4.2 Exponential speedup by quantum walk
describes a ballistic propagation with speed 2 [26]; thus the quantum walk moves a distance proportional to t, which is quadratically faster than the classical random walk in which x/p
t.
4.2 Exponential speedup by quantum walk
We have seen that the behavior of a quantum walk can be dramatically di↵erent from that of its classical counterpart. In Ref. [27] a finite graph is given to demonstrate an even stronger example of the power of quantum walk, which will be discussed in this section.
The so-called glued trees graph GL, studied in Ref. [27], consists of two bal-anced L - level binary trees with the 2L leaves of the left tree identified with the 2L leaves of the right tree according to the structure shown in Fig. 4.2 (for G4).
The number of vertices in GL is 2L+1+ 2L 2. We are interested in the dynamics of both the classical and quantum random walks from the leftmost vertex (the root of the left tree) to the rightmost vertex (the root of the right tree).
It is not hard to see that a classical random walk on the graph starting from the leftmost vertex will get trapped in the middle of the graph, and never has a considerable probability of reaching the right root for L 1. Due to the symmetry of the graph, it is more convenient to consider the probabilities, P`, for the walker to be on each column indexed by ` = 0, 1, 2,· · · , 2L (starting from the root of the left tree) while analyzing the dynamics of the walk. These probabilities at time t obey the di↵erential equations
dP0
which are obtained by grouping the master equation for the probability at each vertex in the given column. The elements of the corresponding transition matrix
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Figure 4.2: Glued trees graph G4. The bottom panel shows the reduction to the column space for both the classical and quantum walks.
are depicted in Fig. 4.2. The probabilities distribution over columns of G50 at di↵erent time steps, obtained by solving this set of di↵erential equations with the initial condition P`(t = 0) = `,0, are shown in Fig. 4.4 (a); the figure shows that the probability accumulates rapidly in the middle of the graph, but the probability of reaching the right root evolves extremely slowly. As can be seen from Eqs. (4.16), in the left tree (0 < ` < L), the probability of moving from column ` to column ` + 1 is twice as great as the probability of moving from column ` to column ` 1; on the other hand, in the right tree (L < ` < 2L), the probability of moving from column ` to column ` + 1 is half as great as the probability of moving from column ` to column ` 1. This asymmetry of the probability implies that the probability of reaching column 2L in a time that is polynomial in L is exponentially small as a function of L.
We now analyze the quantum walk on the glued trees graph, which is described
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4.2 Exponential speedup by quantum walk
(a) Quantum (b) Classical
Figure 4.3: Comparison of the probability distributions at time t = 4 of continuous-time quantum walk (panel (a)) and classical random walk (panel (b)) starting from the left root of a glued trees graph G5. It is evident that the spread of the quantum walk is faster than the analogous classical spread. The data obtained from our simulations are plotted using the software package qwViz [28].
by the Schr¨odinger equation i@
@t| (t)i = H| (t)i, (4.17)
where the Hamiltonian is given by the transition rate matrix. Instead of working with the (2L+1+2L 2)-dimensional Hilbert space spanned by all vertices, one de-fines a (2L+1)-dimensional ”column subspace” spanned by the ”column-vectors”
|c`i (where 0 ` 2L) that are the equal superpositions of vertex-states on a given column `, that is
|c`i = 1 pN`
X
n2 column `
|ni, (4.18)
where N` is the number of the vertices in column ` and is given by
N` = 8<
:
2`, 0 ` L,
22L `, ` ` 2L. (4.19)
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(a) classical random walk
0
(c) quantum vs. classical
Figure 4.4: Propagation in G50 starting at the left root. (a) Probability dis-tribution P`(t) of a classical walk at time t = 25, 35, 45, 55, 65; (b) distribution of a quantum walk at time t = 15, 20, 25, 30, 35; (c) a comparison of the classical random walk and the quantum walk at time t = 25.
In this basis, the Hamiltonian has non-zero matrix elements given by hc`|H|c`±1i = p
can be formally expressed as
P`(t) =|hc`|e iHt|c0i|2. (4.21)
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4.2 Exponential speedup by quantum walk
Denoting the eigenvalues of H by Ek and the eigenvectors of H by | ki, the probability P`(t) is then given by
|hc`|e iHt|c0i|2 = X
k
e iEkthc`| kih k|c0i
2
. (4.22)
In Fig. 4.3(a) and Fig. 4.4(b) we show the numerical results for P`(t), obtained by solving the eigenvalue problem of H; the quantum walk shows a remarkable speedup of propagation through the glued trees, as compared with the classical random walk.
Figure 4.5: Propagation in a large graph G500 starting at the left root. The column located approximately at 2p
2t is indicated by the red dashed line; this coincides with the location of the wavefront, implying that the wave packet prop-agates with speed 2p
2.
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Indeed, by identifying the subspace of column-states |c`i, the quantum walk on the L - level glued trees graph starting from the left root is e↵ectively the same as a quantum walk on a line with 2L + 1 vertices, with all edge weights the same (see Fig. 4.2, bottom panel). In the limit of L ! 1, the walk on GL is nearly identical to a quantum walk on the infinite, translationally invariant line. The probability amplitude to go from column ` to column `0 for L ! 1 in a time t is then (cf. Eq. 4.14)
hc`0|e iHt|c`i = e i3 ti`0 `J`0 `(2p
2t), (4.23)
with J`0 ` being a Bessel function of order `0 `; this corresponds to propaga-tion with speed 2p
2 . To verify this, we numerically compute the probability
|hc`|e iHt|c0i|2 for a large system with L = 500 and = 1 at t = 100, 175, 250 and 354. The results are shown in Fig. 4.5. The wavefront of the distribution at t is located approximately 2p
2 t away from the left root.