Spontaneous moralization problem in quantum stochastic walk Krzysztof Domino1 , Adam Glos∗1,2 and Mateusz Ostaszewski1,2
arXiv:1701.04624v1 [quant-ph] 17 Jan 2017
1
Institute of Theoretical and Applied Informatics, Polish Academy of Sciences, Bałtycka 5, 44-100 Gliwice, Poland 2 Institute of Informatics, Silesian University of Technology Akademicka 16, 44-100 Gliwice, Poland
Abstract The spontaneous moralization of the quantum stochastic walk is the effect of propagation on the moral graph instead of the original graph. We argue that this effect is undesired, especially in the case of directed graphs. We also propose a procedure to correct this effect. We analyse the corrected model using the Hurst exponent. We show that the fast propagation of the walk, results from the model itself and not from the existence of the additional amplitude transitions. We demonstrate that the obtained result can be used for arbitrary directed graphs.
Keywords Quantum stochastic walks, Hurst exponent, Moral graphs.
1
Introduction
Quantum stochastic walks (QSW) model continuous quantum walks, which generalize both random walks and quantum walks [1]. QSW are governed by the Gorini-Kossakowski-SudarshanLindblad (GKSL) master equation [2–4], which is a general description of a continuous quantum evolution of an open system. The motivation to investigate quantum stochastic walks comes from the fact that such walks model a quantum propagation with an environmental interaction [1]. For this reason, quantum stochastic walks have a potential for modelling real-world quantum systems and quantum computers. Moreover, there are currently known applications of quantum stochastic walks [5] such as: the interpolation between quantum and classical dynamics for the one-dimensional tight-binding model in the electron transport theory [6], the pure de-phasing scattering process [7], the photosynthetic light-harvesting [8, 9], and the quantum page rank algorithm [10, 11]. Generally the advantage of the quantum walk comes from the ballistic propagation regime, with Hurst exponent h = 1. The significance of quantum walks for quantum algorithms is reviewed in [12, 13]. However, in the case of quantum stochastic walks, the propagation speed depends on the choice of Lindblad operator. In [14] we have shown that in the case of a local interaction where each Lindblad operator corresponds to a single edge, the ballistic propagation is broken due to decoherence and the Hurst exponent equals h = 12 , analogically to a classical random walk. However, in the case of a global interaction where adjacency matrix is the only one Lindblad operator, we have proven analytically in [14] that fast propagation property is preserved, henceforth we find this case particularly interesting. In this paper we consider the spontaneous moralization problem. In the case of the global interaction, which has ballistic propagation, there appears additional transitions between disconnected vertices, which have at least one common child. While in the case of the walk on ∗
[email protected]
1
undirected graphs such classically forbidden links may be treated as advantage, in the case of directed graphs it is an evident drawback of the model. Furthermore, it is beneficial to determine, if the fast propagation of the walk comes from the fact that additional links are allowed, or precisely from the quantum stochastic walk model. To do so, we considered an undirected line graph after our correction scheme and recomputed the result from [14]. The last but not least, it is also beneficial to propose the model, which bounds the walk evolution to the original graph. The main result of this work is a fact that quantum stochastic walk maintains fast propagation in the global interaction case, despite correction schemes, and hence can be used for the quantum search algorithm on directed graphs. Such propagation appears to be ballistic, at least for some intensities of interaction. The paper is organized as follows. In Sec. 2 we provide preliminaries and describe the research problem. In Sec. 3 we propose a procedure which prohibits the spontaneous moralization effect. In Sec. 4 provide evidence that our model maintains fast propagation. In Sec. 5 we conclude our results.
2
Preliminaries
2.1
GKSL master equation and quantum stochastic walks
To define quantum stochastic walks in general, let us start with the Gorini-Kossakowski-SudarshanLindblad (GKSL) master equation, that is a differential equation of the form [2–4] X d 1 † † % = M[%] = −i[H, %] + L%L − {L L, %} , (1) dt 2 L∈L
where {A, B} is the anticommutator and M is the evolution superoperator. Here H is the Hamiltonian, which describes the evolution of the closed system, and L is the collection of Lindblad operators, which describes the evolution of the open system. This master equation describes general continuous evolution of mixed quantum states. In the case where H and L do not depend on time, we say that Eq. (1) describes the Markovian evolution of the system. Henceforth, we can solve the differential equation analytically: if we choose initial state %(0), then |%(t)ii = St |%(0)ii, (2) where "
# X 1 † † ¯ +t ¯− St = exp −it H ⊗ 1l − 1l ⊗ H L⊗L L L ⊗ 1l + 1l ⊗ LL , 2
(3)
L∈L
and |·ii denotes the vectorization of the matrix (see eg. [?]). Using Eq. (1) we can define the propagation from |vihw| to |v 0 ihw0 | as [1] ! ! X X X 1 1 0 0 Mvvww = δww0 hv 0 | −iH − L† L |vi + δvv0 hw| iH − L† L |w0 i + hv 0 |L|vihw|L† |w0 i, 2 2 L∈L
L∈L
L∈L
(4) where δij is the Kronecker delta. The GLKS master equation was used for defining quantum stochastic walks. They are the generalization of both classical random walks and quantum walks [1]. Both H and L correspond to the graph structure, however one may verify that at least a choice of Lindblad operators may be non-unique [1, 14]. Suppose we have the undirected graph G = (V, E). Two main models can be distinguished. In both models we choose the Hamiltonian H to be the adjacency matrix of the graph. In the local environment interaction case each Lindblad operator corresponds to a 2
single edge, L = {|wihv| : {v, w} ∈ E}, and in the global environment interaction case we choose a single Lindblad operator L = {H}. In [14] we have shown by analysing the Hurst exponent that the local case leads to the classical propagation of the walk. Oppositely, in the case of the global interaction, the ballistic propagation was obtained. Henceforth, we find the choice of the global interaction case particularly interesting and only this case will be analysed.
2.2
Research problem and motivation
To provide the motivation let us consider the following example. Suppose we choose the GKSL master equation which consist of the Lindblad operators part only and let us consider the graph presented in Fig. 1(a). We choose the natural Lindblad operator describing the evolution on this graph as 0 0 0 L = 0 0 0 = |3ih1| + |3ih2|, (5) 1 1 0 and the initial state %(0) = |1ih1|. If the Lindblad operator describes the evolution on the graph presented in Fig. 1(a), we should expect h2|%(t)|2i = 0 for arbitrary t ≥ 0. However, simple calculations shows that 1 −2t 2 1 + et 41 −1 + e−2t 0 4e . %(t) = 14 −1 + e−2t (6) e−t sinh2 2t 0 −t 0 0 e sinh(t) In a time limit we obtain
1 4
lim %(t) = − 14 0
t→∞
− 14 1 4
0
0 0 ,
(7)
1 2
hence we have non-zero probability of measuring the state in node 2. The analysis of Eq. (4), which is demonstrated in detail in Sec. 3.1, shows that an operator from Eq. (5) corresponds to a graph with the new link created between v1 and v2 , see Fig. 1(b). It is because of the common child v3 , which allows additional propagation inconsistent with the original graph structure. For an arbitrary graph, an additional link appears between every two disconnected vertices, if they have at least one common child. We call this effect the spontaneous moralization of a graph. While in some cases this may be beneficial to use a moralized graph, in our opinion it is necessary to propose a new model, which preserves the original graph structure. In particular, the model should work properly on an arbitrary directed graph. In this case the moralization process destroys the crucial structure of the graph, namely the one-directional edges.
3
Quantum stochastic walks on authentic graphs
In Sec. 2.2 considered the spontaneous moralization problem, caused by the Lindbladian part of the GLKS equation. In this section we propose the new model which preserves the original graph structure, and discuss it using only the Lindbladian part of the GLKS equation. We start by suppressing the propagation between parents against the graph structure (Sec. 3.1). Next, in Sec. 3.2 we correct the premature localization. Finally, in Sec. 3.3 we correct asymmetry of frequency distribution of stochastic walks on symmetric graphs.
3
v1
v2
v1
v2
v3
v3
(a)
(b)
Figure 1: Visualisation of a directed graph (a) and its spontaneous moralization (b).
3.1
Spontaneous moralization
Let us recall the transition rate formula of Lindblad operator with the global environment interaction 1 1 0 0 Mvvww = − δww0 hv 0 |L† L|vi − δvv0 hw|L† L|w0 i + hv 0 |L|vihw|L† |w0 i. 2 2
(8)
One can notice that the only not-intuitive propagation is due to terms including L† L. Kronecker deltas cause that one of those terms is non-zero if: 0
1. w = w0 , where we have propagation Mvvww , 2. v = v 0 , where we have propagation Mvw vw , 3. both v = v 0 and w = w0 , which results in Mvw vw . The last case describes the amplitude preservation on the off-diagonal element |vihw| of the density matrix. Such preservation of the amplitude is consistent with structure of the graph. Moreover, first two cases are equivalent, hence we focus on the first case only. We conclude, that we need to zero the addend hv|L† L|wi between disconnected vertices v and w in any direction. ˜ which is homomorphic to G, To achieve this, for a given graph G we construct a new graph G ˜ ˜ to adjacent i.e. for which there exists function from V to V that maps adjacent vertices of G vertices of G. We propose the following approach. Suppose we have a directed connected graph G = (V, E) ˜ = (V˜ , E) ˜ with where V = {v0 , . . . , vn−1 }. We construct the new graph G V˜ =
n−1 [
[vi ]
(9)
i=0 indeg(vi )−1
where [vi ] = {vi0 = vi , vi1 , . . . , vi
} iff indeg(vi ) > 0 and [vi ] = {vi0 } otherwise, and
˜ ⇐⇒ (vi , vj ) ∈ E. (vik , vjl ) ∈ E
(10)
Here indeg(vi ) is a number of directed lines incoming to a vertex vi . In other words we create additional copies of vertices and make a connection between them iff their representatives from original graph G are connected. ˜ to be simply an adjacency matrix of G, ˜ again we may have additional coherence If we choose L ˜ between the disconnected nodes. We need to choose such entries of the Lindblad operator L
4
such that vil -th and vjk -th column are orthogonal, if they are from different abstraction classes, ˜ of the form i.e. i 6= j. We choose Lindblad operator L ( hk|Ai |ji, (j, i) ∈ E, k ˜ l hvi |L|vj i = (11) 0, otherwise, where Ai is square matrix of the size indeg(vi ), such that columns form an orthogonal basis. The evolution takes the form d ˜ %˜}. ˜ %L ˜ † − 1 {L ˜ † L, (12) %˜(t) = L˜ dt 2 We define the probability of measuring the state in vi after time t as X hv|˜ %(t)|vi, (13) p(vi , t) = v∈[vi ]
Lemma 1. Let us take Lindblad operator defined in Eq. (11). Suppose vik , vjl ∈ V˜ are arbitrary ˜ † L|v ˜ l i = 0. vertices such that i 6= j. Then hvik |L j Proof. We have ˜ † L|v ˜ jl i = hvik |L
X
˜ † |vihv|L|v ˜ jl i hvik |L
v∈V˜
=
n−1 X indeg(v Xp )−1 p=0
˜ k ihvpr |L|v ˜ jl i hvpr |L|v i
(14)
r=0 indeg(vp )−1
=
X
X
p:((j,p)∈E∧(i,p)∈E)
r=0
hr|Ap |iihr|Ap |ji.
Columns Ap |ii and Ap |ji are orthogonal which confirms the claim. Theorem 1. Let us take two different vertices v, v 0 from different abstraction classes, such ˜ v0 w = M ˜ wv0 = 0 for that the representatives of abstraction classes are not connected. Then M vw wv arbitrary w. Proof. We have ˜ v0 w = − 1 hv 0 |L ˜ + hv 0 |L|vihw| ˜ ˜ † |wi ˜ † L|vi M L vw 2 (15) 1 0 ˜† ˜ = − hv |L L|vi = 0, 2 where the second equality follows from the fact that v and v 0 are not connected in original graph 0 ˜ wv G, and the last equality follows from Lemma 1. Proof of M wv = 0 is analogical. From Theorem 1, we conclude that the model defined by Eq. (12) evolves on the original 2πikl graph. Throughout this paper we choose Ai to be a Fourier matrix, i.e. hk|Ai |li = exp indeg vi . Let us recall the example given in Fig. 1(a). After the correction we obtain the Lindblad operator of the form ˜ = |v30 ihv10 | + |v31 ihv10 | − |v31 ihv20 | + |v31 ihv20 |. L (16) The operator is presented in Fig. 2. One can verify that, if %(0) = |v10 ihv10 |, then the evolution described by Eq. (12) yields 1 %(t) = e−2t |v10 ihv10 | + e−2t −1 + e2t (|v30 i + |v31 i)(hv30 | + hv31 |). (17) 2 The probability of measuring the state in v2 equals 0. 5
v31 −1
1 v10
v20 1
1 v30
Figure 2: Visualization of the Lindblad operator after the correction. The original graph is prestented in Fig. 1(a).
3.2
Premature localization
Through numerical analysis of new model we noticed undesired phenomenon: premature localization. For example, let us consider the graph presented in Fig. 3(a). According to our correction scheme, the corrected Lindblad operator will „represent” the graph presented in Fig. ˜ has the form 3(b). Its Lindblad operator L 0 1 1 0 0 1 1 1 0 1 0 1 0 1 1 1 0 0 1 1 0 ˜ = 1 0 , 0 0 1 0 0 L 0 1 −1 0 0 1 −1 1 0 −1 0 1 0 −1 1 −1 0 0 1 −1 0
(18)
with order v10 , v20 , v30 , v40 , v11 , v21 , v31 . It is expected that starting from arbitrary proper mixed state (at least in a vertex, i.e. %(0) = |vij ihvij |), we should obtain limt→∞ %(t) = |v4 ihv4 |. Oppositely, one can verify that 5 1 1 0 −5 −1 −1 1 1 1 0 −1 −1 −1 1 1 1 0 −1 −1 −1 1 0 0 0 2 0 0 0 %¯ = (19) , 16 1 1 −5 −1 −1 0 5 −1 −1 −1 0 1 1 1 −1 −1 −1 0 1 1 1 with order v10 , v20 , v30 , v40 , v11 , v21 , v31 is a stationary state of the evolution. Such state can be obtained starting from initial state %(0) = |v10 ihv10 |. ˜ rot which changes the state To correct this problem, we propose to add the Hamiltonian H l k ˜ within the abstraction class, i.e. if i 6= j then hvi |Hrot |vj i 6= 0. Now the evolution takes the form d ˜ rot , %˜] + L˜ ˜ %L ˜ † − 1 {L ˜ † L, ˜ %˜}. %˜ = −i[H (20) dt 2 We call this Hamiltonian the locally rotating Hamiltonian, since it acts only locally on abstraction classes. We have verified numerically that the appropriate Hamiltonian corrects the premature localisation problem. Application of the rotating Hamiltonian in the evolution on the graph, presented in Fig. 3(b), gives new stationary state %¯ = |v40 ihv40 | 6
(21)
v30
v31
v11
v21
v10
v20
v3
v1
v2
v40
v4 (a)
(b)
˜ Figure 3: The original graph with the premature localization (a), and the corrected graph G 0 0 based on the original graph (b). Starting from |v1 ihv1 | we obtain a stationary state which is not fully localised in the vertex v40 . Furthermore, numerical analysis show that any random non real-valued Hamiltonian corrects the problem for an arbitrary random digraph. In the scope of this paper we will use the Hamiltonian i = j and l = k + 1 mod (vi ), i, k ˜ l hvi |Hrot |vj i = −i, i = j and l = k − 1 mod (vi ), (22) 0, otherwise.
3.3
Lack of symmetry on symmetric graphs
A further problem has occurred for some symmetric graphs, where we observe the lack of symmetry of the probability distribution. Let us analyse the walk described in Eq. (20) on undirected segment graph. In Fig. 4(a) we present the probability distribution of the position measurement and the reflection of distribution according to the initial position. We observe that probability distribution is not symmetric with respect to the initial position. Our analysis of transition rates suggests that such behaviour results from non-symmetry of ˜ ln = hl|L|mihm| ˜ ˜ † |ni. In the case of the Fourier matrix used in spontaneous the propagation M L mm ˜ ln belong to a complex unit circle. Since the columns of moralization correction values of M mm ˜ ln matrices Ai , need to form an orthogonal basis, the values M mm will be different in general. Our idea is to add another global interaction Lindblad operator, with different Ai matrices which ˜ 1, L ˜ 2 } with will remove the side-effect. In the case of undirected segment, we choose L = {L (1) (2) ˜ 1 we choose for each vertex matrices Ai and Ai for the ith vertex. Henceforth, for L 1 1 (1) Ai = (23) 1 −1 ˜ 2 we choose for each vertex and for L (2) Ai
1 1 = . −1 1 7
(24)
0.14
0.10
0.12 0.10
0.08
probability
probability
0.12
0.06 0.04 0.02 0.00
0.08 0.06 0.04 0.02
20
10
0
position
10
0.0030
20
(a)
20
10
0
position
10
20
30
(b)
Figure 4: Probability distribution of the measurement in the standard basis and its reflection by the initial position for t = 20 on the line segment of length 51, before procedure application Fig. (a) and after procedure application Fig. (b). Both cases start in equal mixture of the class abstraction of 0, i.e. 12 (|v00 ihv00 | + |v01 ihv01 |). Finally, the evolution takes the form d ˜ rot , %˜] + %˜ = −i[H dt
X ˜ L ˜ 1 ,L ˜2} L∈{
1 ˜† ˜ † ˜ ˜ L˜ %L − {L L, %˜} . 2
(25)
Numerical analysis (see Fig. 4(b)) show, that we obtain symmetric distribution of probability for arbitrary time t. It is possible, that for the general graph the choice of different permutations (1) (2) of columns in Ai matrix as Ai , Ai , . . . would suffice.
4
The Hurst exponent analysis
In [14] we have shown that quantum stochastic walks with the Hamiltonian operator and the global interaction Lindblad operator yield a ballistic propagation. However, because of the spontaneous moralization, the graph which was actually analysed was an undirected line with additional edges between every two vertices (see Fig. 5). Such observation does not diminish the results of [14], but gives additional incentive to recompute the Hurst exponent of stochastic walks on a line by using the corrected model. Hence, in this section we reproduce those results using our correction scheme, to verify if the fast propagation recorded in [14] for the global interaction case is due to the additional amplitude transitions or due to the quantum stochastic walk itself. To do so we use the Hurst exponent h that is a useful measure of the propagation speed. In the case of the walk on a line, Hurst exponent can be defined as the scaling parameter of the second moment µ2 (t), i.e. h is a number such that lim
t→∞
µ2 (t) =1 Ct2h
(26)
for some positive C or equivalently µ2 (t) = Θ t2h .
(27)
It is called a standard scaling law that originated from a Fractional Brownian Motion [15] idea, and a large time approach [16]. Generally, the greater the value of h, the faster the walk 8
propagates for large times. In the case of the walk on the line, it is well known that for the classical random walk we have h = 12 [17], and for the quantum walk h = 1 [13]. In [14] we have analysed evolution of the form d 1 % = −i(1 − ω)[H, %] + ω L%L† − {L† L, %} , (28) dt 2 where both L and H are adjacency matrix on the undirected line, and ω ∈ [0, 1] is a free parameter of interaction strength. In [14] we demonstrated that for ω ∈ [0, 1) the Hurst exponent equals one, i.e. the propagation will follow the ballistic regime. In our case, we consider the model based on the symmetrized quantum stochastic walk given in Eq. (25). We introduce a Hamiltonian, which is an adjacency matrix of the increased matrix, i.e. ( ˜ 1 |ji = 1, hi|L 6 0, ˜ hi|H|ji = (29) ˜ 0, hi|L1 |ji = 0, The evolution takes the form d ˜ %] + ω i[H ˜ rot , %˜] + %˜ = −i(1 − ω)[H, dt
X ˜ L ˜ 1 ,L ˜2} L∈{
1 ˜ † L, ˜ %˜} . ˜ %L ˜ † − {L L˜ 2
(30)
In [14] we examined quantum stochastic walks on an undirected line. For the global interaction case we demonstrated that if ω ∈ [0, 1), the Hurst exponent equals one, and thus the propagation if ballistic. As it was shown in previous sections, in such case, the walk does not propagate on the line graph, but on the graph with additional edges between every two nodes, see Fig. 5. Such observation does no diminish the results of [14], but gives additional incentive to recompute the Hurst exponent of stochastic walks on a line, by using our corrected model. To determinate the Hurst exponent we use a formula 1 log µ2 (t) 1 log µ2 (t) − log C ≈ lim . t→∞ 2 t→∞ 2 log t log t
h = lim
(31)
First, we compute the second moment for ω ∈ {0.5, 0.6, 0.7, 0.8, 0.9} for different times from interval [0, 2400]. Then, we chose 10 consecutive points and compute the linear regression of 1 2 log µ2 (t) vs log(t). The slope of the regression will approximate the Hurst exponent. The result of numerical analysis is shown in Fig. 6. We can see that for each value of ω the slope increase in time and exceeded the value 21 , which is the upper bound for classical propagation. The result suggests that, at least for some values of ω, we have reproduced the ballistic propagation regime h ≈ 1. Moreover one can conclude, that the Hurst exponent exceeds 12 even for ω near 1, if not ballistic regime, the super–diffusive one (with 12 < h < 1) may be suggested. Both results confirm that the fast propagation (ballistic or at least super–diffusive) is the property of global interaction case of the quantum stochastic walk and not from the fact that the original model allows additional transitions not according to the graph structure. Moreover, it confirms that our correction scheme preserves the ballistic propagation of the walk, at least for some interaction strength ω.
5
Conclusions
In this paper we have show that original quantum stochastic walks do not evolve on the original graph, but on the moralized graph. While for some reason it may be convenient to use such an approach, our goal was to remove the observed invalid propagations. To make a propagation exactly on the graph model, we provide the correction procedure, which is based on the 9
···
−3
−2
−1
0
1
2
3
···
Figure 5: Line graph. Dashed lines correspond to additional amplitude transitions every two nodes coming from the GKSL model.
1.0 0.9
slope
0.8 0.7 0.6 0.5
ω = 0.5 ω = 0.8
0
500
1000
ω = 0.6 ω = 0.9
1500
ω = 0.7
2000
2500
log(t)
Figure 6: Slope of the local regression line for different values of ω, respectively from top to bottom: ω = 0.5, 0.6, 0.7, 0.8, 0.9. We can see that for each value of the ω slope exceeds 21 and increases in time.
10
proper system enlargement. We correct further two side-effects caused by the correction procedure, i.e. by adding so-called locally rotating Hamiltonian and by increasing the collection of Lindblad operators. We have recomputed the results from [14] in context of our model and we have confirmed that our corrections do not disturb the ballistic propagation of the walk. This demonstrates, the fast propagation corresponds not to classically forbidden transitions, but to the quantum stochastic walks model itself. It is worth noting that for some graphs the symmetrization procedure is not necessary. For example, if we evolve quantum stochastic walks on undirected perfect k-ary tree, with the initial state localized in the root, we can observe that the distribution is symmetric due to arbitrary automorphism A such that A(root) = root. We haven not found an explanation for this phenomenon yet.
Acknowledgements Krzysztof Domino acknowledges the support of the National Science Centre, Poland under project number 2014/15/B/ST6/05204. Adam Glos and Mateusz Ostaszewski were supported by the Polish Ministry of Science and Higher Education under project number IP 2014 031073.
References [1] J. D. Whitfield, C. A. Rodríguez-Rosario, and A. Aspuru-Guzik, “Quantum stochastic walks: A generalization of classical random walks and quantum walks,” Physical Review A, vol. 81, no. 2, p. 022323, 2010. [2] A. Kossakowski, “On quantum statistical mechanics of non-hamiltonian systems,” Reports on Mathematical Physics, vol. 3, no. 4, pp. 247–274, 1972. [3] V. Gorini, A. Kossakowski, and E. C. G. Sudarshan, “Completely positive dynamical semigroups of n-level systems,” Journal of Mathematical Physics, vol. 17, no. 5, pp. 821–825, 1976. [4] G. Lindblad, “On the generators of quantum dynamical semigroups,” Communications in Mathematical Physics, vol. 48, no. 2, pp. 119–130, 1976. [5] P. E. Falloon, J. Rodriguez, and J. B. Wang, “Qswalk: a mathematica package for quantum stochastic walks on arbitrary graphs,” arXiv preprint arXiv:1606.04974, 2016. [6] S. Datta, “Electronic transport in mesoscopic systems (cambridge studies in semiconductor physics and microelectronic engineering),” Cambridge University Press, vol. 40, pp. 10011– 4211, 1997. [7] V. Kendon, “Decoherence in quantum walks–a review,” Mathematical Structures in Computer Science, vol. 17, no. 06, pp. 1169–1220, 2007. [8] M. Mohseni, P. Rebentrost, S. Lloyd, and A. Aspuru-Guzik, “Environment-assisted quantum walks in photosynthetic energy transfer,” The Journal of chemical physics, vol. 129, no. 17, p. 174106, 2008. [9] R. E. Blankenship, Molecular mechanisms of photosynthesis. John Wiley & Sons, 2013. [10] E. Sánchez-Burillo, J. Duch, J. Gómez-Gardenes, and D. Zueco, “Quantum navigation and ranking in complex networks,” arXiv preprint arXiv:1202.3471, 2012. [11] T. Loke, J. Tang, J. Rodriguez, M. Small, and J. Wang, “Comparing classical and quantum pageranks,” arXiv preprint arXiv:1511.04823, 2015. 11
[12] V. M. Kendon, “A random walk approach to quantum algorithms,” Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, vol. 364, no. 1849, pp. 3407–3422, 2006. [13] J. Kempe, “Quantum random walks: an introductory overview,” Contemporary Physics, vol. 44, no. 4, pp. 307–327, 2003. [14] K. Domino, A. Glos, M. Ostaszewski, Ł. Pawela, and P. Sadowski, “Properties of quantum stochastic walks from the hurst exponent,” arXiv preprint arXiv:1611.01349, 2016. [15] B. B. Mandelbrot and J. W. Van Ness, “Fractional brownian motions, fractional noises and applications,” SIAM review, vol. 10, no. 4, pp. 422–437, 1968. [16] J. Haus and K. Kehr, “Diffusion in regular and disordered lattices,” Physics Reports, vol. 150, no. 5, pp. 263–406, 1987. [17] A. O. Caldeira and A. J. Leggett, “Path integral approach to quantum brownian motion,” Physica A: Statistical mechanics and its Applications, vol. 121, no. 3, pp. 587–616, 1983. [18] P. Sadowski, J. A. Miszczak, and M. Ostaszewski, “Lively quantum walks on cycles,” Journal of Physics A: Mathematical and Theoretical, vol. 49, no. 37, p. 375302, 2016. [19] A. Montanaro, “Quantum walks on directed graphs,” Quantum Information & Computation, vol. 7, no. 1, pp. 93–102, 2007.
12
