0 0.5 1 1.5 2 -2 0 2 –2 0 2 kappa2 –3 –2 –1 0 1 2 3 kappa1 –3 –2 –1 0 1 2 3 1 /a /a k mod 2 /a /a /a /a /a 2 | > | $_{o}$> | > − | > + a) c) b) /a /a k mod 2 /a 2 model consisting of a single open channel between nearby lattice sites ( m = 2 d ) and we assume Kirchhoff bound- ary conditions at each vertex. The Hilbert space is then effectively N = 2 d n d dimensional. This physical model captures the essence behind the effect described below. We label incoming wave components from each unit cell as | i $^{±}$〉 ⊗  | ⃗ x 〉 = | i ± , ⃗ x 〉 , where ⃗ x specifies the vertex in position space and i ± , for i = 1 , . . . , d gives the ± direction in dimension i . Incoming waves at vertex ⃗ x are mapped onto outgoing waves by a scattering matrix σ . For Kirchhoff boundary conditions, one obtains σ = 2 | s 〉 〈 s | − GLYPH<BD> 2 d and | s 〉 is the uniform distribution | s 〉 = 1 √$_{2}$$_{d}$ ∑ d i $_{=1}$( | i $^{+}$〉 + i $^{−}$〉 ) [15]. Outgoing waves in direction i ± are now identified with incoming waves at an adjacent vertex ⃗ x ± a ⃗ e$_{i}$ where ⃗ e$_{i}$ is the unit vector in direction i . The local scattering processes is described in terms of a (global) scattering (or coin) matrix C = σ ⊗ GLYPH<BD> n d . The full wave propagator (or quantum walk) U$_{0}$ is obtained from C after identifying incoming and outgoing waves of adjacent unit cells accordingly. The spectrum of the unperturbed walk U$_{0}$ exhibits a band structure as shown in Fig. 2 a), here for d = 2. (For a finite lattice, the quasi-momenta κ$_{x}$, κ$_{y}$ are discretised according to κ$_{x,y}$ = 2 πj/ ( na ); j = 0 . . . n − 1.) FIG. 2: a) The band structure at λ = 0 for d = 2 and in the limit n → ∞ with ⃗ κ , the Bloch wave numbers; b) the eigenphases of U λ for n = 11, d = 2; c) the avoided crossing with spectral gap Δ at λ = 1 and k mod 2 π/a = 0 together with the approximate eigenstates | ν λ 〉 and | Φ 0 〉 (dashed lines). invariant two-level subspace near the crossing spanned by | Φ$_{0}$ 〉 and | ν$_{λ}$$_{=1}$ 〉 . The technique developed in [14] for the hypercube has been adapted to regular grids. We will only give the result here, further details will be presented elsewhere [18]. One finds for the normalised vector | ν$_{λ}$$_{=1}$ 〉 Following Ambainis, Kempe and Rivosh (AKR), the quantum walk U$_{0}$ acts as a search algorithm after mark- ing a target vertex | v 〉 by a modified scattering matrix σ ′ [15], that is, one considers C ′ = C − ( σ − σ ′ ) ⊗ | v 〉 〈 v | . The AKR search uses σ ′ = − GLYPH<BD> 2 $_{d}$. Since | s 〉 is an eigen- vector of σ , we may write U ′ = U$_{0}$ (1 − 2 | sv 〉 〈 sv | ), where | sv 〉 = | s 〉 ⊗ | v 〉 . The search algorithm is initialised in the uniform state | Φ$_{0}$ 〉 = 1 $_{/}$√$_{N}$ ∑ ⃗ x | s⃗ x 〉 , and the walk ( U ′ ) T | Φ$_{0}$ 〉 localizes at | v 〉 after T ∝ √$_{N}$ steps. The AKR search can be analysed by defining a one parameter family of unitary operators [14] | ν$_{λ}$$_{=1}$ 〉 = −〈 sv | v$_{λ}$$_{=1}$ $_{〉}$√ 2 N ∑ ⃗ $^{κ}$̸ = ⃗ 0 e − 2 $^{π}$ı n ⃗ κ⃗ v × (3) ( e ı θ ⃗ κ 1 − e ı θ ⃗ κ ∣ Φ + ⃗ κ 〉 + e − ı θ ⃗ κ 1 − e − ı θ ⃗ κ ∣ $_{Φ}$− ⃗ κ 〉 ) , U$_{λ}$ = U$_{0}$ + ( e ı πλ − 1 ) U$_{0}$ | sv 〉 〈 sv | ; (2) 〈 sv | ν$_{λ}$$_{=1}$ 〉 = { O (1 / √ log N ) for d = 2 , (1) for d > 2 . H $^{2}$×2 = ( k$_{l}$a − ı ϵ ı ϵ k$_{l}$a ) (4) where ⃗ v is the position of the target vertex and | $_{Φ}$± ⃗ κ 〉 and ± θ$_{⃗}$ κ are the eigenvectors and eigenphases of the un- perturbed walk U$_{0}$ [15]. The d -dimensional label ⃗ κ with κ$_{i}$ = 0 , . . . , n − 1 is equivalent to the (discretised) Bloch wave number, see Fig. 2 a). The eigenphases are ex- plicitly given as cos θ$_{⃗}$ κ = 1 d ∑ d i $_{=1}$cos 2 πκ i n . The overlap- matrix element 〈 sv | ν$_{λ}$$_{=1}$ 〉 can be estimated [18]: one obtains U$_{0}$ for λ = 0 or 2 and the AKR search for λ = 1. The part of the eigenfrequency spectrum of U$_{λ}$ in- teracting with the perturbation is shown in Fig. 2 b). The spectrum is periodic in k with period 2 π/a independent of λ . When varying λ , a “perturber state” | ν$_{λ}$ 〉 emerges which crosses the k mod 2 π/a = 0 axis at λ = 1. The re- sulting avoided crossing between the initial state | Φ$_{0}$ 〉 and | ν$_{λ}$ 〉 is shown in Fig. 2 c). Note that | Φ$_{0}$ 〉 is the fully sym- metric eigenstate of U$_{0}$ corresponding to a d -dimensional Bloch-vector ⃗ κ = 0 of the unperturbed spectrum with eigenvalue k mod 2 π/a = 0. Like in Grover’s algorithm, the quantum search U ′ = U$_{λ}$$_{=1}$ rotates the initial state | Φ$_{0}$ 〉 into a localised state | ν$_{λ}$ 〉 which has here a strong overlap with the target state | sv 〉 . The search time T$_{0}$ is inversely proportional to the gap at the avoided crossing Δ, that is, T$_{0}$ ≈ π/ Δ. Detailed expressions for the leading order coefficients for d = 2 and 3 are given in [18]. We find that | ν$_{λ}$$_{=1}$ 〉 is ex- ponentially localised on the marked vertex ⃗ v ; note, that the overlap of | sv 〉 with a typical eigenstate of the unper- turbed spectrum | Φ$_{κ}$ 〉 is of the order O ( N − 1 2 ) ≪ O (1). Near the avoided crossing, the level dynamics can be described in terms of the two-level sub-space spanned by the orthogonal vectors | ν$_{λ}$ 〉 and | Φ$_{0}$ 〉 . Writing the unitary operator U$_{λ}$ as U$_{λ}$ = e − ı H λ , one obtains at λ = 1 in the {| Φ$_{0}$ 〉 , | ν$_{λ}$$_{=1}$ 〉} basis an effective two-dimensional Hamiltonian H$_{λ}$$_{=1}$ of the form In order to obtain an estimate for the search time T$_{0}$ as well as the efficiency of the search, that is, the matrix element 〈 sv | ν$_{λ}$ 〉 , it is essential to find the approximately