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2.3 Mapping between random walks and quantum spin systems
Fourier transforming both sides of Eq. (2.26), we obtain
@ ˜Pk(t)
@t =⇥
(eik+ e ik) 2⇤ ˜Pk(t). (2.28) We suppose that the initial condition is Pn(0) = n,0. Since ˜Pk(0) = 1, the solution to Eq. (2.28) is
P˜k(t) = e2(cos k 1)t. (2.29) By comparing Eq. (2.29) with the Jacobi-Anger expansion [11] in terms of the n-th Bessel function of first kind, Jn(z),
eiz cos k = X1 n= 1
inJn(z)eink, (2.30)
we obtain the solution
Pn(t) = In(2t) e 2t, (2.31) where In(t) = (i) nJn(it) is the modified Bessel function of order n. In the long time limit, this expression is approximately a Gaussian function of widthp
2t Pn(t)! 1
p4⇡te n2/4t, (2.32)
in agreement with the long-time behavior of the discrete time version.
2.3 Exact mapping between classical random walks and equilibrium quantum spin systems
Great progress has been achieved in solving certain stochastic processes by the realization of the connection between the formalism for those problems and that for quantum many-body systems. More specifically, the master equation (2.19) of a stochastic system may be reformulated as a Schr¨odinger equation
1 i
@
@t|P (t)i = H|P (t)i (2.33)
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in terms of a Hamiltonian H of a quantum spin system. This has allowed the stochastic models to be solved by exact analytic techniques of equilibrium sta-tistical mechanics. Or, conversely, the corresponding quantum problems can be solved by techniques for stochastic models.
In this section, we will show an example of this type of approach [9, 12]:
the mapping of the one-dimensional random walk to the quantum Ising chain described by the Hamiltonian
H = X
in terms of the Pauli spin operators{ x,zn } located on the sites (vertices) {vn} of a one-dimensional lattice. Here{Jn} are the nearest-neighbor ferromagnetic cou-plings, and { n} represent on-site transverse fields. In the quantum computing terminology, we can say that the system is a set of interacting ”qubits”. Without the transverse fields, the spin Hamiltonian is reduced to a classical Ising Hamil-tonian since each spin is in one of the eigenstates {| "i, | #i} of z; in this case, the Hamiltonian has two possible ferromagnetic ground states
|*i =O
n
|"in and |+i =O
n
|#in, (2.35)
with all spins parallel. The non-commuting transverse field term induces quan-tum fluctuations changing the system’s ground state to a non-trivial quanquan-tum superposition of all possible spin configurations.
One of the standard analytic techniques for solving the quantum Ising chain is free-fermion diagonalization [13]. Using the Jordan-Wigner transformation [14] and a subsequent canonical transformation [13], we can map the interacting quantum Ising chain into spinless free fermions
Hfermion =X
where ⌘q†and ⌘qare fermion creation and annihilation operators, respectively. The fermionic excitation energies "q for a chain of length N ( 1) with free boundary
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2.3 Mapping between random walks and quantum spin systems
condition (i.e. JL= 0) can be obtained by solving the eigenvalue equation T | qi = "2q| qi (2.37)
for the 2L⇥ 2L symmetric matrix:
T =
Here we are particularly interested in the general case, in which the couplings and fields are random variables.
Consider now a one-dimensional random walk in a random environment, char-acterized by transition rates wn,n±1for a move from site vnto site vn,n±1. The walk is confined in a finite segment between site v1 and site vL; the one-dimensional lattice has two adsorbing sites located at v0 and vL+1 so that w0,1 = 0 and wL+1,L = 0. The time evolution of the probability Pn(t) for the walker to be on site vn at time t is governed by a master equation (cf. (2.19)) with a transition rate matrix, M , whose elements take the form
Mn,n±1 = wn±1,n and Mn,n = wn,n 1+ wn,n+1. (2.39) Solving the master equation amounts to solving the eigenvalue problem for the transition rate matrix
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in terms of the new variables
eu(n) = u(n)
Through this transformation the transition rate matrix fM has become symmetric and is given by
M =f
Comparing Eqs. (2.38), (2.37), (2.43) and (2.41), we notice that the exact map-ping between the quantum Ising chain and the 1D random walk with the following correspondences:
Jn , pwn+1,n n , pwn,n+1
"2q , q.
(2.44)
Aside from the results for the 1D random walks we discussed in the previous sections, one can extract many other exact results from this mapping, for example, the probability for a single walker not to return to the starting point up to t.
This survival probability, calculated with adsorbing boundary conditions w0,1 = wL+1,L = 0, converges to a finite value denoted by Ps(L) in the long time limit t! 1, which is given by an exact expression [12, 15]
Ps(L) = 1 +
In the quantum Ising chain mapping, the expression (2.45) corresponds to an
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2.3 Mapping between random walks and quantum spin systems
exact formula for the surface magnetization of the chain:
(ms(L))2 , Ps(L) . (2.46)
For a homogeneous medium with wn,n+1 = wn+1,n = const, we obtain
Ps(L)/ (L + 1) 1. (2.47)
For a random medium without drift, an exact result has been found for the disorder-average survival probability [9, 12]
[Ps(L)]av/ L ✓, with ✓ = 1
2. (2.48)
Combining the size-dependent survival probabilities (Eqs. (2.47) and (2.48)) with the known relations between the characteristic time and length, L ⇠ p
t for a homogeneous random walk, and L ⇠ ln2(t) for a disordered random walk, one can further deduce the finite-size scaling form for the time-dependent survival probability in a homogeneous medium
Ps(L, t) = L ePs(t/L2) , (2.49) and in a random environment [12]
[Ps(L, t)]av= L 1/2Pes(ln t/L1/2) . (2.50) We have verified these results by recursive implementations of the master equa-tions for discrete random walks; the finite-size scaling is depicted in Fig. (2.6) and Fig. (2.7).
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Figure 2.6: A scaling plot of the time-dependent survival probability Ps(L, t) for a 1D walk with homogeneous transition rates. The plot assumes the form in Eq. (2.49)
Figure 2.7: A scaling plot of the disorder-averaged survival probability Ps(L, t) for a 1D walk with asymmetric random transition rates drawn from a uniform dis-tribution. The data are averaged over 105 disorder realizations. The plot assumes the scaling form in Eq. (2.50)
.
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3
Discrete-Time Quantum Walks
Quantum walks were initially introduced in both continuous [17] and discrete [18]
time, in analogy with their classical counterparts. The major di↵erence between these two types of quantum walks lies in the definition of the evolution operator.
A quantum computer would work with discrete registers; discretizing the position space of the quantum walk will allow us to map it to algorithms on such a machine.
Therefore, we will only focus on quantum walks discrete in space, i.e. quantum walks on graphs. In this chapter we will first consider discrete-time quantum walks.
In the simplest discrete-time classical random walk on a graph G = (V, E), at each time step the walker simply moves from any given vertex to each of its neighbors with equal probability. Thus the walk is governed by the |V | ⇥ |V | matrix W with elements
Wmn = 8<
:
1/dn (vm, vn)2 E
0 otherwise, (3.1)
where dn is the degree of the vertex vn. After one step of the walk, an initial probability distribution ~P over the vertices evolves to ~P0 = W ~P .
The quantum version of the discrete-time random walk can broadly be defined as the repeated application of a unitary evolution operator at each time step, without performing intermediate measurements. To define a quantum walk, we need to specify a unitary operator U with the property that an input state |ni,
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corresponding to the vertex vn 2 V , evolves to a superposition of the neighbors of vn. It has turned out that to define such a unitary operator one needs to enlarge the Hilbert space by adding an ancillary system storing the direction in which the walk is moving [19]. The Hilbert space of discrete-time quantum walks is often the tensor product of the position space Hp and a so-called ”coin space” Hc. The position space is a |V |-dimensional Hilbert space spanned by orthonormal basis vectors, {|ni : n = 1, · · · , |V |}, corresponding to the vertices of the graph. Associated with each vertex vn is also a set of ”coin states” {|cmi : m = 1,· · · , dn}, where dnis the degree of the vertex vn; these coin states represent the outgoing edges of each vertex and span the coin space. Basis vectors of the Hilbert space of the walk H = Hp ⌦ Hc are then product states of the form
|ni ⌦ |cmi ⌘ |n, cmi. This type of discrete-time quantum walk is sometimes referred to as the coined quantum random walk.
Being an element in H, a general state of the quantum walker can be written as
| (t)i =X
m,n
↵m,n(t) n, cm
↵, (3.2)
where ↵m,n(t) (2 C) is the probability amplitude associated with the walker being at vertex vn with the coin state |cmi at time t. One step of the walk is the application of a given unitary time evolution operator U on | (t)i. The unitary time evolution of a discrete-time quantum walk is generally defined as [19]
U = S · Ip⌦ C , (3.3)
where S is the shift operator, C is the so-called coin operator, and Ip denotes the identical operator on the position space. For a vertex vnof degree dn the coin operator C defined on Hc can be represented by a dn⇥dnmatrix. Various choices of the coin operator are possible as long as the operator is unitary. For a given graph, one can define a rich family of walks with di↵erent behavior by changing C.
After the application of the operator C, the shift operator S applies a conditional translation to the position state according to the coin state. A quantum walk
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with the initial state | (0)i after t steps is then described by the equation
| (t)i = Ut| (0)i, (3.4)
resulting in propagation of probability amplitudes across the graph.
For a discrete-time quantum walk on a line, we need to specify a 2⇥ 2 coin matrix since the coin space Hc for this case must be spanned by two basis states, denoted by | i and | !i, corresponding to two possible outgoing edges on a vertex. This coin operator may be any U(2) operator (i.e. 2⇥2 unitary matrices) parameterized by
After ”tossing” the coin with the coin operator, a conditional translation is made by the shift operator in the way that the walker moves one step to the left if the accompanying coin state is | i, or to the right if the accompanying coin state is | !i. The shift operator for this operation is expressed mathematically as
S =✓X
One example for the coin operator is the Hadamard coin given by
CH(2) = 1 the quantum walk governed by the Hadamard coin is thus
|ni ⌦ | !i CH! 1
The position state resulted from the action of the unitary operator U is an equal superposition of the left vertex and the right vertex, which is fundamentally
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ferent from an equal mixture of these two components. Indeed, the interference between di↵erent positions in the graph will cause a behavior radically di↵erent from that of a classical random walk. If we, however, perform a measurement at some point in order to know the outcome of the walk, quantum interference will be destroyed. With a measurement at each step of the walk, we will revert to the classical ransom walk.
To illustrate the feature of the quantum walk, we perform the walk starting with the initial state |n = 0i ⌦ | !i without intermediate measurements up to t steps. A measurement on the position is performed at time t by using the projection operator
⇤n=|nihn| ; (3.9)
the probability that the walker is in the vertex state |ni at time t is given by Pn(t) =h (t)|⇤n| (t)i . (3.10) The probability distribution over the graph after t = 100 steps is shown in Fig. 3.1.
The distribution looks markedly di↵erent from the analogous classical distribu-tions. Peculiar features include the asymmetry structure and the relative uni-formity of the central portion of the distribution, in contrast to the classical Gaussian distribution. The asymmetry of the quantum distribution arises from the fact that the Hadamard coin operator does not treat | !i and | i coin states in the same way (cf. Tab. 3.1).
t n -5 -4 -3 -2 -1 0 1 2 3 4 5
0 1
1 12 12
2 14 12 14
3 18 18 58 18
4 161 18 18 58 161
5 321 325 18 18 1732 321
Table 3.1: The probability of being found at position |ni after t steps of the quantum random walk on the line with a Hadamard coin and the initial state
| (0)i = |0, !i. Note that this distribution starts to show a drift to the right from t = 3 on.
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Figure 3.1: Position probability distribution after 100 time steps for a quantum walk on a line using a Hadamard coin defined in Eq. (3.7), and the initial state
|0, !i. Only even positions are shown, since odd positions have probability zero.
To obtain a symmetric position distribution, we can use an initial state as a superposition of| !i and | i, for example,
| (0)i = 1
p2 0,!↵
+ i 0, ↵
. (3.11)
Alternatively, we can use an unbiased coin, like
Csymm(2) = 1
to generate a symmetric distribution. The position probability distribution at t = 100 using the Hadamard coin operator with the initial state (3.11) (or the unbiased coin Csymm with the initial state |0, !i) is depicted in Fig. 3.2. It is apparent that the distribution is almost uniform over the interval [ t/a, t/a] (it can be shown analytically [20, 21] that a =p
2), and strongly peaked at the edges at±t/a. Thus, the variance 2 is approximately given by
2 =hn2i ⇡ a t
Xt/a 1
n2 / t2. (3.13)
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-80 -60 -40 -20 0 20 40 60 80
position
0 0.02 0.04 0.06 0.08
Probability
quantum classical
(a) (b)
Figure 3.2: (a) Position probability distribution (black line) after 100 time steps for a quantum walk on a line using the Hadamard coin operator, and the symmetric initial state given in Eq. (3.11). Only even positions are shown, since odd positions have probability zero. The red line corresponds to the distribution for a classical random walk; (b) evolution of the probability distribution from time t = 1 up to t = 100.
This linear time dependence of the variance implies that the quantum walk prop-agates quadratically faster than the classical symmetric random walk (which has
2 / t).
A discrete-time quantum random walk can be defined for arbitrary undirected graphs by using di↵erent position and coin spaces [19]. We consider here a two-dimensional (2D) lattice. Again, the position space is spanned by all vertices of the graph. We often use a pair notation |nx, nyi to identify the x- and y-components of vertex |ni. Each vertex in a 2D lattice has four edges connected to it, so the coin operator is now a four-dimensional unitary operator. We label the four directions | i, | !i corresponding to the two directions on the x-axis and | "i, | #i for two directions on the y-axis. A common choice of coin operator is the Grover coin CG, which has elements [22]
(CG)mn = mn+ 2/d (3.14)
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for a vertex of degree d; for our 2D lattice it is given by
CG(4) = 1
The unitary operator U constituting a step of the random walk is defined again by the composition of a coin operator C and a shift operator S. The shift is defined by its action on each basis vector |nx, ny, cmi in the joint space of position and
We note that this shift operator translates the position of the walker to an ad-jacent vertex depending on the coin state, and inverts the coin state after every move.
Like in the 1D case, the initial coin state for a given coin operator can be used to control the evolution of the 2D quantum walk. The spatial probability distributions shown in Fig. 3.3 demonstrate that di↵erent choices of initial state can make the Grover coin quantum walk spread fast or slow; the initial states used are
The root mean square displacement measured byp
hr2i =q
hn2x+ n2yi is approx-imately 33.1 at t = 40 for the fast spreading using the initial state | 1(0)i, while it is only about 7.2 for the slow spreading using | 2(0)i. The root mean square distance after t steps for the 2D classical walk is p
hr2i =p
t [25], thus only 6.3
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after 40 steps.
Figure 3.3: Spatial probability distribution for quantum walk on 2D lattice run for 40 steps with initial state given in Eq. (3.17) (top) and in Eq. (3.18)(bottom).
Only even positions are shown, since odd positions have probability zero.
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4
Continuous-Time Quantum Walks
The discrete-time version of quantum walks discussed in the previous chapter is only one way to introduce quantum e↵ects into random walks. Another route to utilize quantum mechanics to move through graphs was introduced by Farhi and and Gutmann [17], referred to as continuous-time quantum walk.
Recall that a continuous-time classical random walk on an N -vertex graph G = (V, E) with constant transition rates can be represented by an N ⇥ N symmetric transition rate matrix M
Mmn = 8>
>>
<
>>
>:
dn, vn = vm
, (vm, vn)2 E 0, otherwise.
(4.1)
The probability distribution {Pn(t)} over the vertices {v1, v2,· · · , vN} at time t are the solution of the di↵erential equation
@
@tPn(t) = X
m
MnmPm(t). (4.2)
We notice that this di↵erential equation is very similar to the Schr¨odinger equa-tion
i@
@t| (t)i = H| (t)i, (4.3)
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except that it lacks the factor of i.
The key idea of Farhi and Gutmann [17] is to use the transition rate matrix as the generator of time evolution, i.e. as the Hamiltonian H. In this definition, the quantum walk on an undirected graph is obtained by replacing the di↵usion equation (4.2) with the Schr¨odinger equation.
The set of vectors{|ni} associated with all vertices forms an orthonormal basis for the Hilbert space. In the basis{|ni}, the Schr¨odinger equation describing the quantum walk is then written as
i@
as given in Eq. (4.1). The solution of this di↵erential equation, known as the probability amplitude, can be given in closed form as
hn| (t)i = hn|e iHt| (0)i, (4.6)
where the matrix exponential generated by the Hamiltonian corresponds to the time evolution operator
U (t) = e iHt. (4.7)
Since the Hamiltonian is a Hermitian operator, this time evolution is unitary, U†U = I, and preserves normalization of the probability in the sense that
@
@t X
n
|hn| (t)i|2 = 0 . (4.8)
Denoting the eigenvectors of H by | ki and the corresponding eigenvalues by Ek, the probability Pn!m(t) that the walker starting at t = 0 at site vn arrives
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4.1 Continuous-time quantum walk on a line
Using the same notation for the quantum case (in fact, the eigenvalues and eigen-vectors of the transition rate matrix M and the quantum Hamiltonian H are the same), the probability Pncl!m(t) in the classical case is given by
Pncl!m(t) =hm|e M t|ni = X
k
e Ekthm| kih k|ni. (4.10)
Since the eigenvalues Ek are non-negative (the matrix M given in Eq. (4.1) is nonnegative-definite), for t 1 all exponential terms with Ek > 0 in the sum of Eq. (4.10) decay to zero; the probability in the long time limit is then determined by the term with Ek= 0, leading to
tlim!1Pncl!m(t) = 1
N. (4.11)
Unlike the classical case, the unitarity of the time evolution operator prevents the probability for the quantum walk from a definite limit when t! 1.
In principle, we could define a continuous-time quantum walk using any Her-mitian Hamiltonian that respects the structure of the graph. For example, we could use the the adjacency matrix A of the graph, whose matrix elements Amn
are equal to one if vertices vm and vnare connected, and zero otherwise; although this matrix cannot be used as the generator of a continuous-time classical random walk.
4.1 Continuous-time quantum walk on a line
Let us first consider a quantum walk on an infinite integer line, with V =Z and nearest-neighboring edges.
The Hamiltonian for this problem is defined by (cf. Eq. (2.26))
H|ni = |n 1i + |n + 1i 2|ni , (4.12) where the transition rate is set to . To obtain the probability amplitudehn + x|e iHt|ni for the walk to move a distance x (2 N) in time t, one can actually simply use the analytic continuation t! it of the exact result for the corresponding continuous
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-20 -10 0 10 20
n
0 0.02 0.04 0.06 0.08 0.1
P n(t=10)
quantum walk classical walk exact solutions
Figure 4.1: Probability distributions for continuous-time random walks (classical and quantum) on an integer line at time t = 10; the transition rate is = 1.
The data sets denoted by squares and circles are obtained by diagonalizing the transition rate matrix M (of the classical random walk) as well as the Hamiltonian H (of the quantum walk) for a finite line with 801 vertices. The data are compared with the exact solutions given in Eq. (4.13) and in Eq. (4.14) for an infinite line.
The grey dashed line is the Gaussian approximation for the classical case
time classical random walk. In the classical version (discussed in Sec. 2.2), the probability of moving a distance x in time t is
Px(t) = e 2 tIx(2 t), (4.13) where Ix is the modified Bessel function of order x. By replacing t with it in Eq. (4.13), we obtain the amplitude
hn + x|e iHt|ni = e i2 tixJx(2 t), (4.14) and the corresponding probability
hn + x|e iHt|ni 2 = Jx(2 t) 2 (4.15) for the quantum walk. Here Jx(t) is the Bessel function of order x. Eq. (4.14)