n ∈ Z n r P ( X t = n )= (cid:90) π − π dk π (cid:104) ˜Ψ( t, k ) | (cid:16) D r | ˜Ψ( t, k ) (cid:105) (cid:17) = ( t ) r (cid:90) π − π dk π ∆ (cid:88) j =1 (cid:16) i λ (cid:48) j λ j (cid:17) r (cid:12)(cid:12)(cid:12) (cid:104) λ j | ˜Ψ(0 , k ) (cid:105) (cid:12)(cid:12)(cid:12) + O ( t r − ) , where D = i ( d/dk ) and ( t ) r = t ( t − · · · ( t − r + 1). Tohave X t spatially rescaled by time, we divide both sidesof the above equation by t r and take a limit on t lim t →∞ E (cid:104)(cid:16) X t t (cid:17) r (cid:105) = (cid:88) j =1 (cid:90) π − π dk π (cid:16) i λ (cid:48) j λ j (cid:17) r (cid:12)(cid:12)(cid:12) (cid:104) λ j | ˜Ψ(0 , k ) (cid:105) (cid:12)(cid:12)(cid:12) + ∆ (cid:88) j =3 (cid:90) π − π dk π (cid:12)(cid:12)(cid:12) (cid:104) λ j | ˜Ψ(0 , k ) (cid:105) (cid:12)(cid:12)(cid:12) . (21)As F j has no impact on the first two components of | ˜Ψ(0 , k ) (cid:105) for all j ≥
4, the second term in Eq. 21 canbe easily calculated by making use of the transformationmatrix F defined in Eq. 14 ∆ (cid:88) j =3 (cid:90) π − π dk π (cid:12)(cid:12)(cid:12) (cid:104) λ j | ˜Ψ(0 , k ) (cid:105) (cid:12)(cid:12)(cid:12) = (cid:90) π − π dk π (cid:12)(cid:12)(cid:12) (cid:104) λ | ˜Ψ(0 , k ) (cid:105) (cid:12)(cid:12)(cid:12) = (cid:104) ˜Ψ(0 , k ) | · F · | ˜Ψ(0 , k ) (cid:105) = Θ + 2Θ R ( α † β ) . (22)Then we calculate the first term. As λ j = e iω j , we canget the derivation of λ j using the expressions for dω j /dk obtained in Eq. 18 for j = 1 , i λ (cid:48) j λ j = − dω j dk = ( − j − τ sin k (cid:112) τ (1 − cos k )( τ cos k + τ + 4) . Putting iλ (cid:48) j /λ j = x in the integrals of Eq. 21 and aftersome tedious calculations, we are able to show that (cid:88) j =1 (cid:90) π − π dk π (cid:16) i λ (cid:48) j λ j (cid:17) r (cid:12)(cid:12)(cid:12) (cid:104) λ j | ˜Ψ(0 , k ) (cid:105) (cid:12)(cid:12)(cid:12) = (cid:90) √ ττ +2 − √ ττ +2 x r f ( x ) dx, (23)where f ( x ) satisfies f ( x ) = 1 π (1 − x ) (cid:112) τ − τ + 2) x (cid:110) R ( α † β ) + 2( | β | − | α | ) x + (cid:16) − R ( α † β ) τ + 4 τ (cid:17) x (cid:111) . (24)Substitute Eq. 22 and 23 into Eq. 21, we obtain the r -thmoment of the quantum walkers probability distributionlim t →∞ E (cid:104)(cid:16) X t t (cid:17) r (cid:105) = ˆ P + (cid:90) √ ττ +2 − √ ττ +2 x r f ( x ) dx = (cid:90) ∞−∞ x r (cid:110) δ ,x ˆ P + f ( x ) I ( − Ω , Ω) ( x ) (cid:111) dx. As a corollary of the weak limit theorem, we prove allLQWs with system initial states defined in Eq. 9 spreadballistically and obtain an analytical expression for thevariance of the walker’s probability distribution. Thevariance of a walker’s probability distribution is definedas σ = E ( X t ) − E ( X t ) ∼ ct α , where c is the spread coefficient, and α is the spreadexponent. The spread behavior of a quantum walk isdetermined by the spread exponent of the variance. If α = 2, we say that the walk spreads ballistically; If α = 1,we say that the walk spreads diffusively. It has beenproved that for standard quantum walks α = 2 [29], whilefor random walks α = 1 [18]. COROLLARY 1.
For a LQW whose laziness factor is τ , if the walker starts with the system initial state given inEq. 9, the variance of the walker’s probability distributionsatisfies σ = c ( τ, α, β ) t , where c ( τ, α, β ) is the spread coefficient defined in Eq. 26. We can see easily from Corollary 1 that all lackadaisi-cal quantum walks spread ballistically for system initialstates defined in Eq. 9 as the spread exponent is 2. More-over, we obtain an analytical solution for the spread coef-ficient c ( τ, α, β ) of the variance in Eq. 26, from which wefind that it is dependent on both τ and coin initial state α , β and the laziness factor τ . By tuning the parameters τ , α and β , we can achieve arbitrary spread coefficientsin the range (0 , Proof.
By Theorem 2 it is easy to see that σ = E ( X t ) − E ( X t ) ∼ t (cid:90) Ω − Ω x f ( x ) dx − t (cid:110) (cid:90) Ω − Ω xf ( x ) dx (cid:111) = c ( τ, α, β ) t , where the coefficient function c ( τ, α, β ) is defined as c ( τ, α, β ) = (cid:90) Ω − Ω x f ( x ) dx − (cid:110) (cid:90) Ω − Ω xf ( x ) dx (cid:111) . (25)Solving Eq. 25, we obtain the analytical solution for c ( τ, α, β ) which has the form of c ( τ, α, β ) = 1 − (5 τ + 8) √ τ + 4(2 τ + 4) + (cid:40) τ + 12 τ + 16) √ τ + 4 τ (2 τ + 4) − τ (cid:41) R ( α † β ) − (cid:40)(cid:16) − √ τ + 4 τ + 2 (cid:17) ( | β | − | α | ) (cid:41) . (26) VI. CONCLUSION
In this paper, we analyze in detail the properties ofLQWs on a line for arbitrary laziness factor τ . First, westudy the localization phenomenon shown in the walks. With the discrete Fourier transformation method, we areable to present an explicit form for the localization prob-ability of the walker in the limit of t → ∞ . The limitingcoin state is obtained by a set of linear maps F j on the ini-tial coin state. This set of F j contain all information re-quired to depict the walker’s behavior at the origin. Thelocalization probability is the inner product of the limit-ing coin state, which is shown independent on the initialcoin state, and totally determined by τ . We also calculate FIG. 4: Numerical and theoretical spread coefficients for dif-ferent laziness factors. The system initial state is α = √ i , β = √ i )4 . This state is designated to guarantee that both R ( α † β ) and | β | − | α | do not equal to 0. The theoreticalresults are obtained by Eq. 26. The numerical results are gotby fitting the numerical data to function ct α in Matlab. Theslight difference between two curves is due to that the nu-merical data is obtained by running the walks for only 1000steps. the velocities of the left and right-travelling probabilitypeaks appeared in the walker’s probability distribution.We can control the spread behavior the quantum walksand achieve faster spreading than the standard quantumwalks by tuning the laziness factor. Furthermore, we show that when τ approaches infinity, the LQW degen-erates to a trivial walk. At last, we calculate rigorouslythe system state and get a long time approximation forthe entire probability density function. The density func-tion has both the Dirac δ -function and a continuous func-tion with a compact support whose domain is determinedby the peak velocities. As an application of the densityfunction, we prove that all LQWs spread ballistically, andgive an analytic solution for the variance of the walker’sprobability distribution. The analytical results we obtainillustrate interesting characteristics of LQWs comparedto standard quantum walks and the corresponding lazyrandom walks. For example, it is obvious that the greaterthe τ is, the more the walker prefers to stay in lazy ran-dom walks. However, a lackadaisical quantum walkerspread even faster with the increment of τ . That’s whywe say the lackadaisical quantum walker is not lazy atall. Acknowledgement
The authors want to thank Haixing Hu, QunyongZhang, Xiaohui Tian and Huaying Liu for the insight-ful discussions. K. W. wants to thank Takuya Machidafor his kind help. This work is supported by the Na-tional Natural Science Foundation of China (Grant Nos.61300050, 91321312, 61321491) and the Chinese NationalNatural Science Foundation of Innovation Team (GrantNo. 61321491). [1] Thomas G Wong. Grover search with lackadaisical quan-tum walks.
Journal of Physics A: Mathematical and The-oretical , 48(43):435304, 2015.[2] Yakir Aharonov, Luiz Davidovich, and Nicim Zagury.Quantum random walks.
Physical Review A , 48(2):1687,1993.[3] David A Meyer. From quantum cellular automata toquantum lattice gases.
Journal of Statistical Physics ,85(5-6):551–574, 1996.[4] Edward Farhi and Sam Gutmann. Quantum computa-tion and decision trees.
Physical Review A , 58(2):915,1998.[5] Frank Spitzer.
Principles of random walk , volume 34.Springer Science & Business Media, 2013.[6] Andris Ambainis, Eric Bach, Ashwin Nayak, AshvinVishwanath, and John Watrous. One-dimensional quan-tum walks. In
Proceedings of the thirty-third annual ACMsymposium on Theory of computing , pages 37–49. ACM,2001.[7] Dorit Aharonov, Andris Ambainis, Julia Kempe, andUmesh Vazirani. Quantum walks on graphs. In
Pro-ceedings of the thirty-third annual ACM symposium onTheory of computing , pages 50–59. ACM, 2001.[8] Cristopher Moore and Alexander Russell. Quantumwalks on the hypercube. In
Randomization and Approxi- mation Techniques in Computer Science , pages 164–178.Springer, 2002.[9] Andrew M Childs and Jeffrey Goldstone. Spatial searchby quantum walk.
Physical Review A , 70(2):022314,2004.[10] Hari Krovi and Todd A Brun. Hitting time for quan-tum walks on the hypercube.
Physical Review A ,73(3):032341, 2006.[11] Scott Aaronson and Yaoyun Shi. Quantum lower boundsfor the collision and the element distinctness problems.
Journal of the ACM (JACM) , 51(4):595–605, 2004.[12] Andris Ambainis. Quantum walk algorithm for elementdistinctness.
SIAM Journal on Computing , 37(1):210–239, 2007.[13] Mario Szegedy. Quantum speed-up of markov chainbased algorithms. In
Foundations of Computer Science,2004. Proceedings. 45th Annual IEEE Symposium on ,pages 32–41. IEEE, 2004.[14] Andrew M Childs, Richard Cleve, Enrico Deotto, Ed-ward Farhi, Sam Gutmann, and Daniel A Spielman. Ex-ponential algorithmic speedup by a quantum walk. In
Proceedings of the thirty-fifth annual ACM symposiumon Theory of computing , pages 59–68. ACM, 2003.[15] Andrew M Childs. Universal computation by quantumwalk.
Physical review letters , 102(18):180501, 2009. [16] Neil B Lovett, Sally Cooper, Matthew Everitt, MatthewTrevers, and Viv Kendon. Universal quantum compu-tation using the discrete-time quantum walk.
PhysicalReview A , 81(4):042330, 2010.[17] Julia Kempe. Quantum random walks: an introductoryoverview.
Contemporary Physics , 44(4):307–327, 2003.[18] Salvador Elias Venegas-Andraca. Quantum walks: acomprehensive review.
Quantum Information Processing ,11(5):1015–1106, 2012.[19] Norio Inui, Norio Konno, and Etsuo Segawa. One-dimensional three-state quantum walk.
Physical ReviewE , 72(5):056112, 2005.[20] Stefan Falkner and Stefan Boettcher. Weak limit of thethree-state quantum walk on the line.
Physical ReviewA , 90(1):012307, 2014.[21] M ˇStefaˇn´ak, I Bezdˇekov´a, and Igor Jex. Limit distri-butions of three-state quantum walks: the role of coineigenstates.
Physical Review A , 90(1):012342, 2014.[22] Kun Wang, Nan Wu, Parker Kuklinski, Ping Xu, HaixingHu, and Fangmin Song. Grover walks on a line withabsorbing boundaries.
Quantum Information Processing ,pages 1–25, 2016.[23] Norio Inui, Yoshinao Konishi, and Norio Konno. Lo- calization of two-dimensional quantum walks.
PhysicalReview A , 69(5):052323, 2004.[24] Martin ˇStefaˇn´ak, I Bezdˇekov´a, and Igor Jex. Continuousdeformations of the grover walk preserving localization.
The European Physical Journal D , 66(5):1–7, 2012.[25] Martin ˇStefaˇn´ak, Iva Bezdˇekov´a, Igor Jex, andStephen M Barnett. Stability of point spectrum for three-state quantum walks on a line.
Quantum Information &Computation , 14(13-14):1213–1226, 2014.[26] Takuya Machida. Limit theorems of a 3-state quantumwalk and its application for discrete uniform measures.
Quantum Information & Computation , 15(5-6):406–418,2015.[27] Changyuan Lyu, Luyan Yu, and Shengjun Wu. Localiza-tion in quantum walks on a honeycomb network.
PhysicalReview A , 92(5):052305, 2015.[28] Geoffrey Grimmett, Svante Janson, and Petra F Scudo.Weak limits for quantum random walks.
Physical ReviewE , 69(2):026119, 2004.[29] CM Chandrashekar, R Srikanth, and RaymondLaflamme. Optimizing the discrete time quantum walkusing a su (2) coin.