TTHE QUANTUM WALK OF F. RIESZ
F. A. GR ¨UNBAUM, L. VEL ´AZQUEZ
Abstract.
We exhibit a way to associate a quantum walk (QW)on the non-negative integers to any probability measure on theunit circle. This forces us to consider one step transitions that arenot traditionally allowed. We illustrate this in the case of a veryinteresting measure, originally proposed by F. Riesz for a differentpurpose.For a review of Riesz’s construction and its many uses, see [28,22, 11]. For reviews of quantum walks, see [1, 12, 13]. Introduction and contents of the paper
The purpose of this note is to consider the probability measureconstructed by F. Riesz, [18], back in 1918, and to study a quantumwalk naturally associated to it.The measure on the unit circle that F. Riesz built is formally givenby the expression(1) dµ ( z ) = ∞ (cid:89) k =1 (1 + cos(4 k θ )) dθ π = ∞ (cid:89) k =1 (1 + ( z k + z − k ) / dz πiz = ∞ (cid:88) j = −∞ µ j z j dz πiz . Here z = e iθ . If one truncates this infinite product the correspondingmeasure has a nice density. These approximations converge weakly tothe Riesz measure. Mathematics Subject Classification.
Key words and phrases.
Riesz measure, Laurent orthogonal polynomials, CMVmatrices, quantum random walks.The research of the first author was supported in part by the Applied MathSciences subprogram of the Office of Energy Research,USDOE, under contract DE-AC03-76SF00098.The research of the second author was partly supported by the Spanish grantsfrom the Ministry of Education and Science, project code MTM2005-08648-C02-01,and the Ministry of Science and Innovation, project code MTM2008-06689-C02-01,and by Project E-64 of Diputaci´on General de Arag´on (Spain). a r X i v : . [ m a t h - ph ] N ov GV The recent paper [6] gives a natural path to associate to a quan-tum walk on the non-negative integers a probability measure on theunit circle. This construction is also pushed to quantum walks on theintegers. The traditional class of coined quantum walks considered inthe literature allows for certain one step transitions and this leads to arestricted class of probability measures.In this paper we take the attitude that for an arbitrary probabilitymeasure a slightly more general recipe for these transitions gives riseto a quantum walk. The measure considered by Riesz falls outsideof the more restricted class considered so far, and is used here as aninteresting example.There is an obvious danger that having gone beyond the traditionalclass of quantum walks some of the appealing properties of these walksmay no longer hold. From this perspective we consider the examplediscussed here as a laboratory situation where we will test some of thesefeatures.There is an extra reason for looking at this special example: Riesz’smeasure is one of the nicest examples of purely singular continuousmeasures. This means that the unitary operator governing the evolu-tion of the corresponding quantum walk has a pure singular continuousspectrum. Hence, Riesz’s quantum walk becomes an ideal candidate toanalyze the dynamical consequences of such a kind of elusive spectrum.We first show how to introduce a quantum walk given a probabilitymeasure on the unit circle and then we analyze in more detail the caseof Riesz’s measure, and give some exploratory results pertaining to thelarge time behaviour of the “site distribution” for this non-standardwalk.We are grateful to Prof. Reinhard Werner for pointing out thatF. Riesz actually started the infinite product (1) with k = 0 . There aretwo well known references [22, 8] that use the convention used here.Each choice has its own advantages as will be seen in section 5.For a review of Riesz’s construction and its many uses, see [28, 22,11]. For reviews of quantum walks, see [1, 12, 13].This paper will appear in the Proceedings of FoCAM 2011 heldin Budapest, Hungary, to be published in the London MathematicalSociety lecture Note Series.2.
Szeg˝o polynomials and CMV matrices
Let µ be a probability measure on the unit circle T = { z ∈ C : | z | = 1 } , and L µ ( T ) the Hilbert space of µ -square-integrable functions HE QUANTUM WALK OF F. RIESZ 3 with inner product ( f, g ) = (cid:90) T f ( z ) g ( z ) dµ ( z ) . For simplicity we assume that the support of µ contains an infinitenumber of points.A very natural operator to consider in our Hilbert space is given by(2) U µ : L µ ( T ) → L µ ( T ) f ( z ) −→ zf ( z ) . Since the Laurent polynomials are dense in L µ ( T ), a natural basis toobtain a matrix representation of U µ is given by the Laurent polyno-mials { χ j } ∞ j =0 obtained from the Gram–Schmidt orthonormalization of { , z, z − , z , z − , . . . } in L µ ( T ).The matrix C = ( χ j , zχ k ) ∞ j,k =0 of U µ with respect to { χ j } has theform(3) C = α ρ α ρ ρ . . .ρ − α α − α ρ . . . ρ α − α α ρ α ρ ρ . . . ρ ρ − α ρ − α α − α ρ . . . ρ α − α α ρ α ρ ρ . . . ρ ρ − α ρ − α α − α ρ . . .. . . . . . . . . . . . . . . . . . . . . . . . , where ρ j = (cid:112) − | α j | and { α j } ∞ j =0 is a sequence of complex numberssuch that | α j | <
1. The coefficients α j are known as the Verblunsky(or Schur, or Szeg˝o, or reflection) parameters of the measure µ , andestablish a bijection between the probability measures supported onan infinite set of the unit circle and sequences of points in the openunit disk. The unitary matrices of the form above are called CMVmatrices, see [22, 23, 27].The problem of finding the sequence { α j } for a given measure µ or, more generally, that of relating properties of the measure and thesequence is a central problem, see [22], where a few explicit examplesare recorded. Even in cases when the measure is a very natural one,this can be a hard problem. Back in the 1980’s one of us formulateda conjecture based on work on the limited angle problem in X-raytomography. The same conjecture was also made in a slightly differ-ent context in work of Delsarte, Janssen and deVries. The conjectureamounts to showing that the Verblunsky parameters of a certain mea-sure are all positive. This was finally established in a real tour-de-forcein [14]. GV One of the results of this paper consists of finding these parametersin the case of F. Riesz’s measure. In the process of finding these pa-rameters we will need to invoke some other sequences. Some of thesewill be subsequences of { α j } , and some other ones will only have anauxiliary role. We will propose an ansatz for the Verblunsky param-eters of the Riesz measure that have been checked so far for the first6000 non-null Verblunsky parameters. This is enough for computa-tional purposes concerning the related quantum walk. A proof of ouransatz deserves additional efforts.The decomposition of a measure dµ above into an absolutely contin-uous and a singular part can be further refined by splitting the singularpart into point masses and a singular continuous part. The example ofRiesz that we will consider later will consist only of this third type ofmeasure, and it is (most likely) the first known example of a measureof this kind, built in terms of a formal Fourier series. For the case ofthe unit interval there is a construction of such a singular continuousmeasure in the classical book by F. Riesz and B. Sz-Nagy which is mostlikely due to Lebesgue. Notice that the method of “Riesz products”introduced in [18] can be used to produce measures such as the one thatlives in the Cantor middle-third set. However in this case, as well as inthe one due to Lebesgue, one loses the tight connection with Fourieranalysis that makes the example of Riesz easier to handle.A very important role will be played by the Carath´eodory function F of the orthogonality measure µ , defined by(4) F ( z ) = (cid:90) T t + zt − z dµ ( t ) , | z | < .F is analytic on the open unit disc with McLaurin series(5) F ( z ) = 1 + 2 ∞ (cid:88) j =1 µ j z j , µ j = (cid:90) T z j dµ ( z ) , whose coefficients provide the moments µ j of the measure µ .Another useful tool in the theory of orthogonal polynomials on theunit circle is the so called Schur function related to µ by means of F through the expression f ( z ) = z − ( F ( z ) − F ( z ) + 1) − , | z | < . Since the Schur function, and its Taylor coefficients will play such animportant role, see [9], we will settle the issue of names by sticking tothe name Verblunsky parameters for those that could be also called bythe names of Schur or Szeg˝o.
HE QUANTUM WALK OF F. RIESZ 5
These functions obtained here by starting from a probability mea-sure on T can be characterized as the analytic functions on the unitdisk D = { z ∈ C : | z | < } such that F (0) = 1, Re F ( z ) > | f ( z ) | < z ∈ D , respectively.Starting from f = f , the Verblunsky parameters α k = f k (0) canbe obtained through the Schur algorithm that produces a sequence offunctions { f k } by means of(6) f k +1 ( z ) = 1 z f k ( z ) − α k − α k f k ( z ) . By using the reverse recursion(7) f k ( z ) = zf k +1 ( z ) + α k α k zf k +1 ( z ) = α k + ρ k α k + zf k +1 ( z ) one can obtain a continued fraction expansion for f ( z ). This is calleda “continued fraction-like” algorithm by Schur, [21], and made into anactual one by H. Wall in [26]. See also [22]. We will illustrate thepower of this way of computing these parameters by using it in ourexample to compute (with computer assistance, in exact arithmetic)enough of them so that we can formulate an ansatz as to the form ofthese parameters.3. Traditional quantum walks
We consider one-dimensional quantum walks with basic states | i (cid:105) ⊗|↑(cid:105) and | i (cid:105) ⊗ |↓(cid:105) , where i runs over the non-negative integers, and witha one step transition mechanism given by a unitary matrix U . This isusually done by considering a coin at each site i , as we will see below.One considers the following dynamics: a spin up can move to theright and remain up or move to the left and change orientation. A spindown can either go to the right and change orientation or go to the leftand remain down.In other words, only the nearest neighbour transitions such that thefinal spin (up/down) agrees with the direction of motion (right/left) areallowed. This dynamics bears a resemblance to the effect of a magneticinteraction on quantum system with spin: the spin decides the directionof motion. This rule applies to values of the site variable i ≥ i = 0 to get a unitary evolution. GV Schematically, the allowed one step transitions are | i (cid:105) ⊗ |↑(cid:105) −→ (cid:40) | i + 1 (cid:105) ⊗ |↑(cid:105) with amplitude c i | i − (cid:105) ⊗ |↓(cid:105) with amplitude c i | i (cid:105) ⊗ |↓(cid:105) −→ (cid:40) | i + 1 (cid:105) ⊗ |↑(cid:105) with amplitude c i | i − (cid:105) ⊗ |↓(cid:105) with amplitude c i where, in the case i = 0, the unitarity requirement forces the identifi-cation |− (cid:105) ⊗ |↓(cid:105) ≡ | (cid:105) ⊗ |↑(cid:105) . For each i = 0 , , , . . . ,(8) C i = (cid:18) c i c i c i c i (cid:19) is an arbitrary unitary matrix which we will call the i th coin.If we choose to order the basic states of our system as follows(9) | (cid:105) ⊗ |↑(cid:105) , | (cid:105) ⊗ |↓(cid:105) , | (cid:105) ⊗ |↑(cid:105) , | (cid:105) ⊗ |↓(cid:105) , | (cid:105) ⊗ |↑(cid:105) , | (cid:105) ⊗ |↓(cid:105) , . . . then the transition matrix is given below U = c c c c c c c c c c c c
0. . . . . . . . . . . . . . . and we take this as the transition matrix for a traditional quantumwalk on the non-negative integers with arbitrary (unitary) coins C i asin (8) for i = 0 , , , . . . .The reader will notice that the structure of this matrix is not toodifferent from a CMV matrix for which the odd Verblunsky parametersvanish. This feature will guarantee that in the CMV matrix the central2 × × × µ on the unit circle with the above matrix U . Indeed, U becomes the matrix representation of the operator U µ , defined in (2),with respect to an orthonormal basis of Laurent polynomials X j dif-fering only by constant phase factors e iθ j from the standard ones { χ j } giving the CMV matrix. HE QUANTUM WALK OF F. RIESZ 7
In [6] one considers the case of a constant coin C i for which themeasure µ and the function F ( z ) are explicitly found. In this case,after the conjugation alluded to above, the Verblunsky parameters aregiven by(10) a, , a, , a, , a, . . . for a value of a that depends on the coin, and the function F ( z ) is, upto a rotation of the variable z , given by the function F ( z ) = − z − z − − i Im a (cid:112) ( z − z − ) + 4 | a | − a . The corresponding Schur function is the even function of zf ( z ) = z − (cid:112) ( z − + 4 | a | z az . It is easy to see that, in general, the condition f ( − z ) = f ( z ) isequivalent to requiring that the odd Verblunsky parameters of µ shouldvanish. Traditional coined quantum walks are therefore those whoseSchur function is an even function of z . In terms of the Carath´eodoryfunction the restriction to a traditional quantum walk amounts to F ( − z ) F ( z ) = 1.4. Quantum walks resulting from an arbitraryprobability measure
One of the main points of [6] was to show that the use of the measure µ allows one to associate with each state of our quantum walk a complexvalued function in L µ ( T ) in such a way that the transition amplitudebetween any two sates in time n is given by an integral with respectto µ involving the corresponding functions and the quantity z n . Moreexplicitly we have(11) (cid:104) ˜Ψ | U n | Ψ (cid:105) = (cid:90) T z n ψ ( z ) ˜ ψ ( z ) dµ ( z ) , where ψ ( z ) = (cid:80) j ψ j X j ( z ) is the L µ ( T ) function associated with state | Ψ (cid:105) = (cid:80) j ψ j | j (cid:105) . Here | j (cid:105) is the j -th vector of the ordered basis con-sisting of basic vectors as given in (9), i.e. | j (cid:105) stands for a site and aspin orientation, while X j ( z ) are the orthonormal Laurent polynomialsrelated to the transition matrix of the quantum walk. Similarly ˜ ψ ( z )is the function associated to the state | Ψ (cid:105) .This construction will now be extended to the case of any transi-tion mechanism that is cooked out of a CMV matrix as above. More GV explicitly, we allow for the following dynamics | i (cid:105) ⊗ |↑(cid:105) −→ | i + 1 (cid:105) ⊗ |↑(cid:105) with amplitude ρ i +2 ρ i +3 | i − (cid:105) ⊗ |↓(cid:105) with amplitude ρ i +1 α i +2 | i (cid:105) ⊗ |↑(cid:105) with amplitude − α i +1 α i +2 | i (cid:105) ⊗ |↓(cid:105) with amplitude ρ i +2 α i +3 | i (cid:105) ⊗ |↓(cid:105) −→ | i + 1 (cid:105) ⊗ |↑(cid:105) with amplitude − α i +2 ρ i +3 | i − (cid:105) ⊗ |↓(cid:105) with amplitude ρ i +1 ρ i +2 | i (cid:105) ⊗ |↑(cid:105) with amplitude − α i +1 ρ i +2 | i (cid:105) ⊗ |↓(cid:105) with amplitude − α i +2 α i +3 . The expressions for the amplitudes above are valid for any even i withthe convention |− (cid:105) ⊗ |↓(cid:105) ≡ | (cid:105) ⊗ |↑(cid:105) . If i is odd then in every amplitudethe index i needs to be replaced by i − The Schur function for Riesz’s measure
From the expression for the Riesz measure given earlier we see thatthe expansion dµ ( z ) = ∞ (cid:88) j = −∞ µ j z j dz πiz leads to the moments µ j of the measure µ . Apart form the first one, µ = 1, if j (cid:54) = 0 can be written, in the necessarily unique form, as j = ± k ± k ± · · · ± k p , k > k > · · · > k p ≥ , then µ j = 1 / p . For values of j that cannot be written in the form above we have µ j = 0.In particular for j = 4 k we have µ j = 1 / HE QUANTUM WALK OF F. RIESZ 9
The moments of µ provide the Taylor expansion of the Carath´eodoryfunction F ( z ) = 1 + 2 ∞ (cid:88) j =1 µ j z j , and from this it is not hard to compute the first few terms of the Taylorexpansion of the Schur function f ( z ) around z = 0. Indeed, from F ( z ) = 1 + z + z z + z z z z z z + · · · we get that f ( z ) has the expansion f ( z ) = z − z z z − z − z − z − z
256 + · · · . Only powers differing in multiples of 4 appear in the Taylor ex-pansion of both functions, F and f . This follows from the fact that dµ ( z ) = dν ( z ) with ν given by the same infinite product (1) as µ butstarting at k = 0, i.e. dν ( z ) = (1 + ( z + z − ) / dµ ( z ). From thiswe find that F ( z ) = G ( z ) and f ( z ) = z g ( z ) where G and g are theCarath´eodory and Schur functions of ν respectively.It is now possible, in principle, to compute as many Verblunskyparameters for the function g as one wishes; they are given by thecontinued fraction algorithm given at the end of section 2. The firstfew ones are given below, arranged for convenience in groups of eight.We list separately the first four parameters.1 / − / / − / / − / − / − / / − /
11 21 / − / / − / − / − /
49 1 / − /
51 5 / − / / − / − / − /
57 1 / − / − / − / / − / − / − /
33 1 / − /
35 1 / − / . . . It is clear that we get the Verblunsky parameters of f by introducingthree zeros in between any two values above (a consequence of theargument z in g above) and then shifting the resulting sequence byadding three extra zeros at the very beginning (a consequence of thefactor z in front of g above), yielding finally the following sequence ofVerblunsky parameters for f , where each row contains eight coefficients starting with α in the first row, α in the second one, etc.0 0 0 1 / − /
30 0 0 5 / − /
130 0 0 1 /
14 0 0 0 − /
150 0 0 − / − /
90 0 0 1 /
10 0 0 0 − /
110 0 0 21 /
32 0 0 0 − /
530 0 0 1 /
54 0 0 0 − /
550 0 0 − /
52 0 0 0 − /
490 0 0 1 /
50 0 0 0 − /
510 0 0 5 /
56 0 0 0 − / . . . The non-zero Verblunsky parameters of f are given by ξ m ≡ α m − , m = 1 , , , . . . where the sequence { ξ m } will be determined below. In fact it will beenough to determine the subsequence { ξ n } since all the other valuesof ξ m can be given by simple expressions in terms of these.The expression for these ξ n ≡ α n is given by α n = − /A n +1 , n = 0 , , , . . . for a sequence of integer values { A n } to be described below. We willlater give a different description of the complete sequence { ξ m } whichmakes clear what its limit points are and obviates the need to considerthe subsequence { ξ n } .We will first describe a procedure that allows us to generate theinfinite sequence of integers { A n } ∞ n =1 of which the first ones are13 , , , , , , , , , , , , , , , , , . . . Once this sequence is accounted for, i.e. if the Verblunsky parametersof the form α n are known, we will see that all the remaining onesare determined by simple explicit formulas in terms of { A n } . For thisreason we will refer to the sequence { A n } to be constructed in the nextsection as the backbone of the sequence { α n } we are interested in.As we noted at the end of introduction, F. Riesz included the factorcorresponding to k = 0 in the infinite product (1). That is, the measureconsidered by Riesz is the measure ν giving our measure µ (starting theinfinite product with k = 1) by replacing z by z , and the correspondingSchur function is g . Therefore, the Verblunsky parameters { α Rn } thatF. Riesz would have are those obtained deleting in the sequence { α n } the groups of three consecutive zeros, so α Rn = α n +3 = ξ n +1 , and all ofthem should be computed from α R n − = − /A n . HE QUANTUM WALK OF F. RIESZ 11
The main difference between including the factor k = 0 in (1) orleaving it out is the inclusion of many zeros in the list of Verblunskyparameters in the second case, which is the one we choose. This makesfor a much sparser CMV matrix which is easier to analyze than itwould be in the original case of F. Riesz. On the other hand his choiceis better for computational purposes when, of necessity, one has todeal with truncated matrices. In Riesz’s case there is more informationpacked in the same size finite matrix. This point is exploited in someof the graphs displayed at the end of the paper.6. Building the backbone
Consider the sets v j , j ≥
0, defined as the ordered set of integers ofthe form (( − j − / k j +1 where k runs over the integers. As an illustration we give a few elementsof the sets v , v , v , v , . . . , v , namely v = . . . , − , − , − , , , , , . . . ; v = . . . , − , − , − , − , , , , , . . . ; v = . . . , − , − , − , , , , , . . . ; v = . . . , − , − , − , , , , . . . ; v = . . . , − , − , − , , , , , . . . ; v = . . . , − , − , − , − , − , , , , . . . ; v = . . . , − , − , − , − , , , , , . . . ; v = . . . , − , − , − , − , − , , , , . . . ; v = . . . , − , − , − , − , , , , . . . ; v = . . . , − , − , − , , , , . . . ; v = . . . , − , − , − , , , , . . . . These sets v j , j ≥
0, are disjoint and their union gives all integers.A simple argument to prove this was kindly supplied by B. Poonen.We observe that d j , defined as the first positive element in theinfinite set v j (corresponding either to the choice k = 0 or k = 1) isgiven as follows: if j = 0 then d = 2 otherwise, for n ≥ d j = (( − j − / − ( − j )2 j . Define now, for n ≥ c n = 8 + (( − n − − / c , c , c , c , . . . are given by8 , − , , − , , − , , − , , . . . a sequence whose first differences are given by( − n +1 , n = 4 , , , . . . . For each pair j, n , j ≥ n ≥
0, define w j,n as the number of positiveelements in the sequence { v j } that are not larger than n . Notice that (cid:80) ∞ j =0 w j,n = n and that w j, is zero for all j. To be very explicit, we have for instance w , = 20, w , = 10, w , = 5, w , = 2, w , = 2, w , = 0, w , = 1, and all values of w j, after these ones vanish.It is possible to give an expression for w j,n in terms of the sequence { d j } defined above. In fact one has w j,n = (cid:22) n + 2 j +1 − d j j +1 (cid:23) where we use the notation (cid:98) x (cid:99) to indicate the integer part of the quan-tity x. We are finally ready to put all the pieces together and define, for n ≥ u n = ∞ (cid:88) j =0 w j,n c j . Notice that by definition this is a finite sum since, for a given n theexpression w j,n vanishes if j is large enough.The reader will have no difficulty verifying that we get(13) A i = 13 + u i − , i ≥ . Recall that we have, for n ≥ α n = − A n +1 and, as mentioned earlier, we will see how all other values of α j can bedetermined from these ones.7. Building the sequence from its backbone
We have observed already that the first non-zero Verblunsky pa-rameters of our measure are given by α = 1 / , α = − / , α = 5 / , α = − / , . . . . We will see now that, for i ≥
15 and using the sequence { A i } builtabove, we have a way of computing the non-zero values of α j . Start HE QUANTUM WALK OF F. RIESZ 13 by observing that, after α = − /A we get for values of j between j = 16 and j = 47 the following non-zero Verblunsky parameters: α = 11 + A , α = −
12 + A , followed by α = − A − , α = 1 A − , α = − A − , and finally α = − A . The reader will have noticed that we did not give a prescription for α or for α . This is done now: α = − A − , α = A − A + 2 A + A − . We have seen that the non-zero values of α j for j in between j = 15and j = 47 are all obtained from the values of A and A . We claimthat exactly the same recipe apply for values of j in the range from16(2 p − − p + 1) −
1, with p ≥
1, namely we set α p − − = − A p , α p +1) − = − A p +1 and fill in the SEVEN non-zero values of α j in between these two byusing expressions that are extensions of the ones above, namely, α p − = 11 + A p , α p − = −
12 + A p ,α p − = − A p − , α p − = 1 A p − ,α p − = − A p − , α p − = − A p +1 , and, just as above, the missing Verblunsky parameters are given by α p − = − A p − , α p − = A p +1 − A p + 2 A p +1 + A p − . A different expression for the Verblunsky parameters
The construction above gives as many non-zero Verblunsky param-eters for our measure as one wants, starting with α , α , α , α , . . . for which we get the values 1 / , / , / , − / , . . . .In this section we give an explict formula for these Verblunsky pa-rameters in terms of a sequence of constants { K i } , i = 0 , , , . . . , which are closely related to the sequence { A i } introduced above. Oneof the advantages of this new expression is that the set of limit values ofthe sequence { α j } becomes obvious and is given by the union of threeinfinite sets, namely − K i , i = 1 , , , . . . , K i + 3 , i = 0 , , , , . . . , − K i + 6 , i = 0 , , , , . . . . where the constants K i are given by K = 3 ,K i − = 3 A i , K i = 3( A i − , i = 1 , , , . . . . One needs to add the limit points of the three infinite sets given aboveto get all limit points of the sequence { α j } .We are now ready to give the alternative expressions for the non-zero Verblunsky parameters alluded to above, i.e. ξ n so that ξ = 1 / , ξ = − / , ξ = 5 / , . . . , and in general α m − = ξ m . For this purpose we consider a disjointunion of the set of all non-negative integers into sets B n , where theindex n runs over the set 1 , , , , , , . . . , i.e. all positive n (cid:54) = 3(mod 4) . For each such n , define B n as the set of integers of the form13 + 4 p n − , p = 0 , , , , . . . . Once again, a simple proof of these properties of the sets B n was sup-plied by B. Poonen.The sets B n break naturally into three classes, with n ≡ n ≡ n ≡ ξ +4 p n − = − K s (cid:18) p (cid:19) , n = 4 s, s = 1 , , , . . . ,ξ +4 p n − = 1 K s + 3 (cid:18) − p (cid:19) , n = 4 s + 1 , s = 0 , , , . . . ,ξ +4 p n − = − K s + 6 (cid:18) p (cid:19) , n = 4 s + 2 , s = 0 , , , . . . . HE QUANTUM WALK OF F. RIESZ 15
From the expressions above it follows that we have identified thelimit values of the sequence { α m − } . The largest one is 2 / − / Some properties of the Riesz quantum walk
Once we get our hands on the Verblunsky parameters correspond-ing to the Riesz measure we construct the corresponding CMV matrixand we can compute different quantities pertaining to the associatedquantum walk. In the rest of the paper we choose to illustrate some ofthese results with a few plots.Figures 1 and 2 display the Verblunsky parameters themselves, keyingredients in the one-step transition amplitudes of the Riesz quan-tum walk. They show an apparent chaotic behaviour which is actu-ally driven by the rules previously described which generate the fullsequence of Verblunsky parameters. This is in great contrast to thetranslation invariant case of a constant coin.Figures 3 and 5 display the Taylor coefficients of the Schur functionfor Riesz measure and the Hadamard quantum walk with constant coin C = 1 √ (cid:18) − (cid:19) on the non-negative integers. These coefficients have an importantprobabilistic meaning discussed in great detail in the forthcoming paper[9]: the n -th Taylor coefficient is the first time return amplitude in n steps to the state with spin up at site 0.The first time return amplitudes for the Riesz walk seem to fluctuatein an apparent random way around a mean value which must decreasestrongly enough to ensure that the sequence is square-summable, asFigure 6 makes evident. This is because the sum of the first returnprobabilities is the total return probability, which cannot be greaterthan one. Equivalently, any Schur function is Lebesgue integrable onthe unit circle with norm bounded by one, and its norm is the sum ofthe squared moduli of the Taylor coefficients.In the Hadamard case the behaviour of the first time return am-plitudes is much more regular and the convergence to the total returnprobability, depicted in Figure 4, holds with a much higher speed. Weshould remark that the plot for the Hadamard walk picks up only thefirst 70 non-null coefficients, while the Riesz picture represents the firstnon-null 7000 coefficients. Hence, the differences between these twoexamples are not only in the more regular pattern that the Hadamardreturn probabilities exhibit, but also in the much higher tail for theRiesz return probabilities. Figures 7 and 8 give the probability distribution of the randomvariable X n /n , where X n stands for the position (regardless of spinorientation) after n steps of the quantum walk started at position 0with a spin pointing up. The plots given here correspond to the value n = 800 for both, the Riesz quantum walk and the Hadamard constantcoin on the non-negative integers.The figures show that, in contrast to classical random walks forwhich X n behaves tipically as √ n , the position in a quantum walk cangrow linearly with n . Nevertheless, Figure 8 shows a striking behaviourof the Riesz walk compared to the more regular asymptotics of theHadamard walk reflected in Figure 7. This should be viewed as aclear indication of the anomalous behaviour that can appear underthe presence of a singular continuous spectrum. In particular, theseresults make evident that quantum walks with a singular continuousmeasure can not exhibit nice limit laws as other toy models do. Forthe case of translation invariant ones it is known that obey invertedbell asymptotic distributions (see for instance [13]).These results should motivate a more detailed analysis of quantumwalks associated with singular continuous measures. This could leadto the discovery of new interesting quantum phenomena. HE QUANTUM WALK OF F. RIESZ 17 y x
Figure 1.
Riesz’s measure: The first 30000 non-zeroVerblunsky parameters. y x
Figure 2.
Riesz’s measure: The non-zero Verblunskyparameters for indices between 30000 and 60000.
HE QUANTUM WALK OF F. RIESZ 19 y x
Figure 3.
Hadmard’s walk: The first 70 non-zero Tay-lor coefficients of the Schur function. y x
Figure 4.
Hadamard’s walk: The cumulative sums ofthe squares of the first 70 non-zero Taylor coefficients ofthe Schur function.
HE QUANTUM WALK OF F. RIESZ 21 y x
Figure 5.
Riesz’s measure: The first 7000 non-zeroTaylor coefficients of the Schur function. y x
Figure 6.
Riesz’s measure: The cumulative sums of thesquares of the first 7000 non-zero Taylor coefficients ofthe Schur function in steps of 100.
HE QUANTUM WALK OF F. RIESZ 23 y x
Figure 7.
Hadamard’s walk on the non-negative inte-gers: The probability of X n /n for 800 iterations startingat | (cid:105) ⊗ |↑(cid:105) . y x Figure 8.
Riesz’s walk on the non-negative integers:The probability of X n /n for 800 iterations starting at | (cid:105) ⊗ |↑(cid:105) . HE QUANTUM WALK OF F. RIESZ 25
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Department of Mathematics, University of Cali-fornia, Berkeley, CA, 94720 (L. Vel´azquez)(L. Vel´azquez)