Conditional Probability Distributions of Finite Absorbing Quantum Walks
CConditional Probability Distributions of Finite Absorbing Quantum Walks
Parker KuklinskiJanuary 30, 2020
Abstract
Quantum walks are known to have nontrivial interactions with absorbing boundaries. In particular it hasbeen shown that an absorbing boundary in the one dimensional quantum walk partially reflects information, asobserved by absorption probability computations. In this paper, we shift our sights from the local phenomena ofabsorption probabilities to the global behavior of finite absorbing quantum walks in one dimension. We conductour analysis by approximating the eigenbasis of the associated absorbing quantum walk operator matrix Q n where n is the lattice size. The conditional probability distributions of these finite absorbing quantum walksexhibit distinct behavior at various timescales, namely wavelike reflections for times t = O ( n ), fractional quantumrevivals for t = O ( n ), and stability for t = O ( n ). At the end of this paper, we demonstrate the existence offractional quantum revivals in other sufficiently regular quantum walk systems. The quantum walk is a unitary analogue of the classical random walk. Where classical random walks have beenused in algorithms for classical computers [34] [17] [37] [18], quantum walks can be used in algorithms for quantumcomputers, providing various degrees of speedup over their classical counterparts [23] [15] [14] [3] [4]. As opposedto the standard deviation of the random walk which grows at O ( √ t ), the standard deviation of the quantum walkgrows at O ( t ) and the wave function has highly oscillatory behavior close to the wave fronts. The quantum walkhas been studied in a variety of purely mathematical contexts [25] [43] and has also been physically implementedin a number of settings [40] [47]. In this paper, we will explore the asymptotic behavior of finite quantum walkswith absorbing boundaries.Quantum walks with absorbing boundaries were first studied in relation to absorption probabilities [5] [27]. Let p ∞ be the probability that a Hadamard walk particle initialized in | (cid:105)| R (cid:105) is eventually absorbed at | (cid:105) , and let p n bethe probability that this particle is absorbed at | (cid:105) and not by an additional absorbing boundary at | n (cid:105) . Ambainiset. al. [5] found that p ∞ = π and lim n →∞ p n = √ . This paradoxical result (i.e. lim n →∞ p n > p ∞ ) indicates thatabsorbing boundaries in the quantum walk setting partially reflect information. A sharper result conjectured bythese authors and later proved by Bach and Borisov [6] states that p n +1 = p n p n . These results were extended tothe three-state Grover walk [44], two-state quantum walks [31], and general discrete quantum mechanical systems[28] [30]. Other authors have considered hitting times, or the mean expected time that a quantum walk particleis first observed at an absorbing boundary [20] [42] [46]. Absorption probabilities and hitting times, however, areconcerned with local behavior at an absorbing boundary as opposed to the global behavior that we wish to study.The quantum walk can be defined as a linear combination of translations, so it is natural to view the quantumwalk operator on a finite domain as a matrix. In the case of the quantum walk with absorbing boundaries,the quantum walk operator matrix is a composition of a unitary operator with a Hermitian projection operatorwhich projects off of the absorbing boundary locations. The asymptotic behavior of a finite quantum walk is bestunderstood by computing the eigenbasis of the operator matrix. To compute the eigenvalues of a quantum walkmatrix, we first calculate its characteristic polynomial p n ( λ ) where n is the size of the domain. These characteristicpolynomials satisfy a second order recursion which we exploit to compute the eigenvalues up to polynomial order.Similar techniques are used in the computation of the eigenvectors.The eigenvalues of Q n uniformly approach two sectors of the unit circle at O ( n − ) as n increases. At t = O ( n ),the top eigenvalues of Q tn begin to dominate the system. More interestingly, the minimum phase difference betweenthe k th top eigenvalue of Q n and one of the points ±| a | ± i | b | ( a, b determined by Q n ) is roughly k n α for small1 a r X i v : . [ qu a n t - ph ] J a n and some constant α . Thus, there exist times t for which the top eigenvalues of Q tn approximately align inthe complex plane in various patterns. More specifically, there exists a value τ ∈ R + such that Q τn n becomes acrude approximation of the identity matrix. For sufficiently simple rational multiples z ∈ Q of τ (i.e. z = p/q such that p, q are small), the matrix Q zτn n becomes a weighted sum of approximations of vector reversals andtranspositions. As z increases, the granularity of these approximations decreases. For an initial state Ψ describinga delta potential directly between the absorbing boundaries, the situation becomes more visually striking. For m ∈ N sufficiently small, Q mτ/ n Ψ reproduces an approximation of Ψ . Furthermore for p, q ∈ N sufficientlysmall, Q pτ/ qn Ψ produces an approximation of q evenly spaced delta potentials in the domain. This behavior isan approximation of fractional quantum revivals which have been observed in several quantum settings. [9] [10]The rest of the paper is organized as follows: section 2 is dedicated to defining the finite absorbing quantumwalk and its matrix representation. In section 3 we approximate the eigensystem of the one-dimensional absorbingquantum walk operator. In section 4 we use these approximations to describe the approximate fractional quantumrevival behavior at t = O ( n ). In section 5 we demonstrate the existence of these quantum revivals in a twodimensional absorbing quantum walk. To begin, we recount the quantum walk on groups as first defined by Acevedo et. al. [1]. The following definitionshave appeared in previous works by Kuklinski [29] [31] [30].
Definition 2.1
Let ( G, · ) be a group, let Σ ⊂ G where | Σ | = n , and let U ∈ U ( n ) where U ( n ) is the set of n × n unitary matrices. The quantum walk operator Q : (cid:96) ( G × Σ) → (cid:96) ( G × Σ) corresponding to the triple ( G, Σ , U ) maybe written as Q = T ( I ⊗ U ) where for g ∈ G and σ ∈ Σ , T : | g (cid:105)| σ (cid:105) (cid:55)→ | g + σ (cid:105)| σ (cid:105) . We denote this correspondenceas Q ↔ ( G, Σ , U ) . The pair ( G, Σ) can be thought of as an undirected Cayley graph which admits loops [16].We must also define an absorption unit for quantum walks. To this end, we formally define the measurementoperator. Let b ∈ G × Σ be the location of an absorption unit. The measurement operator Π b yes : (cid:96) ( G × Σ) → (cid:96) ( G × Σ) is a projection onto | b (cid:105) while Π b no is a projection onto the the subspace spanned by elements in ( G × Σ) \ b .The probabilistic interpretation of quantum mechanics dictates that if we measure a state ψ ∈ (cid:96) ( G × Σ) at b , theresulting state becomes Π b yes ψ/ (cid:107) Π b yes ψ (cid:107) with probability (cid:107) Π b yes ψ (cid:107) and Π b no ψ/ (cid:107) Π b no ψ (cid:107) with probability (cid:107) Π b no ψ (cid:107) .If B ⊂ G × Σ, let Π B no be the composition of no measurement projections for all b ∈ B . In this way, we can definean operator for the absorbing quantum walk. Definition 2.2
Let Q ↔ ( G, Σ , U ) be a quantum walk operator and let B ⊂ G × Σ be a collection of absorptionunits. Then we say that Π B no Q is the absorbing quantum walk operator corresponding to the ordered quadruple ( G, Σ , U, B ) and we denote this correspondence as Π B no Q ↔ ( G, Σ , U, B ) . We use the no operator in our definition because if we observe the particle somewhere in B , then the experimentis terminated, while if the particle is not observed in B (i.e. we are in the range of Π B no ) the experiment continues.Note that we speak of the absorption units as being elements of the classical space and not as members of thecorresponding orthonormal basis.In this paper, we are interested in the computing the probability distribution of an absorbing quantum walkΠ B no Q ↔ ( G, Σ , U, B ) on G × Σ conditioned on the particle not being absorbed by B . If ψ ∈ (cid:96) ( G × Σ) is an initialcondition, then we are interested in calculating the following function P t : G × Σ → [0 , P t ( x ) = |(cid:104) x | (Π B no Q ) t | ψ (cid:105)| (cid:107) (Π B no Q ) t ψ (cid:107) . (1)However, for the absorbing quantum walk it is more natural to perform analysis on the probability amplitudespace before converting to the probability space, bypassing the need for renormalization at every time t . If G isfinite, we can represent the operator Π B no Q ↔ ( G, Σ , U, B ) as a | G | · | Σ | × | G | · | Σ | matrix.It will often occur that we need to take large powers of 2 × emma 2.1 Let M = (cid:20) a bc d (cid:21) , λ ± = (cid:104) a + d ± (cid:112) ( a + d ) − ad − bc ) (cid:105) , and F n = λ n + − λ n − . Then M n = 1 F (cid:20) F n +1 − dF n bF n cF n F n +1 − aF n (cid:21) (2) and F n +2 − ( a + d ) F n +1 + ( ad − bc ) F n = 0 . We now compute the eigensystem of the one-dimensional two-state absorbing quantum walk. The correspondingquantum walk operator can be written as Q n = Π Π n no Q ↔ ( Z , C , U, { , n } ), which we represent as a 2 n × n unitary matrix acting as Q n Ψ = U − . . . U + U − U + U − U + ψ ......... ψ n (3)where C = {− , } , U = U + + U − , U + = (cid:20) a b (cid:21) , U − = (cid:20) − ¯ b ¯ a (cid:21) , | a | + | b | = 1, and ψ k = [ ψ R ( k ) , ψ L ( k )] (cid:48) . Tocompute the eigenvalues of Q n we first calculate the characteristic polynomial. Proposition 3.1
Let p n ( λ ) = det ( λI − Q n ) . These characteristic polynomials satisfy the following recursion: p n +1 ( λ ) = ( λ + 1) p n ( λ ) − | a | λ p n − ( λ ) (4) Here, p ( λ ) = 1 and p ( λ ) = λ . Proof:
We conduct a recursive cofactor expansion on the matrix A n = λI − Q n . Let [ M ] ij be the ij - minor of M , or the matrix resulting from the deletion of the i th row and j th column. Let B n = [ A n ] , C n = [ B n +1 ] , and D n = [ C n ] . Letting A (cid:48) n = det A n and likewise for the other matrices, we find the following recursions: A (cid:48) n = λB (cid:48) n , B (cid:48) n = λA (cid:48) n − + bC (cid:48) n − , C (cid:48) n = ¯ bB (cid:48) n + ¯ aD (cid:48) n D (cid:48) n = aA (cid:48) n − We arrive at the result by rewriting these recursions strictly in terms of A (cid:48) . (cid:3) While in the case of a Chebyshev recursion we are able to use a trigonometric substitution to easily facilitatelocating the roots of the polynomial, this procedure will not work here due to the initial conditions [21] [39] [8].We instead must reference a set with no elementary analytic representation to describe the eigenvalues.
Theorem 3.1
Let Θ ± n = { θ ∈ C : sin nθ = − y sin θ, < Re θ < π, ± Im θ > } with y = | a || b | . Then the set Λ n of eigenvalues of Q n may be written as Λ n = { } ∪ Λ + n ∪ Λ − n where Λ ± n = {| a | cos θ ± i (cid:112) − | a | cos θ : θ ∈ Θ ± n } . (5) The eigenvalue λ = 0 has multiplicity 2. Proof:
Using Lemma 2.1, we can derive a closed form for the characteristic polynomial: p n ( λ ) = λ F ( λ ) (cid:2) F n ( λ ) − | a | F n − ( λ ) (cid:3) . (6)Here, F n ( λ ) = ω + ( λ ) n − ω − ( λ ) n and ω ± ( λ ) = (cid:104) λ + 1 ± (cid:112) ( λ + 1) − | a | λ (cid:105) . The factor of λ accounts forthe eigenvalue at λ = 0 of multiplicity 2. Using the substitution λ ± ( θ ) = | a | cos θ ± i (cid:112) − | a | cos θ , we have F n ( λ ± ( θ )) = ± i ( | a | λ ± ( θ )) n sin nθ . Substituting this into the nontrivial factor and squaring gives us the equation:sin nθ = − y sin θ θ ∈ Θ ± n satisfy F n ( λ ± ( θ )) − | a | F n − ( λ ± ( θ )) = 0. (cid:3) We provide additional results to better visualize the location of the eigenvalues, the first of which gives auniform bound on Λ n . Proposition 3.2
Let S n be the set defined as follows: S ± n = { } ∪ { λ ∈ C : r ( n ) < λ < , − φ < arg ( ± λ ) < φ } (7) where r ( n ) = 12( c ( n ) − (cid:2) (4 | a | − − (4 | a | − c + (cid:112) [(2 | a | + 1) c − (4 | a | + 4 | a | − | a | − c − (4 | a | − | a | − (cid:105) ,c ( n ) = (cid:16) | a | −| a | (cid:17) n − , and e iφ = | a | + i | b | . Then Λ n ⊂ ( S + n ∪ S − n ) . Proof:
Following the argument of Proposition 2.2 in Kuklinski [30], suppose v is an eigenvector of P U witheigenvalue | λ | = 1 where U is unitary and P is a projection. Then v must also be an eigenvalue of U . FromKuklinski [29], the eigenvectors of the corresponding quantum walk operator with periodic boundary conditionssatisfy (cid:107) P v (cid:107) < (cid:107) v (cid:107) where P is the projection associated with the absorbing boundaries at | (cid:105) and | n (cid:105) and (cid:107)·(cid:107) isthe (cid:96) norm. Therefore, all eigenvalues of the absorbing quantum walk operator must satisfy | λ | < p n ( λ ) = 0 and λ (cid:54) = 0. By expanding F n in terms of ω ± in equation (5), we have:0 = (cid:12)(cid:12)(cid:12)(cid:12) ω − ( λ ) − | a | ω + ( λ ) − | a | (cid:12)(cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12)(cid:12) ω + ( λ ) ω − ( λ ) (cid:12)(cid:12)(cid:12)(cid:12) n − = | f ( λ ) | − | g ( λ ) | n − . If we can find a set S ⊂ D such that max λ ∈ S | f ( λ ) | ≤ min λ ∈ S | g ( λ ) | n − , then λ / ∈ S . Since f ( λ ) is analytic on theunit disk, it obtains its maximum absolute value on the unit circle [2], and this value is obtained at λ = ± i suchthat | f ( λ ) | ≤ | a | −| a | . We also find that for fixed | λ | , the function g ( λ ) = ω + ( λ ) ω − ( λ ) achieves its minimum absolute valueat ± i | λ | such that min | λ |≤ R | g ( λ ) | = − g ( iR ). The formula for r ( n ) follows by solving ( − g ( iR )) n − = (cid:16) | a | −| a | (cid:17) for R . Through direct computation, we can prove that | f ( λ ) | < | g ( λ ) | > A ± = { λ ∈ C : − φ < arg ( ± λ ) < φ } , thus completing the proof. (cid:3) By the proposition, the eigenvalues of the absorbing quantum walk operator uniformly limit to the unit circleas n increases. Using the identity lim n →∞ n ( x /n −
1) = log x , we can show that the following approximation onour bound holds: r ( n ) = 1 − | a | n log 1 + | a | − | a | + O ( n − )Furthermore, the eigenvalues are restricted to two sectors of the unit disk symmetric about the real axis; as | a | increases these sectors become larger. These features of the eigenvalues can be observed in Figure 1.We now make asymptotic approximations on the eigenvalues of Q n . First we write an asymptotic descriptionof elements of Θ n , which may be verified via direct computation. Lemma 3.1
Consider the set Θ ± n from Theorem 4.1 with elements θ ± k,n ∈ Θ ± n for k ∈ { , ..., n − } . For fixed k ,let x = πk and y = | a || b | such that: θ ± k,n = xn ± ixyn − xy n ∓ ixyn (cid:18) iy + x x y (cid:19) + O ( n − ) . (8) For α = kn fixed, we have: θ ± αn,n = πα ± in sinh − ( y sin πα ) + O ( n − ) . (9)4 roof: Recall that we are solving the equation sin nθ = − y sin θ , where the roots of positive imaginary realpart belong to Θ + n and the roots of negative imaginary real part belong to Θ − N . Otherwise, we are solving twoequations sin nθ = ± iy sin θ (here, ± is independent of Θ ± n ) and separating solutions into the sets Θ ± n afterwards.By using the representation θ = θ n + O ( n − ), we find that sin θ + O ( n − ) = O ( n − ), and therefore θ = πk forsome k ∈ Z . By further considering an N th order approximation of θ = (cid:80) Nj =1 θ j n − j + O ( n − ( N +1) ) and using anangle sum identity, we have( − k sin N − (cid:88) j =1 θ j +1 n j + O ( n − N ) = ± iy sin N − (cid:88) j =1 θ j n j + O ( n − N )Each choice of k corresponds to two root approximations dictated by the ± sign on the right hand side. If weeliminate the factor of ( − k on the left side, then the two root approximations indexed by the ± sign correspondto roots with positive and negative imaginary part respectively, and thus are in direct correspondence with thesets Θ ± n . By choosing N = 4 and using a Taylor expansion of sin θ , we can write: θ n + θ n + θ n − (cid:18) θ n (cid:19) + O ( n − ) = ± iy (cid:32) θ n + θ n + θ n − (cid:18) θ n (cid:19) (cid:33) + O ( n − )By equating like factors of n − j , we have the following system of three equations: θ = ± iyθ , θ = ± iyθ , θ − θ ± iy (cid:18) θ − θ (cid:19) Solving this system gives the first result. If at the outset we instead fix α = kn , we instead have the expansion θ = πα + θ n + O ( n − ), thus giving the equationsin (cid:18) θ n (cid:19) = ± iy sin πα Solving this for θ gives us the second result. (cid:3) For the remainder of the paper we let y = | a | / | b | . This lemma allows us to index our approximations ofthe nonzero eigenvalues of Q n as λ ± k,n = λ ± ( θ ± k,n ). It should be noted that θ = π + θ ± k,n is also a solution tosin nθ = ± iy sin θ . These solutions, when passed through the function λ ± ( θ ), represent eigenvalues which convergeto − e ± iφ . Since the characteristic polynomial p n ( λ ) in equation (3) is an even function, we will omit mention ofthese solutions without loss of generality. When presenting asymptotic characterizations of these eigenvalues, wedistinguish between the convergence of the k th eigenvalue to ± e ± iφ (as given by λ k,n ) and the uniform convergence ofthe collection of eigenvalues to arcs of the unit circle as described by Proposition 3.2 (these eigenvalues representedas λ αn,n ). We summarize this in the following proposition: Theorem 3.2
For fixed k , we write λ ± k,n ∈ Λ n from Theorem 4.1 as follows: λ ± k,n = e ± iφ (cid:18) ∓ ix y n − x y n − x yn (cid:20) ± y + x (cid:0) y ± i (3 y + 1) (cid:1)(cid:21)(cid:19) + O ( n − ) . (10) For | α | < with αn ∈ Z , we can write λ ± αn,n = e ± if ( α ) (cid:32) − n | a | sin πα (cid:112) − | a | cos πα sinh − (cid:18) | a || b | sin πα (cid:19)(cid:33) + O ( n − ) (11) where e ± if ( α ) = | a | cos πα ± i (cid:112) − | a | cos πα . Proof:
We can formally write θ ± k,n = θ ± (cid:0) n (cid:1) where the coefficient of n − m in equation (8) is represented by θ ( m ) ± (0) /m !. In this way, we can write λ ± k,n = λ ± (cid:0) θ ± (cid:0) n (cid:1)(cid:1) . If we abuse notation and define, for the moment,5igure 1: Plots of the eigenbasis for n = 200 and a = √ ( Left ) Location of eigenvalues (
Right ) Plots of | v ± k,n | λ = λ ± (0), θ = θ ± (0), and likewise for higher derivatives, then by the chain rule we have: λ ± k,n = λ + θ (cid:48) λ (cid:48) n + θ (cid:48)(cid:48) λ (cid:48) + ( θ (cid:48) ) λ (cid:48)(cid:48) n + θ (cid:48)(cid:48)(cid:48) λ (cid:48) + 3 θ (cid:48) θ (cid:48)(cid:48) λ (cid:48)(cid:48) + ( θ (cid:48) ) λ (cid:48)(cid:48)(cid:48) n (12)+ θ (cid:48)(cid:48)(cid:48)(cid:48) λ (cid:48) + (4 θ (cid:48) θ (cid:48)(cid:48)(cid:48) + 3( θ (cid:48)(cid:48) ) ) λ (cid:48)(cid:48) + 6( θ (cid:48) ) θ (cid:48)(cid:48) λ (cid:48)(cid:48)(cid:48) + ( θ (cid:48) ) λ (cid:48)(cid:48)(cid:48)(cid:48) n + O ( n − )Lemma 3.1 gives us the “derivatives” θ ( m ) , while implicit differentiation of the equation λ ± ( θ ) − | a | λ ± ( θ ) cos θ +1 = 0 with respect to θ gives us the derivatives λ ( m ) . Since λ ( θ ) is an even function, we need only consider theeven derivatives: λ = e ± iφ , λ (cid:48)(cid:48) = ± iye ± iφ , λ (4) = − ye ± iφ (cid:2) y ± i (3 y + 1) (cid:3) Making these substitutions gives us the first result. To arrive at the second, we use a Taylor approximation of λ ± ( θ ) at θ = πα instead of at θ = 0 as in the previous case. Combining equation (12) with the approximations inLemma 3.1 gives us the second result. (cid:3) With the results from Theorem 3.2, we can prove that the absorbing quantum walk reaches a steady state attime t = O ( n ). Proposition 3.3
Let (cid:15) > be fixed. There exists T = log(1 /(cid:15) ) π y ( k − n + O ( n ) such that for all t > T , we have (cid:32) | λ ± k,n || λ ± ,n | (cid:33) t < (cid:15) This proposition shows that the magnitude of the ratio between the largest and k th largest eigenvalues of Q tn becomes arbitrarily small at times t = O ( n ). This implies that the largest eigenmode becomes dominant at thistimescale. Compare this behavior to the classical random walk which reaches a stable distribution at t = O ( n ).Using similar techniques we can compute entries of the corresponding eigenvectors. Theorem 3.3
Let v ± k,n = [ r ± k,n, , l ± k,n, , ..., r ± k,n,n , l ± k,n,n ] (cid:48) be the eigenvector of Q n corresponding to eigenvalue λ ± k,n .By letting j = nβ for | β | < fixed, we can write these eigenvectors as (cid:34) r ± k,n,βn l ± k,n,βn (cid:35) = (cid:18) a | a | (cid:19) nβ (cid:18)(cid:20) ¯ ab ± i | a || b | (cid:21) sin πkβ + πkn (cid:20) ¯ ab ( ± iβy − | a | ( β − (cid:21) cos πkβ + O ( n − ) (cid:19) (13)6 roof: We find that the entries of v ± k,n satisfy a matrix multiplication recursion: (cid:34) r ± k,n,j +1 l ± k,n,j +1 (cid:35) = 1 λ ± k,n (cid:34) a b a ¯ b ¯ a ( λ ± k,n ) + | b | ¯ a (cid:35) (cid:34) r ± k,n,j l ± k,n,j (cid:35) Using Lemma 2.1 gives us the expression: (cid:34) r ± k,n,j l ± k,n,j (cid:35) = 1(¯ aλ ) j F (cid:20) F j − ( λ + | b | ) F j − ¯ abF j − a ¯ bF j − F j − | a | F j − (cid:21) (cid:34) r ± k,n, l ± k,n, (cid:35) Since the nonzero eigenvalues must satisfy r ± k,n, = 0, we have: (cid:34) r ± k,n,j l ± k,n,j (cid:35) = 1(¯ aλ ) j F (cid:20) ¯ abF j − F j − | a | F j − (cid:21) By making the substitutions j = nβ and λ ( θ ), and using equations (8) and (10) to make approximations for θ and λ respectively, we arrive at the result. (cid:3) On the right side of Figure 1, we see that the top eigenvectors are are approximations of sine waves up to phase,as described by Theorem 3.3. We pause for a moment to consider these results as an analogy to the partical inan infinite well [22]. The approximately sinusoidal top eigenvectors of the presently defined absorbing quantumwalk operator Q n are in agreement with eigenfunctions of the particle in an infinite potential well. Moreover, thephase difference between the k th top eigenvalue of Q n and one of the points ±| a | ± i | b | is proportional to k for n sufficiently large. This same energy spacing is present in the infinite potential well particle and leads to the revivalbehavior of the following section. Previous studies have illustrated that the quantum walk has wave-like behavior for t = O ( n ) [5], and in particularthat quantum walks appear to partially reflect off absorbing boundaries [35] [31]. In the previous section, weshowed that the absorbing quantum walk reaches a stable state for t = O ( n ). At the intermediate timescale t = O ( n ), the absorbing quantum walk exhibits fractional quantum revivals. [9] [10]First, let us consider an approximation of ( λ ± k,n e ∓ iφ ) τn for a specific value of τ : Proposition 4.1
For τ = 4 / ( πy ) , the following result holds: ( λ ± k,n e ∓ iφ ) τn = 1 + O ( n − ) (14) Proof:
Using Theorem 3.2, we can take a logarithm to find:log (cid:104) ( λ ± k,n e ∓ iφ ) τn (cid:105) = ( τ n ) log( λ ± k,n e ∓ iφ )= τ n (cid:18) ∓ i ( πk ) y n + O ( n − ) (cid:19) = ∓ iτ y ( πk ) O ( n − )Letting τ = 4 / ( πy ) results in the main component becoming ± πik and the result immediately follows. (cid:3) For the remainder of the paper, we let τ = 4 / ( πy ). This result shows that the top few eigenvalues of Q τn n areapproximately equal to one. However, these approximations are only suitable for fixed k ; for t = O ( n ) numericsshow us that for (cid:15) > O ( √ n ) eigenvalues with | ( λ ± k,n ) t | > (cid:15) . Thus we would like new approximationfor the eigenvalues λ ± β √ n,n where β is fixed. We write an extension of Lemma 3.1:7 emma 4.1 Consider the set Θ ± n from Theorem 3.1 with elements θ k,n ∈ Θ ± n for k ∈ { , ..., n − } . For β = k √ n fixed, we have: θ ± β √ n,n = x √ n ± ixyn / ± ixyn / (cid:20) ± iy − x y ) (cid:21) + O ( n − / ) (15) where x = πβ . Proof:
We follow the argument of Lemma 3.1 and fix β = k √ n . This allows us to posit an asymptotic expansion θ = (cid:80) Nj =1 θ j n j − / + O ( n − ( N +1 / ), giving us an equation:sin (cid:18) θ √ n + θ n / (cid:19) + O ( n − / ) = ± iy sin (cid:18) θ √ n + θ n / (cid:19) + O ( n − / )The first term becomes θ = πβ . By using a Taylor series expansion of sin θ and matching like powers of n − j , wehave: θ = ± iyθ , θ − θ ± iy (cid:18) θ − θ (cid:19) Solving this system gives the result. (cid:3)
Using Lemma 4.1, we can find the eigenvalues at this new timescale.
Proposition 4.2
For fixed β = k √ n , we write λ ± β √ n,n ∈ Λ n from Theorem 4.1 as follows: λ ± k,n = e ± iφ (cid:18) ± ix y n − x yn (cid:18) y + x (cid:2) y ± i (3 y + 1) (cid:3)(cid:19)(cid:19) + O ( n − ) (16) Proof:
We can formally write θ ± β √ n,n = ϕ ± (cid:16) √ n (cid:17) where the coefficient of n − ( m − / is represented by ϕ (2 m − (0) / (2 m − λ ± β √ n,n = λ ± (cid:16) ϕ ± (cid:16) √ n (cid:17)(cid:17) . By using the same chain rule argument but noting that ϕ (2 m +1) = 0 and λ (2 m ) = 0, we have: λ ± β √ n,n = λ + ( ϕ (cid:48) ) λ (cid:48)(cid:48) n + 4 ϕ (cid:48) ϕ (cid:48)(cid:48)(cid:48) λ (cid:48)(cid:48) + ( ϕ (cid:48) ) λ (cid:48)(cid:48)(cid:48)(cid:48) n + O ( n − )Making the proper substitutions gives us the result. (cid:3) With this new representation of the eigenvalues, we can consider a more detailed approximation of the relevanteigenvalues at the timescale t = O ( n ). Theorem 4.1
For β = k/ √ n fixed, we have: ( λ ± β √ n,n e ∓ iφ ) τn + ρn = e − x y τ exp (cid:20) ± ix y (cid:18) ρ − x τ
12 (3 y + 1) (cid:19)(cid:21) + O ( n − ) (17) Proof:
Taking a logarithm of the left hand side of equation (17) and using the approximation from proposition4.2, we have:log (cid:104) ( λ ± β √ n,n e ∓ iφ ) τn + ρn (cid:105) = ( τ n + ρn ) log( λ ± β √ n,n e ∓ iφ )= ( τ n + ρn ) log (cid:18) ± ix y n − x yn (cid:18) y + x (cid:2) y ± i (3 y + 1) (cid:3)(cid:19) + O ( n − ) (cid:19) = ( τ n + ρn ) (cid:34)(cid:18) ± ix y n − x yn (cid:18) y + x (cid:2) y ± i (3 y + 1) (cid:3)(cid:19)(cid:19) − (cid:18) ± ix y n (cid:19) (cid:35) + O ( n − )= ± ix y τ n − x y τ ± ix y (cid:20) ρ − x τ
12 (3 y + 1) (cid:21) + O ( n − )The result follows from noting that ± ix yτ n/ (cid:3) H [ | Q tn ψ | ] with n = 200, y = 1, and ψ = | (cid:105)| R (cid:105) .This theorem gives us a better perspective on the eigenvalues of Q tn for t = O ( n ). The absolute value ofequation (17) decreases at O ( e − β ) for large β , and the phase is a quartic function of β . More importantly,equation (17) shows that the top O ( √ n ) eigenvalues of Q n are significant for t = O ( n ) and roughly align in thecomplex plane on the real interval [0 , ρn to the exponent is to better “align” the eigenvalues in the complexplane. It is an interpretive task to define alignment, however one particularly compelling definition is to consideralignment as an entropy minimization task. Recall that if p : R → R is a probability distribution function,its Shannon entropy (not to be confused with Von Neumann entropy of a density matrix) is defined as H [ p ] = − (cid:82) ∞−∞ p ( x ) log p ( x ) dx . Loosely speaking, entropy is a measure of uncertainty in the outcome of a probabilisticprocess. For instance, if p ( x ) = δ x ( x ), then H [ p ] = 0, otherwise p describes a deterministic quantity and thereis no uncertainty in the corresponding outcome. Alternatively, if the domain of p is an interval on the real line, H [ p ] is maximized when p is the uniform distribution, or all possible outcomes are equally likely. Suppose webegin our absorbing quantum walk with the initial state ψ = | n (cid:105)| R (cid:105) . Notice that H [ | ψ | ] = 0. We claim thatat time t = τ n + O ( n ), Q tn in some sense approximates the identity matrix; the fitness of this approximationcan be estimated by computing the entropy quantity H [ | Q tn ψ | ]. If Q tn is a good approximation of the identitymatrix, we expect this entropy quantity to be small, and subsequently for the eigenvalues of Q tn to “align” in somecapacity. This method of using entropy to gauge quantum revivals was explored previously in Romera and de LosSantos [36]. We explore this concept visually and computationally in the remainder of the section.Figure 2 contains two plots of the entropy H [ | Q tn ψ | / (cid:107) Q tn ψ (cid:107) ] over time t . The left plot shows that theentropy of this system is a roughly periodic function with decaying amplitude. Notice that the most prominentminima occur at approximately t = τn k where k ∈ N . We deduce that these minima correspond to times at which Q tn ψ approximates a delta function, as shown in the left side of Figure 4. As k increases, these approximationsin some sense lose “higher frequency” components as they eventually approach the steady-state top eigenvector.However, it is not true that Q τn k/ n approximates the identity matrix for all k ; in fact this approximation holdsonly for k divisible by 8. Figure 5 shows this sequence of matrices. Notice that at k = 4, Q τn / n is approximatelyan anti-diagonal matrix which flips the initial condition. For k = 2 , Q τn k/ n becomes a weighted sum of theidentity approximation and the vector flip approximation. The remaining (odd) values of k result in Q τn k/ n becoming a weighted sum of approximations of the identity matrix, the vector flip matrix, a matrix which swapsthe top and bottom halves of the vector, and a matrix which flips the top and bottom half of the vector separately.The right plot of Figure 2 shows significant detail at a smaller scale as well. We observe that entropy minimaoccur approximately at times t = τn p q where p, q ∈ N are sufficiently small, in a manner similar to Thomae’sfunction [7]. Moreover, the right plot of Figure 4 suggests that for gcd( p, q ) = 1, the distribution | Q tn ψ | has q | Q tn | for n = 400, y = 1, and t = τn k with k ∈ { , ..., } .peaks. This observation can be resolved by noting that letting τ = 4 p/πyq leads to an extra factor of exp (cid:0) πik p/q (cid:1) in equations (14) and (17). This causes the eigenvalues to align on a finite set of lines in the complex plane, thenumber of which is equal to the number of quadratic residues in the ring of integers modulo q . By considering theeigenvectors of Q n from equation (13) as crude approximations of sin( πkx ), the mechanism behind these fractionalquantum revivals becomes clear.However, from the estimate of ( λ ± β √ n,n e ∓ iφ ) τn + ρn in equation (17), we see that the phases of the top O ( √ n )eigenvalues never perfectly align. While letting τ = πy pq and ρ = 0 gives a good estimate of the time locationsof minimum entropy, we can see from Figure 5 that the actual locations of minimum entropy occur at slightlylater times as indicated by the improved resolution from the middle to the bottom row of plots. Notice that theeigenvalues at the time of minimum entropy are contained in a narrower band about the real axis than are theeigenvalues at the approximation t = τ n /
2. In Figure 6, we see various estimates of ρ which lead to entropyminimization. The left plot appears to confirm that ρ converges to a finite value as n → ∞ , while the secondplot illustrates how ρ changes as a function of | a | . A true estimation of ρ would require an estimate of entropyin the system, and this requires a better estimate of the eigenvectors than has been given. Further still, no easilydeducible heuristic gives a usable approximation for ρ as a function of y and τ .These fractional quantum revivals are not unique to the absorbing quantum walk and have manifested in avariety of other quantum systems; the bottom plot in Figure 4 is a fractal that is referred to as a quantum carpet[9] [19] [33], and was first observed by Henry Talbot in 1836 in the context of optical science [38]. In the case ofthe particle in an infinite potential well, the spacing of energies arising from the Schr odinger equation preciselyscales at k , leading to exact revivals. In the absorbing quantum walk operator Q n this spacing is not exact,particularly at higher energies. However, only the top O ( √ n ) eigenstates contribute to the conditional probabilitydistribution at times t = O ( n ), and the spacing among these eigenvalues is such that approximate revivals areobtained, though the fidelity of these revivals worsens over time until the stable t = O ( n ) regime is reached.This argument breaks down for several purely unitary quantum walks where all eigenstates are relevant. Here,the eigenvalues irrationally wind around the unit circle such that approximate revivals occur only after extremelylong times scaling with the size of the lattice; approximate revivals of continuous time quantum walks on the cyclehave only been observed for very small lattice sizes. [13]10igure 4: Plots of | Q tn ψ | for n = 400, y = 1, ψ = | (cid:105)| R (cid:105) , and various values of t .11igure 5: Illustrations of entropy minimization for absorbing quantum walks with n = 400 and y = 1 ( Top Left )Entropy correction for | Q tn ψ | with ψ = | (cid:105)| R (cid:105) and t = τn = 25 , t = 25 ,
627 (
Top Right ) Entropy corrected eigenvalues of Q tn with t = τn = 101 , t = 102 , Middle Row ) Heat maps of the matrices | Q tn | for t = τn = 8 , t = τn = 12 , t = τn = 16 ,
977 respectively. (
Bottom Row ) Entropy corrected plots of the matrices | Q tn | at times t = 8 , t = 12 , t = 17 ,
120 respectively. 12igure 6: (
Left ) Estimate of ρ as a function of n for t = τ n / y = 1. ( Right ) Estimate of ρ as a function of | a | with n = 400 and t = τ n / As seen in the previous section, fractional quantum revivals arise because the top eigenvalues of Q n have regularspacing in phase, and the remaining eigenvalues decay to zero exponentially as t = O ( n ). It should not besurprising that other sufficiently symmetric absorbing quantum walk systems also share this property. For example,consider the two-dimensional absorbing Grover walk operator Q x,y = Π B x,y no Q ↔ ( Z , C , G , B x,y ). Here, C = { (0 , , (0 , − , (1 , , ( − , } represents the cardinal directions in Z , G n = n n − I n where n is the n × n matrix filled with ones, and B x,y = { ( a, b ) : a = 1 or a = x, b ∈ Z } ∪ { ( a, b ) : a ∈ Z , b = 1 or b = y } forms anabsorbing “box”. The corresponding 4 xy × xy operator matrix requires tensor methods to decompose and wepostpone this analysis to a later study. From [24] [29], localized eigenvectors with eigenvalue λ = 1 take nonzerovalues on 2 × B x,y . Since these eigenvectors are the only eigenvectors of Q x,y witheigenvalues | λ | = 1, from Proposition 2.2 in Kuklinski [30] an initial condition φ which is orthogonal to each ofthese eigenvectors will eventually decay in norm to zero under Q tx,y . If we let x = y and φ be one such orthogonalinitial conditions localized to ( x , x ), then one of the stable distributions in Figure 7 are eventually achieved. Thesestable distributions correspond to the non-localized top eigenvectors of Q x,y .We document the existence of fractional quantum revivals in the absorbing Grover walk, both for squareabsorbing boxes and also rectangular absorbing boxes. Figure 8 plots entropy of the systems over time; thesegraphs depict a similar periodic stratification of entropy minima as the plot in Figure 2, albeit with less regularity.We speculate that these entropy minima occur at times t = τ zn for sufficiently simple z ∈ Q . Figure 9 displaysthree of these distributions of the Q , absorbing Grover walk at entropy minima; perhaps unsurprisinglythe peaks arise in an evenly spaced square grid pattern, although there also appear to be non-negligable diagonalpatterns. However for the rectangular Q , absorbing Grover walk, the minimum entropy distributions displayedin Figure 10 do not lend themselves so easily to a simple geometric description. In this paper we have computed eigensystems for one-dimensional finite absorbing quantum walks. The eigenvaluesof the corresponding operator Q n uniformly approach two sectors of the unit circle at O ( n − ), while the eigenvectorsare appoximations of sine waves up to phase. As we consider larger powers Q tn , we find that the eigenvalues rotateabout the origin, and the top O ( √ n ) eigenvalues approximately align in the complex plane at regular intervals.This gives rise to fractional quantum revivals described in Section 4. This behavior is found in other sufficiently13igure 7: Non-localized stable distributions of absorbing Grover walk operator Q n,n with n = 201 and initialconditions localized to (101 , Left ) Plot of entropy over time for H [ | Q t , ψ | ] with ψ = | (cid:105)| (cid:105)| R (cid:105) . ( Right ) Plot of entropyover time for H [ | Q t , ψ | ] with ψ = | (cid:105)| (cid:105)| R (cid:105) .Figure 9: Minimum entropy distrubutions of | Q t , ψ | with ψ = | (cid:105)| (cid:105)| R (cid:105) ( Left ) t = 4280 ( Center ) t = 6402 ( Right ) t = 12752. 15igure 10: Minimum entropy distrubutions of | Q t , ψ | with ψ = | (cid:105)| (cid:105)| R (cid:105) ( Top Left ) t = 1812 ( Top Right ) t = 3602 ( Center Left ) t = 4818 ( Center Right ) t = 7202 ( Bottom Left ) t = 9600 ( Bottom Right ) t = 14374.16egular quantum mechanical systems with spacing of eigenvalues proportional to k .Several areas of this paper should be expanded upon in future study. If a more robust approximation of theeigenvectors of Q n are found, one could then make a more informed approximation of entropy for the purpose ofentropy minimization. It may also be possible to compute a characteristic polynomial recursion of Q x,y , whichwould then facilitate computation of the stable distributions in Figure 7. 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