Renormalization Group for Quantum Walks
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1 Renormalization Group for Quantum Walks CompPhys13 Stefan Falkner, Stefan Boettcher and Renato Portugal Physics Department Emory University November 29, 2013 arxiv: Funding: NSF-DMR Grant #
2 Discrete time walks in one dimension ψn 2 t ψ t n 1 ψn t ψ t ψ t n+1 n+2 Discrete Time Evolution ψ t+1 = U ψ t U = P n n 1 + Q n n + 1 n Stefan Falkner (Emory University) Renormalization Group for Quantum Walks November 29, / 17
3 Discrete time walks in one dimension ψn 2 t ψ t n 1 ψn t ψ t ψ t n+1 n+2 Classical p t n = ψ t n 1 P + Q stochastic Discrete Time Evolution ψ t+1 = U ψ t U = P n n 1 + Q n n + 1 n Stefan Falkner (Emory University) Renormalization Group for Quantum Walks November 29, / 17
4 Discrete time walks in one dimension ψn 2 t ψ t n 1 ψn t ψ t ψ t n+1 n+2 Discrete Time Evolution ψ t+1 = U ψ t U = P n n 1 + Q n n + 1 n Classical p t n = ψ t n 1 P + Q stochastic Quantum Mechanical p t n = ψ t n ψ t n U = S (1 C) PP + QQ = 1 PQ = QP = 0 Stefan Falkner (Emory University) Renormalization Group for Quantum Walks November 29, / 17
5 One dimensional walks Probability in % Classical RW Site index Stefan Falkner (Emory University) Renormalization Group for Quantum Walks November 29, / 17
6 One dimensional walks dimesional coin space Probability in % Classical RW Site index Stefan Falkner (Emory University) Renormalization Group for Quantum Walks November 29, / 17
7 One dimensional walks dimesional coin space Probability in % dimesional coin space Classical RW Site index Stefan Falkner (Emory University) Renormalization Group for Quantum Walks November 29, / 17
8 Motivation model for quantum transport (Aharonov PRA 1993) Grover s search algorithm (Grover PRL 1997) ( ) find a marked vertex in a graph in T O N more general quantum search algorithm general quantum computation (Lovett PRA 2010) generally faster spreading compared to the random walk r 2 = t 2 dw with d QW w? = 1 2 d RW w analytic results only on translational invariant lattice Stefan Falkner (Emory University) Renormalization Group for Quantum Walks November 29, / 17
9 The generating function The discrete Laplace transform/ The z-transform The generating function and its backtransform ψ(z) = z t ψ t ψ t = 1 2π i t=0 z =1 z t 1 ψ(z) dz Stefan Falkner (Emory University) Renormalization Group for Quantum Walks November 29, / 17
10 The generating function The discrete Laplace transform/ The z-transform The generating function and its backtransform ψ(z) = z t ψ t ψ t = 1 2π i t=0 z =1 z t 1 ψ(z) dz ψ t+1 = U ψ t = ψ(z) = Ũ(z) ψ(z) + ψ 0 Recurrence equations become algebraic equations. Stefan Falkner (Emory University) Renormalization Group for Quantum Walks November 29, / 17
11 The generating function The discrete Laplace transform/ The z-transform The generating function and its backtransform ψ(z) = z t ψ t ψ t = 1 2π i t=0 z =1 z t 1 ψ(z) dz ψ t+1 = U ψ t = ψ(z) = Ũ(z) ψ(z) + ψ 0 Recurrence equations become algebraic equations. Generating function for p t n in quantum walks p n (z) = t 0 z t p t n = 1 2πi ψ(z/y) ψ(y) d y y Stefan Falkner (Emory University) Renormalization Group for Quantum Walks November 29, / 17
12 Renormalization group in one dimension ψ n 2 ψ n 1 ψ n ψ n+1 ψ n+2 Stefan Falkner (Emory University) Renormalization Group for Quantum Walks November 29, / 17
13 Renormalization group in one dimension ψ n 2 ψ n 1 ψ n ψ n+1 ψ n+2. ψ n 1 = P ψ n 2 + Q ψ n + R ψ n 1 ψ n = P ψ n 1 + Q ψ n+1 + R ψ n ψ n+1 = P ψ n + Q ψ n+2 + R ψ n+1. Stefan Falkner (Emory University) Renormalization Group for Quantum Walks November 29, / 17
14 Renormalization group in one dimension ψ n 2 ψ n 1 ψ n ψ n+1 ψ n+2. ψ n 1 = P ψ n 2 + Q ψ n + R ψ n 1 ψ n = P ψ n 1 + Q ψ n+1 + R ψ n ψ n+1 = P ψ n + Q ψ n+2 + R ψ n+1. Stefan Falkner (Emory University) Renormalization Group for Quantum Walks November 29, / 17
15 Renormalization group in one dimension ψ n 2 ψ n ψ n+2. ψ n 1 P ψ n 2 + Q ψ n + R ψ n 1 ψ n = P ψ n 2 + Q ψ n+2 + R ψ n ψ n+1 P ψ n + Q ψ n+2 + R ψ n+1 Stefan Falkner (Emory University) Renormalization Group for Quantum Walks November 29, / 17
16 The one dimensional classical random walk the recursion equations: p = p2 1 r three fixed points: q = q2 1 r r = r + 2pq 1 r (p, q, r ) = (1 r, 0, r ) t k L d w k with d w = 1 (p, q, r ) = (0, 1 r, r ) d w = 1 (p, q, r ) = (0, 0, 1) d w = 2 Stefan Falkner (Emory University) Renormalization Group for Quantum Walks November 29, / 17
17 The one dimensional quantum walk the recursion (matrix) equations: P = P (1 R) 1 P Q = Q (1 R) 1 Q R = R + P (1 R) 1 Q + Q (1 R) 1 P three different regimes: a non-trivial fixed point (despite unitarity)! quasi periodic/chaotic behavior of the parameters subspace of trivial fixed points: P = Q = 0 we need a generalization of the Jacobian Stefan Falkner (Emory University) Renormalization Group for Quantum Walks November 29, / 17
18 The Lyapunov exponents given a dynamical system evolving according to x(t + 1) = F (x(t)) J t - Jacobian at x(t), i.e.f (x(t) + δx) x(t + 1) + J t δx J t = t J t - relates the change in x(t + 1) when changing x(0) t =0 ) 1 {e λ i,t } - eigenvalue spectrum of Λ = (J t 2(t+1) J t λ 1, - (largest) Lyapunov exponent dominates long term behavior Stefan Falkner (Emory University) Renormalization Group for Quantum Walks November 29, / 17
19 Lyapunov exponent for the line 2 k = arg(z)/π Stefan Falkner (Emory University) Renormalization Group for Quantum Walks November 29, / 17
20 Lyapunov exponent for the line 2 k = arg(z)/π Stefan Falkner (Emory University) Renormalization Group for Quantum Walks November 29, / 17
21 Lyapunov exponent for the line 2 k = arg(z)/π Stefan Falkner (Emory University) Renormalization Group for Quantum Walks November 29, / 17
22 Lyapunov exponent for the line 2 k = arg(z)/π Stefan Falkner (Emory University) Renormalization Group for Quantum Walks November 29, / 17
23 Lyapunov exponent for the line 2 k = arg(z)/π Stefan Falkner (Emory University) Renormalization Group for Quantum Walks November 29, / 17
24 Lyapunov exponent for the line 2 k = arg(z)/π Stefan Falkner (Emory University) Renormalization Group for Quantum Walks November 29, / 17
25 Scaling of the Lyapunov exponent for the line 1 k = arg(z) e log(2) d w k d w =1 Stefan Falkner (Emory University) Renormalization Group for Quantum Walks November 29, / 17
26 Scaling of the Lyapunov exponent for the line 1 k = arg(z) e log(2) d w k d w =1 Stefan Falkner (Emory University) Renormalization Group for Quantum Walks November 29, / 17
27 Scaling of the Lyapunov exponent for the line 1 k = arg(z) e log(2) d w k d w =1 Stefan Falkner (Emory University) Renormalization Group for Quantum Walks November 29, / 17
28 Scaling of the Lyapunov exponent for the line 1 k = arg(z) e log(2) d w k d w =1 Stefan Falkner (Emory University) Renormalization Group for Quantum Walks November 29, / 17
29 Scaling of the Lyapunov exponent for the line 1 k = arg(z) e log(2) d w k d w =1 Stefan Falkner (Emory University) Renormalization Group for Quantum Walks November 29, / 17
30 Scaling of the Lyapunov exponent for the line 1 k = arg(z) e log(2) d w k d w =1 Stefan Falkner (Emory University) Renormalization Group for Quantum Walks November 29, / 17
31 3 regular Hanoi network HN3 hierarchichal network (Boettcher EPL 2008) exact renormalization interpolates between lattices and smallworld networks d RW w = 2 log 2 (φ) = Stefan Falkner (Emory University) Renormalization Group for Quantum Walks November 29, / 17
32 Lyapunov exponent for HN3 2 k = arg(z) π 1 Stefan Falkner (Emory University) Renormalization Group for Quantum Walks November 29, / 17
33 Lyapunov exponent for HN3 2 k = arg(z) π 1 Stefan Falkner (Emory University) Renormalization Group for Quantum Walks November 29, / 17
34 Lyapunov exponent for HN3 2 k = arg(z) π 1 Stefan Falkner (Emory University) Renormalization Group for Quantum Walks November 29, / 17
35 Lyapunov exponent for HN3 2 k = arg(z) π 1 Stefan Falkner (Emory University) Renormalization Group for Quantum Walks November 29, / 17
36 Lyapunov exponent for HN3 2 k = arg(z) π 1 Stefan Falkner (Emory University) Renormalization Group for Quantum Walks November 29, / 17
37 Lyapunov exponent for HN3 2 k = arg(z) π 1 Stefan Falkner (Emory University) Renormalization Group for Quantum Walks November 29, / 17
38 Scaling of the Lyapunov exponent for HN3 0.8 k = (π arg(z)) e d w log(2) k d w = 1 2 (2 log 2(ϕ)) Stefan Falkner (Emory University) Renormalization Group for Quantum Walks November 29, / 17
39 Scaling of the Lyapunov exponent for HN3 0.8 k = (π arg(z)) e d w log(2) k d w = 1 2 (2 log 2(ϕ)) Stefan Falkner (Emory University) Renormalization Group for Quantum Walks November 29, / 17
40 Scaling of the Lyapunov exponent for HN3 0.8 k = (π arg(z)) e d w log(2) k d w = 1 2 (2 log 2(ϕ)) Stefan Falkner (Emory University) Renormalization Group for Quantum Walks November 29, / 17
41 Scaling of the Lyapunov exponent for HN3 0.8 k = (π arg(z)) e d w log(2) k d w = 1 2 (2 log 2(ϕ)) Stefan Falkner (Emory University) Renormalization Group for Quantum Walks November 29, / 17
42 Scaling of the Lyapunov exponent for HN3 0.8 k = (π arg(z)) e d w log(2) k d w = 1 2 (2 log 2(ϕ)) Stefan Falkner (Emory University) Renormalization Group for Quantum Walks November 29, / 17
43 Scaling of the Lyapunov exponent for HN3 0.8 k = (π arg(z)) e d w log(2) k d w = 1 2 (2 log 2(ϕ)) Stefan Falkner (Emory University) Renormalization Group for Quantum Walks November 29, / 17
44 Scaling of the Lyapunov exponent for HN3 0.8 k = (π arg(z)) e d w log(2) k d w = 1 2 (2 log 2(ϕ)) Stefan Falkner (Emory University) Renormalization Group for Quantum Walks November 29, / 17
45 The dual Sierpinsky gasket degree 3 fractal exact renormalization d RW w = log 2 5 one absorbing boundary (square) Stefan Falkner (Emory University) Renormalization Group for Quantum Walks November 29, / 17
46 Lyapunov exponent for the dual Sierpinsky gasket Stefan Falkner (Emory University) Renormalization Group for Quantum Walks November 29, / 17
47 Scaling of the Lyapunov exponent for the DSG Stefan Falkner (Emory University) Renormalization Group for Quantum Walks November 29, / 17
48 Conclusions first attempt to study quantum walks on graphs without translational invariance renormaliztion group leads to potentially chaotic recursion equations scaling determined by non-local properties rather than simple fixed points Lyapunov exponents and observables scale similarly hints for d QW w = d RW w How to analyze situation where no fixed point on the unit circle exists? universality among different coins? problem of localization? Stefan Falkner (Emory University) Renormalization Group for Quantum Walks November 29, / 17
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