Framework for discrete-time quantum walks and a symmetric walk on a binary tree
Zlatko Dimcovic, Daniel Rockwell, Ian Milligan, Robert M. Burton, Thinh Nguyen, Yevgeniy Kovchegov
FFramework for discrete-time quantum walks and a symmetric walk on a binary tree
Zlatko Dimcovic, ∗ Daniel Rockwell, Ian Milligan, Robert M. Burton, Thinh Nguyen, and Yevgeniy Kovchegov Department of Physics, Oregon State University, Corvallis OR 97331 Department of Mathematics, Oregon State University, Corvallis OR 97331 School of Electrical Engineering and Computer Science, Oregon State University, Corvallis OR 97331 (Dated: October 11, 2018)We formulate a framework for discrete-time quantum walks, motivated by classical random walkswith memory. We present a specific representation of the classical walk with memory 2 on whichthis is based. The framework has no need for coin spaces, it imposes no constraints on the evolutionoperator other than unitarity, and is unifying of other approaches. As an example we construct asymmetric discrete-time quantum walk on the semi-infinite binary tree. The generating functionof the amplitude at the root is computed in closed-form, as a function of time and the initial level n in the tree, and we find the asymptotic and a full numerical solution for the amplitude. Itexhibits a sharp interference peak and a power law tail, as opposed to the exponentially decayingtail of a broadly peaked distribution of the classical symmetric random walk on a binary tree. Theprobability peak is orders of magnitude larger than it is for the classical walk (already at small n ).The quantum walk shows a polynomial algorithmic speedup in n over the classical walk, which weconjecture to be of the order 2 /
3, based on strong trends in data.
PACS numbers: 03.67.Ac, 89.70.Eg, 02.50.Ga, 05.40.Fb
I. INTRODUCTION
Random walks on graphs (Markov chains) are usedextensively in science. They provide a number of nowstandard approaches and models in physics. Applicationof such ideas to evolution of quantum systems has ledto the emergence of the field of quantum walks , princi-pally distinguished between discrete-time (DTQW) andcontinuous-time (CTQW) quantum walks. However, be-ing unitary (reversible) processes, quantum walks arevery different from their classical stochastic (Markovian)counterparts.Quantum walks are used to approach varied problems,for instance, on quantum lattice gases, arrow of time,generalized quantum theory, exciton trapping, or topo-logical phases [1–5]. They may become a general tool forbuilding physical models; for example, see a summary in[5] and CTQW in transport phenomena [6]. In quantumcomputing, they are a universal primitive [7], and in thealgorithmic context, the principle alternative to quantumFourier transform. The field has developed since its initi-ation [8–12], with established algorithmic uses, examplesof dramatic superiority over classical approaches [13, 14],and implementations [15]. See [16, 17] for a review anda recent summary.Standard approaches to quantum walks generally stemfrom memoryless classical walks. However, since quan-tum evolution is memoried (unitary), it seems natural toapproach construction of quantum walks from classicalrandom walks with memory. Relation between unitarityand memory in walks has been noted [1, 3]. (Also, incomputer science memoried and biased approaches arecommon and beneficial algorithmically.) ∗ [email protected] It is our observation that DTQW are most directlyrelated to classical walks with memory 2. In this paperwe present a general DTQW framework as a direct analogof a specific representation of memory–2 classical walks.With it we construct a symmetric DTQW on a binarytree, starting from a pure state at an arbitrary level inthe tree, and compute its amplitude at the root.Sec. II starts with a representation for memoried clas-sical walks, that is particularly interesting for quantumwalks due to the specific form of the Markov tensor. In ananalogy with it, we then define a framework for DTQW.Walks are built by choosing the evolution operator withno constraints other than unitarity. They evolve in theproduct of state spaces, while the key component of theoperator acts on single states. There is no need for “coin”degrees of freedom. The framework is flexible and suit-able for general graphs. It is also unifying of other ap-proaches, notably coined and Szegedy’s [18].In Sec. III we apply this framework to a binary tree,a structure with many uses in physics. It is a naturalenvironment for quantum walks, but difficult to utilizewith current techniques for DTQW. A successful specificconstruction exists for CTQW [13].We construct a symmetric walk on the semi-infinite bi-nary tree, and calculate its amplitude at the root, as afunction of time and of the initial level in the tree. Thisinvolves path enumeration using regeneration structures,manipulated with the z –transform. The obtained closed–form generating function yields the analytic asymptoticfor the amplitude, which we also compute numerically.The amplitude has a sharp peak and a power law tail,completely unlike the corresponding classical randomwalk, and shows a polynomial speedup in n . The datastrongly suggests the order of the speedup of 2 / a r X i v : . [ qu a n t - ph ] S e p II. A FRAMEWORK FOR DISCRETE-TIMEQUANTUM WALKS
The main approach to DTQW follows ideas of classicalmemoryless walks, and needs an auxiliary “coin” degreeof freedom. Here we employ a specific representationof memoried Markov chains (II A), as an analogy for ageneral DTQW framework (II B) which does not needcoin spaces. Relation to other approaches is discussed atthe close of this section. We start with an example thatserves as a motivation for using memoried walks.
Memoried walks and coined DTQW.
Classical walkswith memory are walks with internal states: apart fromits state, the walk carries other information. Here weuse one such walk in one dimension, with the followingproperty: the next step depends on the direction of theprevious one. If the walker came to a site i from thesite i −
1, the probability to go to i + 1 (to maintain thedirection) is p , while the probability to go to i − − p . This is often called a persistent walk.We now show that coined DTQW are a special case of aquantum analog of classical persistent walks.Consider a standard coined walk on a d –regular graphwith n vertices (for example, [12]). The state of the walkis in the direct product of two Hilbert spaces: an auxil-iary (“coin”) space H A , spanned by d states | a (cid:105) , in whicha unitary operator C mixes components; and a spaceof vertices, H V , spanned by n states | v (cid:105) . The evolu-tion operator acts in this product space H A ⊗ H V as: U ( | a (cid:105) ⊗ | v (cid:105) ) = S ( C ⊗ I ) ( | a (cid:105) ⊗ | v (cid:105) ). On a cycle with the H A basis {|↑(cid:105) , |↓(cid:105)} , the shift S can be implemented as S = |↑(cid:105)(cid:104)↑| ⊗ (cid:80) j | j + 1 (cid:105)(cid:104) j | + |↓(cid:105)(cid:104)↓| ⊗ (cid:80) j | j − (cid:105)(cid:104) j | . Nowconsider for C a generalized Hadamard coin operator, C = (cid:20) √ p √ − p √ − p −√ p (cid:21) , and evolve the state. Starting from the pure “up” state, S · ( C ⊗ I ) |↑(cid:105) ⊗ | i (cid:105) = S · (cid:16) √ p |↑(cid:105) ⊗ | i (cid:105) + (cid:112) − p |↓(cid:105) ⊗ | i (cid:105) (cid:17) = √ p |↑(cid:105) ⊗ | i + 1 (cid:105) + (cid:112) − p |↓(cid:105) ⊗ | i − (cid:105) . (1)For the pure “down” initial state, S · ( C ⊗ I ) |↓(cid:105) ⊗ | i (cid:105) = (cid:112) − p |↑(cid:105) ⊗ | i + 1 (cid:105) − √ p |↓(cid:105) ⊗ | i − (cid:105) . (2)This is a persistent walk: it maintains the direction withprobability p , and changes it with probability 1 − p . Theobtained walk undergoes the same spectral analysis asits unbiased special case ( p = 1 /
2, the Hadamard walk).Walks with a general coin have been studied (for exam-ple, [19–22]), as well as persistent (correlated) walks andtheir relation to DTQW [23–25]. With this example wepoint out the direct correspondence between them. Notethat the directionality of the walk shows up as soon as the coin transformation is allowed to have p (cid:54) = 0 .
5. Inother words, the standard Hadamard transform generallyimplements a persistent walk (rather than a memorylessone), only with equal probabilities.
A. A representation for classical memory–2 walks
Walks with memory 2 are such Markov processes wherethe next step depends on two states: the current one,and the previous one. Walks with memory are generallystudied by using a suitably enlarged state space. In par-ticular, a memory–2 Markov chain can be represented asa memoryless one over the space with n states. Thetransition matrix is then large ( n × n ) and sparse.Instead, here we represent a Markov chain with mem-ory k by a probability distribution µ ( t ) of dimension k ,while the Markov tensor M is then of dimension k + 1.For a memory–2 walk over n sites, the space has dimen-sion n , and each state is labeled by two indices (the sitethe walker came from, and the current site). So the prob-ability distribution is two–dimensional, µ ( t ) = µ , ( t ) · · · µ ,n − ( t )... ... µ n − , ( t ) · · · µ n − ,n − ( t ) . The matrix µ ij can also be given by a column of rows r i , or by a row of columns c i , which we use below. Thefollowing representation for the third-rank tensor M andits action is convenient.Let p ij | k be the conditional probability for the transi-tion j → k , given that the walk came to j from i . Alltransition probabilities { p ij | k } define the evolution oper-ator M = [ P P . . . P n − ], as n layers of n × n transitionmatrices P j , j = 0 , , . . . , n −
1, one for each site: P j = p ,j | p ,j | · · · p ,j | n − ... ... p n − ,j | p n − ,j | · · · p n − ,j | n − .P j are by construction transition probability matrices,and this is the only requirement imposed on them.The evolution of the state, µ t +1 = µ t M , with M act-ing to the left, is defined as µ ( t ) (cid:55)→ µ ( t + 1) : r j ( t + 1) = c T j ( t ) P j , for each j = 0 , , . . . , n −
1, where r j and c T j are the j -throw and transposed column, respectively, of the matrix µ . In words: At each site j , the P j associated with thatsite acts on the transposed j –th column of µ ( t ) , givingthe j –th row of the evolved µ ( t + 1) . Instead of an n × n probability matrix, we use n of n × n probability matrices P j . They implement the evolu-tion: the j –th column of µ ( t ) has probabilities to arrive to j from any site, and after the action of P j the j –th row of µ ( t + 1) has probabilities to go from j to any site. Thusaction of all transition matrices on all columns evolvesthe probability distribution over all paths. The stochas-tic nature of the process is carried by the assignment of { p ij | k } transition probabilities in P j matrices [26].The P j transition matrices are simple in most cases ofinterest. Consider the cycle, a space { , , . . . , n } withidentified ends (0 and n ), with only nearest-neighbortransitions, ( j ± , j ) → ( j, j ± p to continue, and 1 − p to reverse,( j − , j ) → ( j, j + 1) , with probability p ( j − , j ) → ( j, j − , with probability 1 − p. To obtain this walk, the P j matrices have the followingblock centered at ( j, j ) ( mod n ), P j = . . . 1 1 − p p p − p (3)The rest of the diagonal of P j has 1 s , other elements are0, except for the transitions between sites 0 ≡ n , and n − p ,n − | = 1 − p (reverse), p ,n − | = p (continue), etc. Action of these P j by the above prescription carries the walk. B. Quantum walks: the interchange framework
The above classical procedure for memory–2 walks isdirectly elevated to define quantum processes.Consider a basis in an N –dimensional Hilbert space,with vectors labeled as {| i (cid:105) , i = 1 , , . . . N } . They rep-resent states that the walk is performed on, enumeratedon a general graph. The state of the walk is given in theproduct C N × C N spanned by these bases, by states atthe previous ( | i (cid:105) ) and current ( | j (cid:105) ) step, | ψ ( t ) (cid:105) = (cid:88) ij c ij ( t ) | i (cid:105) ⊗ | j (cid:105) . (4)The evolution is specified by | ψ ( t + 1) (cid:105) = (cid:98) U (cid:98) X | ψ ( t ) (cid:105) , | ψ ( t ) (cid:105) = ( (cid:98) U (cid:98) X ) t | ψ (0) (cid:105) , (5)where (cid:98) X is the interchange operator and (cid:98) U is defined viaunitary operators U j in C N , assigned for each site, (cid:98) X : | i (cid:105) ⊗ | j (cid:105) (cid:55)→ | j (cid:105) ⊗ | i (cid:105) (cid:98) U = N − (cid:88) j =0 Π j ⊗ U j , where Π j = | j (cid:105)(cid:104) j | . (6) Π j selects the first state in the product, and U j acts onthe second. Before explicit examples, we make a fewgeneral comments.Consider a pure state of the walk | i (cid:105) ⊗ | j (cid:105) , representedby an arrow pointing from the previous ( i ) to the current( j ) site. The interchange initiates the walk forward by“reversing the arrow.” Then the U j operator distributesthe “tip of the arrow” to all sites the process can ac-cess (generally in the subspace of adjacent nodes), andthe evolved superposition is obtained. This is best seenin the forthcoming example of the binary tree (Fig. 2).The explicit reversal (cid:98) X is crucial; then U j completelycontrols the evolution over site j , by acting on the origi-nating state(s) | i (cid:105) , and sending the process over all pathsto a new state. The framework does not place any condi-tions on these operators, except for unitarity of quantumevolution. We are free to choose (or construct) them asneeded to implement quantum walks.Note that this construction needs no mention of clas-sical processes. The representation of classical memoriedwalks in Sec. II A is given for motivation and insight,and we now comment on this relation. The discussedinterplay between interchange and (local) U j , criticalfor this formulation, has a clear analog in the classicalrepresentation—recall the transposition before (local) P j evolve the distribution. Also, the freedom to craft any(unitary) U j to implement quantum walks correspondsto a classical property, as P j may be any (probabilistic)matrices. Finally, the classical representation has no ex-plicit coin toss, and there is no need in the quantum caseto mimic randomization via a coin degree of freedom;here U j drive the walk and mix components. Relation to other approaches.
For a comparisonwith a memoryless (coined) approach, consider a walkon the line, over the state space S = {| j (cid:105) , j = 0 , , . . . } .The quantum walk (1) and (2) is obtained with U j ( p ) = . . . 1 √ − p √ p −√ p √ − p , with the block centered at ( j, j ). The rest of the diago-nal has 1 s , and other elements are 0. (On a cycle thereare elements needed for boundary conditions, like in theclassical case.) The square roots provide for probabilitybeing the square of the amplitude, and the −√ p sign isneeded for unitarity. In this simple case, the choice of U j follows from the classical memoried walk (3), but ingeneral a classical analog is not needed.Now look at the evolution steps from pure states.Starting from | i − (cid:105) ⊗ | i (cid:105) (the system is in the state | i (cid:105) , having been in | i − (cid:105) at the previous step), (cid:98) U (cid:98) X | i − (cid:105) ⊗ | i (cid:105) = (cid:98) U | i (cid:105) ⊗ | i − (cid:105) = (cid:16) (cid:88) j ∈ S | j (cid:105)(cid:104) j | ⊗ U j (cid:17) | i (cid:105) ⊗ | i − (cid:105) = (cid:112) − p | i (cid:105) ⊗ | i − (cid:105) + √ p | i (cid:105) ⊗ | i + 1 (cid:105) . Similarly, for the initial | i + 1 (cid:105) ⊗ | i (cid:105) state, (cid:98) U (cid:98) X | i + 1 (cid:105) ⊗ | i (cid:105) = −√ p | i (cid:105) ⊗ | i − (cid:105) + (cid:112) − p | i (cid:105) ⊗ | i + 1 (cid:105) . This is an isomorphism of the memoryless–based walk ofEqs. (1) and (2), via identification | i − (cid:105) ⊗ | i (cid:105) ⇔ |↑(cid:105) ⊗ | i (cid:105) and | i + 1 (cid:105) ⊗ | i (cid:105) ⇔ |↓(cid:105) ⊗ | i (cid:105) . This analysis applies to arbitrary mixed states, as eachcomponent is evolved separately. The choice of p = 1 / U j .Memory in quantum walks is mentioned in literature.For example, it was noted in study of the classical limitvia decoherence and multiple coins [27], and a direct rela-tion between coined walks and classical memoried walkswas observed [28]. Recently a particular “quantum walkwith memory” [29] was studied.An important approach directly resorting to ideas ofmemoried walks is the Szegedy walk [18], which is themost prominent tool in DTQW not using coin degreeof freedom [17]. Its construction starts from a classi-cal Markov chain, and the resulting evolution operatorexplicitly carries classical transition probabilities. It iscontained in the interchange framework via the specificchoice ( U j ) km = 2 (cid:112) p ( j, i k ) p ( j, i m ) − δ km , where p ( i, j ) need to be classical transition probabilities.The present approach does not require a specific formof the evolution operator. It is fully defined by (4)–(6)alone, without reference to classical walks, and quantumprocesses with desired properties are set by choosing U j without constraints. Some benefits of this are seen in thenext section, where we construct a symmetric DTQW ona binary tree. A Szegedy walk on a binary tree cannotbe obtained with equal probabilities for each branch, asthere is no (real) solution for probabilities p ( i, j ) such asto yield a 1 / U j . This demonstrates flexibility, and it seems that theframework can help with problems that so far have beenprohibitively difficult. III. A SYMMETRIC DISCRETE-TIMEQUANTUM WALK ON A BINARY TREE
The binary tree is a common model in physics, anda structure of interest in quantum computing [36]. Oneof the initiating works [8] used it as a model for deci-sion trees, and one of the most successful algorithms [13]solves a particular problem on connected binary trees,both using CTQW. We have not seen such progress inusing DTQW on the binary tree, even though this wouldbe beneficial for many problems. This seems to be duemostly to trouble in handling coin spaces that are nec-essary for (coined) DTQW. In this section we use theestablished framework to set and calculate a symmetricDTQW on the semi-infinite binary tree. We orient ourtree with the root (single starting node) at the left withthe tree spanning to the right (Fig. 1). level from rootroot
FIG. 1. Conventions used for our binary tree.
We focus on the following basic question. The walk isstarted from a pure state at a site in the tree at a level n ,and we examine its amplitude at the root as a functionof time (step) and the initial distance n .The desired symmetric walk has equal probability tostep to either of the connecting nodes, having come fromeither direction. (The analysis remains unchanged fordifferent choices of local U j .) The state of the walk isgiven in the direct product of spaces, each spanned bystates defined at nodes S = {| i (cid:105)} . Label a node in thetree as j , and the nodes connected to it as i (to its left,toward the root), i , and i (to its right, away from theroot), as in Fig. 2. Consider an evolution step from apure state at j , for example, | ψ (cid:105) = | i (cid:105) ⊗ | j (cid:105) . The actionof the interchange (cid:98) X reverses the state. Next we want towrite down the U j matrix, acting on | i (cid:105) , such that theevolved state has equal probabilities for either branch.Formally U j operate in the space of all nodes, but theyare reduced to the subspace of nodes to which transi-tions are allowed; here the adjacent ones. Thus U j canbe written in a block-diagonal form, with the non-trivialtransition matrix U red j in {| i (cid:105) , | i (cid:105) , | i (cid:105)} , and an iden-tity matrix over the remaining dimensions. The matrixelements need to satisfy the unitarity of U j and equalsquared amplitudes of components of the state evolvedby it. The obtained evolution operator, with (reduced)transition matrix U red j in the basis {| i (cid:105) , | i (cid:105) , | i (cid:105)} , is U j = (cid:20) U red j I (cid:21) , with U red j = 1 √ a aa aa a , (7)where a = e π i / . This representation holds for graphsof any degree, where dimensions of U red j and I change.At the root the walk can only get reflected, which is per-formed by interchange (cid:98) X ; then U is the identity ma-trix. This will be accounted for. For all other states, wenow follow the prescription of Eqs. (5) and (6). With (cid:98) U = (cid:80) i ∈ S Π i ⊗ U i , the step is | ψ (cid:105) = (cid:98) U (cid:98) X | ψ (cid:105) = (cid:32)(cid:88) i ∈ S | i (cid:105)(cid:104) i | ⊗ U i (cid:33) (cid:98) X | i (cid:105) ⊗ | j (cid:105) = | j (cid:105) ⊗ U j | i (cid:105) = | j (cid:105) ⊗ √ a , , a , , . . . ) T . (8)Thus the state is evolved by (cid:98) U (cid:98) X to the superposition | i (cid:105) ⊗ | j (cid:105) → | j (cid:105) ⊗ (cid:18) a √ | i (cid:105) + 1 √ | i (cid:105) + a √ | i (cid:105) (cid:19) . (9)Each component of the superposition takes the next stepfrom its node in the same way, and the process spreadsover the tree [37]. FIG. 2. A step taken from a pure state | i (cid:105)⊗| j (cid:105) [Eqs. (7)–(9)].The state is sent by (cid:98) U (cid:98) X over all available paths. Probabilitiesfor either branch are chosen to be equal, regardless of howthe walk approaches the site j (the walk is symmetric). Eachcomponent of a general (mixed) state is evolved this way. In comparison with Markov processes, a quantum walkcan be considered as the evolution of the amplitude dis-tribution. Also, the causality typical of the local dynam-ics of the classical Markov evolution, seen in Sec. II A, is reflected in the quantum walk. Note how the concertedaction of (cid:98) X and U j implements the “arrowed” (memo-ried) nature of the evolution, mentioned in Sec. II B.For organizing the calculation, it is useful to notethe connection between directionality and weights of thecomponents of the evolved state. The component that re-verses the direction of the previous step has the coefficient1 / √
3, while the other two have a/ √ | i (cid:105) ⊗ | j (cid:105) ),since it is symmetric, as explicit in Eqs. (7)–(9). Outline of the calculation.
The amplitude at the rootat time t is computed as a sum of the contributions (am-plitudes) of all possible (classical) paths that are at theroot at that time. This is practically a discrete form ofthe path integral in quantum mechanics, and is a stan-dard technique [1, 11]. So we count all such classicalpaths on this structure, weighted appropriately.The presence of a reflective boundary (the root) com-plicates the classification and counting, and we use re-generation structures, which are then handled via the z transform. The obtained explicit expression for thetransform is complicated, and analytically the asymp-totic of its inverse is found, using the method of steepestdescent. The full amplitude is calculated numerically.For brevity in involved descriptions, we sometimes use“paths h ( t )” to refer to “those paths that contribute tothe part of the amplitude (that is named) h ( t ).” A. Path counting and regeneration sums
Enumeration of paths, weighted with appropriate co-efficients (amplitude), is a combinatorial problem. Giventhe symmetry between up and down directions, the treecan be projected to a line bisecting it. The paths onthe tree can be classified, and this results in rules for anequivalent walk on that line.A component of the state at a site is directed eithertoward or away from the root; and it can either continuein the same direction or reverse it in the next step. Forexample, the component directed toward the root (to theleft) can continue toward the root (taking the branch tothe left), with the amplitude a/ √
3, or it can turn andstep away from the root, by directly reversing or (and)by taking the other branch leading away, with the totalamplitude of (1 + a ) / √ / √ a + a ) / √ a ) / √ a/ √ t . Paths generally reach the root infewer than t steps, then going back and forth in the tree,possibly touching the root again in the process, before fi-nally finding themselves at the root at time t . Wheneverthey touch the root their next step can only be a turnback, with the coefficient 1, and this does not fall intothe above classification. To account for it the paths needbe enumerated particularly carefully.All paths that are at the root at time t have the follow-ing structure. They touch the root for the first time atone point (step s ), and we call the amplitude for this partof the path h n ( s ). Then they go out in the tree, eventu-ally coming back to the root at step t , possibly touchingit multiple times in the process; we call the amplitude ofthis part of the path G ( t − s ). This is encoded by theconvolution over the first contact with the root, and theamplitude, represented by weighted paths that are at theroot at step t , starting from level n , is H n ( t ) = t (cid:88) s ≥ n h n ( s ) G ( t − s ) . (10)After the root is touched for the first time, the remain-der of the walk is a root–to–root path, considered in-dependently as G ( t ) [accounting for the n = 0 case, G ( t ) = H ( t )]. It consists of: a “simple loop” g ( s ), thatgoes from the root into the tree and back to it (reachingit again for the first time), followed by the rest of thepath G ( t − s ), which may touch the root multiple times,so again comprised of simple root–to–root loops (Fig. 3). level rootroot time FIG. 3. The number of paths is a convolution w.r.t. the firstcontact with root, with the regeneration structure of Eqs.(10) and (11).
This is a convolution too, with one adjustment. To cor-rectly account for t = 0, it must be that g (0) = 0, as wellas G (0) = 1 (since t = 0 ⇒ s = t ). Thus δ ( t ) need beadded. The amplitude of such root–to–root multi–loops G ( t ) is G ( t ) = t (cid:88) s =0 g ( s ) G ( t − s ) + δ ( t ) . (11)The regeneration sums, Eqs. (10) and (11), organize path counting. We work with them using the z transform, (cid:98) f ( z ) = ∞ (cid:88) t =0 f ( t ) z t , | z | < . Applying this transformation to Eqs. (10) and (11), (cid:98) H n ( z ) = (cid:98) h n ( z ) (cid:98) G ( z ) , and (cid:98) G ( z ) = (cid:98) g ( z ) (cid:98) G ( z ) + 1 , ⇒ (cid:98) H n ( z ) = (cid:98) h n ( z ) 11 − (cid:98) g ( z ) . (12)We need transforms of amplitudes of paths reaching theroot for the first time [ (cid:98) h n ( z )], and of simple loops [ (cid:98) g ( z )].At this point we note a known combinatorial result.The number of paths on a lattice in two dimensions, goingfrom (0 ,
0) to (2 n, k peaks, is given by Narayana numbers [38], N ( n, k ) = 1 n (cid:18) nk (cid:19)(cid:18) nk − (cid:19) . (13)This expression applies to the number of paths compris-ing the simple loops g , where peaks are positions furthestfrom the root. We first need to identify and enumerate“steps” and “turns” in such paths, so that we can assignweights to them accordingly.A simple loop must take an even number of steps. Thefirst step is a reversal: it starts at the root, having arrivedto it from the first node, and it can only step back ontothe first node, so the coefficient for this step is 1. It isstraightforward to establish that paths with k peaks take k left turns and k − t stepsmust take t − − ( k −
1) right, as well as left, steps. Loopswith t = 2 are different: they can only step away fromthe root and return to it in the next step (left turn); theircoefficient is 1 × / √
3. Thus a simple loop with k peaks,for t ≥ (cid:18) √ (cid:19) k (cid:18) a √ (cid:19) k − (cid:18) a √ (cid:19) t/ − k (cid:18) a √ (cid:19) t/ − k = 1( √ t − (cid:18) a a (cid:19) k − (cid:0) a (cid:1) t/ − = 1( √ t − (cid:18) − (cid:19) k − (cid:0) a (cid:1) t/ − (as a = e πi/ ) . For a = e π i / we have 1 + a + a = 0, used above.Summed over all possible numbers of peaks k , and withthe t = 2 case added, the amplitude of a simple loop is g ( t ) = 1 √ δ ( t − t − (cid:88) k =1 (cid:0) a (cid:1) t/ − ( √ t − (cid:18) − (cid:19) k − N (cid:18) t − , k (cid:19) . Using the Narayana numbers (13), with t = 2 m + 2, g ( m ) = 1 √ δ ( m ) + 1 m √ (cid:18) a (cid:19) m × m − (cid:88) k =0 (cid:18) − (cid:19) k (cid:18) mk + 1 (cid:19)(cid:18) mk (cid:19) . Since we will need the transform of g ( t ), it is helpful towrite the above sum as an integral, using the identity m − (cid:88) k =0 ( αβ ) k (cid:18) mk + 1 (cid:19)(cid:18) mk (cid:19) =12 π (cid:90) π (cid:0) αe ix (cid:1) m (cid:0) βe − ix (cid:1) m e − ix α dx. Employing this, under the constraint αβ = − / g ( m ) = 1 √ δ ( m ) + 12 π m √ (cid:18) a (cid:19) m (14) × (cid:90) π (cid:18)
12 + αe ix + βe − ix (cid:19) m e − ix α dx. It is calculationally convenient to take the z –transformof g ( t ) at this point. Since loops take even number ofsteps, and (cid:98) g ( z ) t =0 = g (0) = 0, with t = 2 m + 2, (cid:98) g ( z ) = ∞ (cid:88) t =0 g ( t ) z t = g (0) + g (2) z + (cid:88) t =4 , ,... g ( t ) z t = (cid:98) g ( z ) m =0 + ∞ (cid:88) m =1 g ( m ) z m +2 . The transform of δ is 1, and Eq. (14) becomes (cid:98) g ( z ) = 1 √ z + 12 π √ z (cid:90) π dx e − ix α × (cid:34) ∞ (cid:88) m =1 m (cid:18) a (cid:19) m (cid:18)
12 + αe ix + βe − ix (cid:19) m z m (cid:35) . Now we make use of (cid:80) ∞ n =1 x n /n = − ln(1 − x ), | x | < α = − β = 1 / √
2, arriving at (cid:98) g ( z ) = z √ z π (cid:114) (cid:90) π dx e − ix × (cid:26) − ln (cid:20) − z a (cid:18)
12 + e ix − e − ix √ (cid:19)(cid:21)(cid:27) . Using ω = e − ix and integrating by parts, (cid:98) g ( z ) = 1 √ z − z πi a × (cid:73) | ω | =1 (cid:0) ω + ω (cid:1) dω − a z (cid:104) + √ (cid:0) ω − ω (cid:1)(cid:105) . Here the Residue Theorem is used. The singularity at ω = 0 is removable, while one of the two zeros of thedenominator is inside the integration contour. Finally, (cid:98) g ( z ) = √ a (cid:34) az ) − (cid:114) −
23 ( az ) + ( az ) (cid:35) . (15)This closed-form expression analytically extends (cid:98) g ( z ) be-yond the disk | z | < (cid:98) h n ( z ).Paths from the n -th level that reach the root for thefirst time in s steps, with amplitude h n ( s ), first reachthe level n −
1, generally going out into the tree in themeanwhile, then the level n −
2, and so forth until the rootis hit. This is organized into paths dropping by one levelcloser to the root (with amplitude h ), convoluted withthe rest of the walk, which itself is comprised of pathsgetting closer to the root by one level, h n = h ∗ h n − = . . . = h ∗ . . . ∗ h ( n times). Then the transform is (cid:98) h n ( z ) = (cid:104)(cid:98) h ( z ) (cid:105) n . (16)Paths h , that for the first time reach one level closer tothe root, are combinatorially equivalent to the paths thatstart at level 1 and touch the root for the first time.Consider such a path, starting at level 1 and findingits way to the root for the first time, in more than onestep (Fig. 4). It must have come to level 1 from level 2,and it first steps back to level 2. For comparison, nowrecall a root–to–root simple loop g ( s ). It differs from h by: the first step of g (with coefficient 1) is not taken by h (which is already at level 1), and the next step of g (for paths with t >
2) is a right step, while the h pathtakes a right turn. loop rootroot level any pathtimepaths FIG. 4. Combinatorial comparison of root–to–root loops ( g )and paths getting closer to root by one level ( h ) (see text). So we divide the expression for g ( s ) by the coefficientassociated with the second step of g that h does nottake, 2 a/ √
3, and multiply it by the coefficient of thestep that h takes instead, (1 + a ) / √
3. We also divideby the coefficient of the first step of g , not taken at all by h , which is 1. In the special case s = 1 a single step istaken to the root from the first level, with a/ √
3. Finally,this path takes one step more as compared to g ( s ), so weuse the expression for g ( s + 1), starting from s = 3 since g (2) corresponds to the special case h (1). This gives usthe expression for the amplitude h , h ( s ) = a √ δ ( s −
1) + a √ × a √ g ( s + 1) × s ∈{ , ,... } . Its transform is, using 1 + a + a = 0 (as a = e π i / ), (cid:98) h ( z ) = a √ z + 1 + a a (cid:34) z (cid:88) t =3 , ,... z t +1 g ( t + 1) (cid:35) (17)= a z √ − a z (cid:18)(cid:98) g ( z ) − z √ (cid:19) = a √ z − a (cid:98) g ( z ) z . The sum is the transform of g ( t ≥ (cid:2)(cid:98) g t ≥ (cid:3) , written as (cid:98) g − (cid:98) g t =2 . With Eqs. (12), (16), and (17), the generatingfunction for the amplitude of the process at the root is (cid:98) H n ( z ) = (cid:104) − a (cid:105) n (cid:34) (cid:98) g ( z ) − √ z z (cid:35) n − (cid:98) g ( z ) , (18)with (cid:98) g ( z ) given in Eq. (15). Now we need to invert this. B. Inverse transform: H n ( t ) asymptotic We take the inverse z –transform via an integral, andusing Laurent expansion and the Residue theorem, H n ( t ) = 12 πi (cid:73) | z | = r< (cid:98) H n ( z ) z t +1 dz = 12 πi (cid:104) − a (cid:105) n (cid:73) z t +1 (cid:34) (cid:98) g ( z ) − √ z z (cid:35) n dz − (cid:98) g ( z ) . This integral is too complicated to yield a closed-formsolution. We look for its asymptotic behavior in the form H n ( t ) = ( − a ) n πi n (cid:73) | z | = r (cid:2)(cid:98) g ( z ) − √ z (cid:3) n − (cid:98) g ( z ) dzz t + n +1 , (19)using the steepest descent method. The calculation isdiscussed in Appendix A. The asymptotic of the ampli-tude of the process at the root, starting from a level n inthe tree, with τ ≡ t − n , is H n ( t ) ∼ ( − a ) n πi n × ( √ n ( − n (20) × (cid:20) c n e − i γn e − i λ τ τ / − c n e i γn e − i λ τ τ / (cid:21) . Constants c n and c n are linear in n , while γ and λ ’s arereal constants (Appendix A). The probability is | H n ( t ) | ∼ C − (cid:110) c c ∗ e − i [ γn +( λ − λ ) τ ] (cid:111) π n τ , (21) where C = | c | + | c | ∼ n . The oscillations of theexponential term are rapid at large times (and/or n ),and this function behaves as ∼ τ − . The n dependenceis ∼ n / n at large times. Also, we see that the walkis transient, in the sense introduced in [39, 40], since (cid:80) ∞ t | H n ( t ) | is finite.The observed power law decay differs sharply fromthe exponential tail of the classical walk (Appendix B).On finite graphs built with binary trees, this exponen-tially slower decay may lead to algorithms with signifi-cant speedups. The treatment in this section is meant tolay the groundwork for such investigations.This behavior should also have general implications forphysics of systems modeled with a binary tree. C. Inverse transform: H n ( t ) computed The transform is defined as (cid:98) H n ( z ) = (cid:80) t ≥ n H n ( t ) z t ,and using its Taylor expansion and equating coefficients, H n ( t ) = (cid:98) H ( t ) n ( z ) | z =0 t ! . (22)This can be evaluated efficiently, for a range of values of t for a fixed n , providing the full amplitude. Symbolic cal-culation of derivatives (with Mathematica ) allows forvalues of n in the thousands. We note that the amplitudeshows an interference pattern, with the main peak fol-lowed by (much) smaller, rapidly diminishing, secondarypeaks. Probability at the root with time is shown inFig. 5, for n = 50. The shape does not depend on n .At long times this exact result can be compared withasymptotics (21), see inset in Fig. 5. The tail exhibits t − dependence, in agreement with Eq. (21).
60 100 2000.51.5 time steps È H n n = æ numerical à asymptotic æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æà à à à à à à à à à à à à à à à à à à à à à à à à
34 450 34 48013 È H n FIG. 5. (Color online) Probability at the root with time( n = 50), computed via Eq. (22). Inset compares this (points)with the steepest descent asymptotic (21) at same time steps(red squares). D. Comparison and algorithmic aspects
We considered an algorithm of finding the root start-ing at the n –th generation in the tree. Here we comparethe quantum and classical walks, and their estimatedrun times, using data calculated via Eqs. (22) and (B3).Probability peaks and times at which they are reached,for quantum and classical walks, are shown in Table Ifor a few initial levels n . Probability at the root for theclassical walk is shown in Fig. 6.
100 150 20014 time @ steps D p t H , L n = FIG. 6. (Color online) Classical walk, probability at the root(Appendix B).
The best run time for a given n is estimated as fol-lows. The inverse of the probability is the average num-ber of times needed to run in order to hit the root; mul-tiplying it by its time gives the total running time tohit the root. We need its minimum over all time steps,min (cid:8) t (cid:14) | H n ( t ) | (cid:9) , usually the values for the peak. Weuse data up to n = 5000 in steps of 100 for the quantumwalk, and up to n = 2000 in steps of 50 for the classicalwalk (numerical integration is more demanding). Usingsmaller increments in n does not affect results. We fitthe natural logarithm (ln) of run times with a polyno-mial and linear (Fig. 7).The polynomial fit establishes a linear trend (reachedat n ∼ a few hundred), so the run time of the quantumwalk is exponential in the initial distance from the root, ∼ e bn . The same holds for the classical walk, and thenthe ratio of slopes of their linear fits compares their runtimes.This ratio does not change much over the whole rangeof data, being within a few percent of 2 /
3. Still, since anexponential complexity is fully felt at large n , the laterportions of data are more relevant for algorithmic com-parison. For the last quarter of data ranges, indicated inFig. 7 by lines fit through data, the ratio of quantum toclassical slopes is ≈ . /
3. Thus it ap-pears reasonable to conjecture the algorithmic speedupof the order of 2 /
3. (For a run time T for the classicalwalk, one expects the run time on the order of T / forthe quantum walk.)We note the behavior of peaks times with n . For the ààààààààààààààààààààææææææææææææææææææææææææææææææææææææææææææææææææææ
100 1000 2000 3000 4000 500012 n H initial level L - l n H T (cid:144) p r ob L m i n polynomial fit, ~ n p p q = ± H qw L p c = ± H cl L linear fit, b ‰ n b q = ± H qw L b c = ± H cl L FIG. 7. (Color online) Natural log (ln) of run time withinitial level in the tree, for quantum (points) and classical(blue squares) walks. For plot clarity not all points are shown.Parameter values for polynomial ( ∼ n p ) and linear ( bn ) fitsare shown, with lines through data used for fits (see text). Therun time is ∼ e bn , and b q /b c → / classical walk, t cl max ( n ) = 3 n − t qw max ( n ) ≈ . n (where data allows for afit of ∼ . n , and for an ∼ ln n correction). IV. CONCLUDING REMARKS
Quantum walks are quantum processes with a specificmixing of states; particular unitary processes. In thisvein, we propose to approach and study them using ideasfrom classical random walks with memory.We have demonstrated how a general framework fordiscrete-time quantum walks arises as a natural analog ofa specific representation of a classical memory–2 Markovchain. Walks are implemented by constructing a localoperator with no restrictions other than unitarity. Theframework needs no “coin” degrees of freedom, is flexible,and is applicable to general graphs. This approach maymake it easier to obtain walks on structures for whichsignificant speedups are expected.The evolution operator works separately on each com-ponent of the amplitude, reducing the state space, andeffectively deconstructing the amplitude [as in in the bi-nary tree example, Eqs. (7–9)]. This makes quantum-mechanical correlations and interference transparent inquantum walks, making their explicit study easier. Itshould also aid the use of quantum walks as a generaltool for exploration and modeling of physical systems.In Sec. III we use the framework to build a symmet-ric discrete-time quantum walk on a semi-infinite binarytree. We start the walk at a level n in the tree and find itsamplitude at the root as a function of time and n . Theconstruction of the walk is simple, but the calculationis complicated by a (reflective) boundary. The gener-ating function of the amplitude is found explicitly, and0 initial level n
10 20 50 100 200 500max | H n | (at t ) 6 . × − (16) 3 . × − (30) 1 . × − (76) 5 . × − (150) 7 . × − (298) 5 . × − (738)max p t ( n ) (at t ) 1 . × − (22) 7 . × − (52) 4 . × − (142) 2 . × − (292) 1 . × − (592) 4 . × − (1492)TABLE I. Probability peaks at the root with their times, for quantum (cid:2) | H n ( t ) | (cid:3) and classical (cid:2) p t ( n, (cid:3) walks, with n . its asymptotic is found via the steepest descent method.The full solution is computed numerically.These solutions show interesting features. The asymp-totic decays in time by the power law (as opposed to theexponential tail of the classical walk), representing long–range correlations. This hints at significant speedups onrestricted structures. The amplitude exhibits a dampedinterference pattern, with a distinct and sharp peak.In comparison with the classical walk, the probabilitypeak is reached more quickly, and is orders of magnitudegreater, already at small n . The run time for hitting theroot on a semi-infinite binary tree is exponential with n for the quantum walk, as it is for the classical walk.It is still clearly slower in n and, following suggestivedata trends, we conjecture the polynomial algorithmicspeedup of the order of 2 / Appendix A: Steepest descent calculation summary
Integrals suitable for analysis by the steepest descentmethod are typically of the form [41] I ( k ) = (cid:90) C f ( ω ) e k Φ( ω ) dω. (A1)We use Eq. (19), where the exponent will be formed frompowers of z . As z is always squared we first change vari-ables via z = ξ . Accounting for the double winding, I ( t ; n ) = 2 × (cid:73) |√ ξ | = r (cid:104) (cid:98) g ( ξ ) ξ − √ (cid:105) n − (cid:98) g ( ξ ) 1 ξ dξξ t − n . We now use (cid:98) g ( ξ ) /ξ = ω , to transfer some of the inte-grand’s complexity into the exponent, (cid:98) g ( ξ ) ξ = ω, ξ = ϕ ( ω ) = a √ ω − √ ( ω + √ )( ω − √ ) . Carrying out the substitution, we have I ( t ; n ) = (cid:73) |√ ϕ | = r ( ω − √ ) n (1 − ω ϕ ) ϕ (cid:48) ϕ ϕ − t − n dω, (A2)in the form (A1), with ϕ − t − n = e t − n ln( ϕ − ) , and f = ( ω − √ ) n (1 − ω ϕ ) ϕ (cid:48) ϕ , Φ = − ln ϕ, k = t − n . (A3)Keeping ϕ (cid:48) /ϕ will be useful. Consider the critical points.A pole of order t − n +22 is at ϕ = 0 ⇒ ω p = 1 / √
3. Two simple poles are at − √ , √ . The logarithm’s branchpoint is at ϕ = 1, and since this is not at ω p = 1 / √ ϕ ) (cid:48) = 0 ⇒ ω s /s = 1 ± i √ √ e ± i arctan √ . The main contribution to this integral comes from sad-dle points. A branch of the original integration contour |√ ϕ | = r can be chosen (via r ) for use with steepestdescent paths. There are no issues with deforming thecontour, as no critical points are in the way, any branchof the logarithm is good, and k = t − n ∈ Z (Fig. 8). - - - Re @ Ω D Im @ Ω D FIG. 8. (Color online) Original integration contour (solid),steepest descent paths (dashed and dotted), and criticalpoints.
In the steepest descent method decreasing orders ofcontribution are computed mostly via expansion aroundsaddles. (There are theorems and formulas for the first–order contribution, but here it is zero.) Around ω s wehave Φ( ω ) = Φ( ω s ) + (cid:2) Φ( ω ) (cid:3) (cid:48)(cid:48) ω s ( ω − ω s ) + o ( ω ) , andthe usual change of variables Φ( ω ) − Φ( ω s ) = − y gives ω = ∓ b √ y + ω s , b = (cid:115) (cid:2) ln ϕ ( ω ) (cid:3) (cid:48)(cid:48) ω = ω s , (A4)where y is zero at the saddle and real along the steep-est descent path. We used (cid:113) (ln ϕ − ) (cid:48)(cid:48) = i (cid:113)(cid:0) ln ϕ (cid:1) (cid:48)(cid:48) .1Now y = ln (cid:2) ϕ ( ω ) /ϕ ( ω s ) (cid:3) and dy = (cid:2) ϕ ( ω ) (cid:48) /ϕ ( ω ) (cid:3) dω ,and with ϕ ( ω ) = ϕ ( ω s ) e y the integral (A2) and (A3)becomes I ( k ) ∼ e k Φ( ω s ) (cid:73) (cid:2) ω ( y ) − √ (cid:3) n − ω ( y ) ϕ ( ω s ) e y e − ky dy. (A5)Now we can directly expand around y = 0 (to any order),restricting integration to the line along the steepest de-scent path, close to saddles. The signs in Eq. (A4) cor-respond to the opposite directions from the saddle; welabel + / − as “R/L.” Substituting ω ( y ) and expanding, I ∼ A ϕ − ks (cid:90) δ (cid:0) ± B √ y (cid:1) e − ky dy, δ ∼ o (1) , (A6) A = (cid:2) ω s − √ (cid:3) n − ω s ϕ s , B = (cid:18) b ϕ s − ω s ϕ s + bω s − √ n (cid:19) . For compactness we use ϕ s ≡ ϕ ( ω s ) = e i λ s , with λ s =arctan [( √ √ / ], λ s = arctan [( √ − √ / ] − π .Note that A ( n ) ∼ ( √ n , as ω s − √ √ e i γ s , with γ s /s = ∓ [arctan(1 / √ ) − π ] ≡ ∓ ( γ − π ), and we willextract π later. The integral is dominated around ω s ( δ ≈ δ → ∞ , and weget I ∼ A (cid:82) ∞ (cid:0) ± B √ y (cid:1) e − ky dy . This results in I R/L ( k ; n ) ∼ A n (cid:18) k ± √ π B n k √ k (cid:19) ϕ ( ω s ) − k . (A7)Subtracting contributions along opposite directions, thefirst non–zero order for either saddle is I s ∼ ( √ n e i γ s n ( a s + d s n ) e − i λ s k k √ k . (A8)We broke up the A n B n term found in I R − I L , to showthe structure of n dependence, where a s = b ϕ s √ π (1 − ω s ϕ s ) and d s = b √ π (1 − ω s ϕ s )( ω s −√ are of order ∼
1. Here we extract π from γ s , and will use e i γ s = ( − e ∓ i γ . Contributionsfor saddles are subtracted (for consistency of ±√ y direc-tions) and, with c s,n ≡ a s + d s n , we get Eq. (20).The full expansion of the integral (A5) results in nestedsums of a Gamma function. This cannot capture thepeak of the amplitude though, and is not needed for ourasymptotic analysis, so we do not pursue it here. Appendix B: Random walk on a semi-infinite line
For completeness here we provide the application of themethod developed in [42, 43] (based on Karlin-McGregorspectral approach to random walks) to a classical walkon a binary tree.If P is a reversible Markov chain over a sample spaceΩ, and π is a reversibility function (not necessarily aprobability distribution), then P is a self-adjoint operatorin (cid:96) ( π ), the space generated by the inner product < f, g > π = (cid:88) x ∈ S f ( x ) g ( x ) π ( x ) induced by π . If P is tridiagonal operator (i.e. a nearest-neighbor random walk) on Ω = { , , , . . . } , then it musthave a simple spectrum, and is diagonalizable via orthog-onal polynomials, as it was studied in the 1950s and 1960sby Karlin and McGregor. There the extended eigenfunc-tions Q j ( λ ) ( Q ≡
1) are orthogonal polynomials withrespect to a probability measure ψ and p t ( i, j ) = π j (cid:90) − λ t Q i ( λ ) Q j ( λ ) dψ ( λ ) ∀ i, j ∈ Ω , where π j ( π = 1) is the reversibility measure of P . Con-sider the following Markov chain P = . . .q p . . . q p . . .... ... . . . . . . . . . p > q. Orthogonal polynomials are obtained via solving a simplelinear recursion: Q = 1, Q = λ , and Q n ( λ ) = c ( λ ) ρ n ( λ ) + c ( λ ) ρ n ( λ ) , where ρ ( λ ) = λ + √ λ − pq p and ρ ( λ ) = λ − √ λ − pq p arethe roots of the characteristic equation for the recursion,and c = ρ − λρ − ρ and c = λ − ρ ρ − ρ . Now π = 1 and π n = p n − q n ( n ≥ | ρ ( λ ) | > (cid:112) q/p on [ − , − √ pq ) , | ρ ( λ ) | < (cid:112) q/p on (2 √ pq, , | ρ ( λ ) | = (cid:112) q/p on [ − √ pq, √ pq ] , and ρ ρ = qp . The above will help us to identify thepoint mass locations in the measure ψ since each pointmass in ψ occurs when (cid:80) k π k Q k ( λ ) < ∞ . Thus weneed to find all λ ∈ (2 √ pq,
1] such that c ( λ ) = 0 andall λ ∈ [ − , − √ pq ) such that c ( λ ) = 0. But thereare no such roots, as c ( −
1) = 0 and c (1) = 0, while − (cid:54)∈ (2 √ pq,
1] and 1 (cid:54)∈ [ − , − √ pq ). Thus there areno point mass atoms in ψ , and the mass of ψ must becontinuously distributed inside [ − √ pq, √ pq ]. In orderto find the density of ψ inside [ − √ pq, √ pq ] we need tofind [ e , ( P − sI ) − e ] for Im( s ) (cid:54) = 0, i.e. the upper leftelement in the resolvent of P .Let ( a ( s ) , a ( s ) , . . . ) T = ( P − sI ) − e , then − sa + a = 1 , and qa n − − sa n + pa n +1 = 0Thus a n ( s ) = α ρ ( s ) n + α ρ ( s ) n , with α = a ( ρ − s ) − ρ ( s ) − ρ ( s ) and α = − a ( ρ − s ) ρ ( s ) − ρ ( s ) . Since ( a , a , . . . ) ∈ (cid:96) ( C , π ), | a n | (cid:114) p n q n → n → + ∞ | ρ ( s ) | (cid:54) = | ρ ( s ) | , either α = 0 or α = 0,and therefore a ( s ) = | ρ ( s ) | < √ qp ρ ( s ) − s + | ρ ( s ) | < √ qp ρ ( s ) − s . (B1)Also dψ ( z ) = ϕ ( z ) dz , where ϕ ( z ) is an atom-less densityfunction over [ − √ pq, √ pq ], and a ( s ) = (cid:90) +2 √ pq − √ pq dψ ( z ) z − s = (cid:90) +2 √ pq − √ pq ϕ ( z ) dzz − s . Next we use the following basic property of Cauchy trans-forms Cf ( s ) = πi (cid:82) R f ( z ) dzz − s that can be derived using theCauchy integral formula, or similarly, an approximationto the identity formula: C + − C − = I. (B2)Observe that the curve in the integral need not be in R for C + − C − = I to hold. Here C + f ( z ) = lim s → z : Im( s ) > Cf ( s ) , and C − f ( z ) = lim s → z : Im( s ) < Cf ( s ) , for all z ∈ R . The relation (B2) implies ϕ ( x ) = 12 πi lim s = x + iε : ε → a ( s ) − lim s = x − iε : ε → a ( s ) , for all x ∈ ( − √ pq, √ pq ). Recalling (B1), we express ϕ as ϕ ( x ) = ρ ( x ) − ρ ( x )2 πi ( ρ ( x ) − x )( ρ ( x ) − x ) for x ∈ ( − √ pq, √ pq ),which in turn simplifies to ϕ ( x ) = (cid:40) √ pq − x πq (1 − x ) if x ∈ ( − √ pq, √ pq ) , ϕ ( x ) always integrates to 1 over ( − √ pq, √ pq ).Now p t ( n,
0) = (cid:90) +2 √ pq − √ pq λ t Q n ( λ ) ϕ ( λ ) dλ = (cid:90) +2 √ pq − √ pq λ t ( c ρ n + c ρ n ) ( ρ − ρ ) dλ πi ( ρ − λ )( ρ − λ ) , and therefore, since c = ρ − λρ − ρ and c = λ − ρ ρ − ρ , p t ( n,
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