Transport Properties of Quantum Walks, Deterministic and Random

Transcription

1 Transport Properties of Quantum Walks, Deterministic and Random Alain JOYE i f INSTITUT FOURIER AQOS, Autrans, 8-19/7/2013 p.1/29

2 Unitary Quantum Walks Interdisciplinary Concept Quantum from Quantum Physics Walk from Probability Theory Well studied in Computer Science Maths QW Computer Science Physics AQOS, Autrans, 8-19/7/2013 p.2/29

3 Unitary Quantum Walks Interdisciplinary Concept Quantum from Quantum Physics Walk from Probability Theory Well studied in Computer Science Maths QW Computer Science Interest for Physics Computer Scientists: elaboration of softwares for Quantum Computers Aharonov et al 93, Grover 96, Childs et al Physicists: effective dynamics of complex Quantum Systems Feynman 82, Chalker-Coddington 88, Meyer Mathematicians: non-commutative extensions of Random Walks Nayak et al 00, Ambainis et al 01, Konno 02,... AQOS, Autrans, 8-19/7/2013 p.2/29

4 Search Algorithm Search in unstructured list Q: Does S N = {x 1,x 2,,x N } contain the elmt. x? If so, find it. Algorithmic Answer: { If x1 =x stop pick x 1 If x 1 x pick x 2 iterate Algorithm typically requires O(N) steps to determine if x S N and find it. AQOS, Autrans, 8-19/7/2013 p.3/29

5 Search Algorithm Search in unstructured list Q: Does S N = {x 1,x 2,,x N } contain the elmt. x? If so, find it. Algorithmic Answer: { If x1 =x stop pick x 1 If x 1 x pick x 2 iterate Algorithm typically requires O(N) steps to determine if x S N and find it. Assume random state a) x S N (with proba p ). N steps required. b) x S N (with proba 1 p ). Stop after k N steps. prob (x found at step k )= 1/N T = N k=1 k N = N+1 2. AQOS, Autrans, 8-19/7/2013 p.3/29

6 Search Algorithm Search in unstructured list Q: Does S N = {x 1,x 2,,x N } contain the elmt. x? If so, find it. Algorithmic Answer: { If x1 =x stop pick x 1 If x 1 x pick x 2 iterate Algorithm typically requires O(N) steps to determine if x S N and find it. Assume random state a) x S N (with proba p ). N steps required. b) x S N (with proba 1 p ). Stop after k N steps. prob (x found at step k )= 1/N T = N k=1 Assuming random choice of states k N = N+1 2. i.e. a random walk on S N same conclusions. AQOS, Autrans, 8-19/7/2013 p.3/29

7 Grover s Algorithm Grover 96 Quantum Version: S N C N Q: How to find the (normalized) state x C N? Assume x C N charact. by an oracle U x U(N) s.t. U x x = x and U x ϕ = +ϕ, for all ϕ span{x} Algorithmic Answer: Let s = 1 N N j=1 j and U s U(N) s.t. U s s = s and U s ϕ = ϕ, for all ϕ span{s} AQOS, Autrans, 8-19/7/2013 p.4/29

8 Grover s Algorithm Grover 96 Quantum Version: S N C N Q: How to find the (normalized) state x C N? Assume x C N charact. by an oracle U x U(N) s.t. U x x = x and U x ϕ = +ϕ, for all ϕ span{x} Algorithmic Answer: Let s = 1 N N j=1 j and U s U(N) s.t. U s s = s and U s ϕ = ϕ, for all ϕ span{s} Sequence of States s s 1 = (U s U x ) 1 s s 2 = (U s U x ) 2 s s r = (U s U x ) r s Fact: For r(n) π N/4, x s r(n) 2 1 O(1/N) (if x s 0 ) Quantum is Faster It takes O( N) steps to find the state x with large prob. by a Quantum Walk in state space AQOS, Autrans, 8-19/7/2013 p.4/29

9 Quantum Walk on Z First Try Hilbert space l 2 (Z) with ONB { j } j Z Unitary op. U on l 2 (Z) Jumps from j to j +1, j 1 only, for all j Z AQOS, Autrans, 8-19/7/2013 p.5/29

10 Quantum Walk on Z First Try Hilbert space l 2 (Z) with ONB { j } j Z Unitary op. U on l 2 (Z) Jumps from j to j +1, j 1 only, for all j Z U j = α j 1 +β j +1, α,β C s.t. α 2 + β 2 = 1 α β l j-1> l j > l j+1 > Z Orthogonality: 0 = j j +2 = Uj U(j +2) = βα Only possibilities: Left (β = 0 ) or Right (α = 0 ) Shifts AQOS, Autrans, 8-19/7/2013 p.5/29

11 Quantum Walk on Z Second Try Add Internal degree of freedom: C 2 a.k.a Spin l j-1> l j > l j+1 > Z Hilbert space l 2 (Z;C 2 ) l 2 (Z) C 2 Ψ = j Z j ψ(j), where ( ) a(j) ψ(j) = = a(j) +1 +b(j) 1 C 2, b(j) s.t. j Z ψ(j) 2 = j Z a(j) 2 + b(j) 2 <. ONB { j τ } τ {+1, 1} j Z QM interpretation: = { j +1, j 1 } j Z prob. (Ψ on "site" j) = ψ(j) 2 (Ψ normalized) AQOS, Autrans, 8-19/7/2013 p.6/29

12 Coined Quantum Walk I l j-1> l j > l j+1 > Z Construction of Unitary op. U on l 2 (Z) C 2 : Jumps from j τ to (j +1) τ, (j 1) τ only, for all j Z,τ,τ,τ {+1, 1} First Step: Let C U(2) to act locally on Spin deg. of freed. (I C) j ϕ = j (Cϕ), ϕ C 2 AQOS, Autrans, 8-19/7/2013 p.7/29

13 Coined Quantum Walk I l j-1> l j > l j+1 > Z Construction of Unitary op. U on l 2 (Z) C 2 : Jumps from j τ to (j +1) τ, (j 1) τ only, for all j Z,τ,τ,τ {+1, 1} First Step: Let C U(2) to act locally on Spin deg. of freed. (I C) j ϕ = j (Cϕ), ϕ C 2 Second Step: Spin-dependent shift op. S B(l 2 (Z) C 2 ) S j +1 = (j +1) +1, S j 1 = (j 1) 1 > I+1> I+1 I+1> I-1 > I-1> I-1> l j-1> l j > l j+1 > i.e. S S + S Z AQOS, Autrans, 8-19/7/2013 p.7/29

14 Coined Quantum Walk II l j-1> l j > l j+1 > Z Definition: U(C) = S(I C) on l 2 (Z) C 2 Remarks: The Coin Matrix C U(2) is a parameter of the QW. ( ) r t Take C =, s.t. 0 r,t and r 2 +t 2 = 1. t r U(C) j +1 = r (j +1) +1 t (j 1) 1 U(C) j 1 = t (j +1) +1 +r (j 1) 1 AQOS, Autrans, 8-19/7/2013 p.8/29

15 Coined Quantum Walk II l j-1> l j > l j+1 > Z Definition: U(C) = S(I C) on l 2 (Z) C 2 Remarks: The Coin Matrix C U(2) is a parameter of the QW. ( ) r t Take C =, s.t. 0 r,t and r 2 +t 2 = 1. t r U(C) j +1 = r (j +1) +1 t (j 1) 1 U(C) j 1 = t (j +1) +1 +r (j 1) 1 QW RW: Starting from j +1 Prob (j j +1) = r2 Prob (j j 1) = t 2 Prob (j j +1) = t2 Starting from j 1 Prob (j j 1) = r 2 ( ) 1 1 "Symmetric Walk": r = t, i.e. Hadamard Matrix C = AQOS, Autrans, 8-19/7/2013 p.8/29

16 QW Dynamics "on Z " U(C) = S(I C) Discrete Time Dynamics Given U(C) and Ψ 0 QW in l 2 (Z) C 2 Ψ 0 Ψ 1 = U(C) 1 Ψ 0 Ψ 2 = U(C) 2 Ψ 0 Ψ n = U(C) n Ψ 0 AQOS, Autrans, 8-19/7/2013 p.9/29

17 QW Dynamics "on Z " U(C) = S(I C) Discrete Time Dynamics Given U(C) and Ψ 0 QW in l 2 (Z) C 2 Ψ 0 Ψ 1 = U(C) 1 Ψ 0 Ψ 2 = U(C) 2 Ψ 0 Ψ n = U(C) n Ψ 0 RW vs QW Random Walker Coin Quantum Walker Coin Matrix C U(2) AQOS, Autrans, 8-19/7/2013 p.9/29

18 QW Dynamics "on Z " U(C) = S(I C) Discrete Time Dynamics Given U(C) and Ψ 0 QW in l 2 (Z) C 2 Ψ 0 Ψ 1 = U(C) 1 Ψ 0 Ψ 2 = U(C) 2 Ψ 0 Ψ n = U(C) n Ψ 0 RW vs QW Random Walker Coin Quantum Walker Coin Matrix C U(2) Position Distribution at time n State at time n : Ψ n = U(C) n Ψ 0 Prob(QW on site j at time n ) = ψ n (j) 2 AQOS, Autrans, 8-19/7/2013 p.9/29

19 Hadamard Walk: C = 1 2 ( ) Ψ 0 = ( ) 1 i Distribution of Ψ n (j) 2 n 75 AQOS, Autrans, 8-19/7/2013 p.10/29

20 Hadamard Walk: C = 1 2 ( Distribution of Ψ n (j) 2 n ) Ψ 0 = 0 ( ) a, a b b AQOS, Autrans, 8-19/7/2013 p.11/29

21 QM Characteristics of { ψ n ( ) 2 }, with Ψ 0 = 0 ϕ Expectation: X n Ψ0 = j Z j ψ n(j) 2 Variance: Var(X n ) Ψ0 = j Z (j X n Ψ0 ) 2 ψ n (j) 2, as n AQOS, Autrans, 8-19/7/2013 p.12/29

22 QM Characteristics of { ψ n ( ) 2 }, with Ψ 0 = 0 ϕ Expectation: X n Ψ0 = j Z j ψ n(j) 2 Variance: Var(X n ) Ψ0 = j Z (j X n Ψ0 ) 2 ψ n (j) 2, as n Special Cases C Diagonal Spin +1 / 1 components shifted to the Left/Right X n Ψ0 = O(n) Var(X n ) Ψ0 = O(n 2 ) > I+1> I+1 I+1> I-1 > I-1> I-1> l j-1> l j > l j+1 > C Off-Diagonal Quantum Walker stuck on sites { 1, 0, 1} Z X n Ψ0 = O(1) Var(X n ) Ψ0 = O(1) l j-1> l j > l j+1 > Z AQOS, Autrans, 8-19/7/2013 p.12/29

23 QM Characteristics of { ψ n ( ) 2 }, with Ψ 0 = 0 ϕ Generic C X n Ψ0 = O(n) and Var(X n ) Ψ0 = O(n 2 ), i.e. Ballistic Behaviour Limiting distribution of X n /n as n. Konno 02 AQOS, Autrans, 8-19/7/2013 p.13/29

24 QM Characteristics of { ψ n ( ) 2 }, with Ψ 0 = 0 ϕ Generic C X n Ψ0 = O(n) and Var(X n ) Ψ0 = O(n 2 ), i.e. Ballistic Behaviour Limiting distribution of X n /n as n. Konno 02 Spectral Method Transl. invar. on Z Fourier transform l 2 (Z 2 ) C 2 L 2 (T k ;C 2 ) ˆψ(k) = j ψ(j)eijk ( ) e ik 0 U(C) = S(I C) C C(k) as a U(2) -valued mult. op. 0 e ik C(k) = e iω(k) v + (k) v + (k) +e iω(k) v (k) v (k) ω(k) = arccos(rcos(k)) σ(u(c)) = σ a.c. (U(C)) ballistic behavior AQOS, Autrans, 8-19/7/2013 p.13/29

25 QM Characteristics of { ψ n ( ) 2 }, with Ψ 0 = 0 ϕ Generic C X n Ψ0 = O(n) and Var(X n ) Ψ0 = O(n 2 ), i.e. Ballistic Behaviour Limiting distribution of X n /n as n. Konno 02 Spectral Method Transl. invar. on Z Fourier transform l 2 (Z 2 ) C 2 L 2 (T k ;C 2 ) ˆψ(k) = j ψ(j)eijk ( ) e ik 0 U(C) = S(I C) C C(k) as a U(2) -valued mult. op. 0 e ik C(k) = e iω(k) v + (k) v + (k) +e iω(k) v (k) v (k) ω(k) = arccos(rcos(k)) σ(u(c)) = σ a.c. (U(C)) ballistic behavior (C n ( )ˆψ)(k) = f (k)e inω(k) v (k) +f + (k) e +inω(k) v + (k), n Z Var(X n ) Ψ0 = j j2 ψ n (j) 2 = n 2 T k ( k ω(k)) 2 ˆψ(k) 2 +O(n) Group velocity: k ω(k) AQOS, Autrans, 8-19/7/2013 p.13/29

26 Random Quantum Walk: Static Disorder Site-dep. Coin Matrices C {C(j)} j Z U j ±1 = C +,± (j) (j +1) +1 +C,± (j) (j 1) 1 Spatial inhomogeneity Coin C ω (j) at each site j is random AQOS, Autrans, 8-19/7/2013 p.14/29

27 Random Quantum Walk: Static Disorder Site-dep. Coin Matrices C {C(j)} j Z U j ±1 = C +,± (j) (j +1) +1 +C,± (j) (j 1) 1 Spatial inhomogeneity Coin C ω (j) at each site j is random Physics Quantum Hall Dynamics Q. Network Model Chalker-Coddington 88 Exp. propagation of polarized photons in networks of coupled wave guides with static random phase shifts Crespi et al 13 AQOS, Autrans, 8-19/7/2013 p.14/29

28 Random Quantum Walk: Static Disorder Site-dep. Coin Matrices C {C(j)} j Z U j ±1 = C +,± (j) (j +1) +1 +C,± (j) (j 1) 1 Spatial inhomogeneity Coin C ω (j) at each site j is random Assumptions: J.-Merkli 09 {C ω (k)} k Z is an i.i.d. U(2) valued random variable The quantum transition amplitudes to the right and left are independent The quantum transition probabilities to the right and left are deterministic. AQOS, Autrans, 8-19/7/2013 p.15/29

29 Random Quantum Walk: Static Disorder Site-dep. Coin Matrices C {C(j)} j Z U j ±1 = C +,± (j) (j +1) +1 +C,± (j) (j 1) 1 Spatial inhomogeneity Coin C ω (j) at each site j is random Assumptions: J.-Merkli 09 {C ω (k)} k Z is an i.i.d. U(2) valued random variable The quantum transition amplitudes to the right and left are independent The quantum transition probabilities to the right and left are deterministic. Consequence Lemma: Up to trivial transformations, r,t [0,1] s.t. r 2 +t 2 = 1 and i.i.d random phases {ω ± k } k Z C ω (k) := ( re iω + k te iω+ k te iω k re iω k ), k Z. AQOS, Autrans, 8-19/7/2013 p.15/29

30 Random Quantum Walk: Static Disorder Structure Property: U(C) U ω (C) Let { k τ } τ {±} k Z be ONB of l 2 (Z) C 2 Set D(ω) = diag(exp(iω τ k)), then random time one dynamics U ω (C) = D(ω)U(C) AQOS, Autrans, 8-19/7/2013 p.16/29

31 Random Quantum Walk: Static Disorder Structure Property: U(C) U ω (C) Let { k τ } τ {±} k Z be ONB of l 2 (Z) C 2 Set D(ω) = diag(exp(iω τ k)), then random time one dynamics U ω (C) = D(ω)U(C) Thm "Anderson Localization" J.-Merkli 09, Ahlbrecht et al 11 If ω ± k have an L density, For any r [0,1), any p N, any Ψ 0 of cpct. supp. : sup Xn ω p ψ0 < C ω, a.s. n Z AQOS, Autrans, 8-19/7/2013 p.16/29

32 Random Quantum Walk: Static Disorder Structure Property: U(C) U ω (C) Let { k τ } τ {±} k Z be ONB of l 2 (Z) C 2 Set D(ω) = diag(exp(iω τ k)), then random time one dynamics U ω (C) = D(ω)U(C) Thm "Anderson Localization" J.-Merkli 09, Ahlbrecht et al 11 If ω ± k have an L density, For any r [0,1), any p N, any Ψ 0 of cpct. supp. : sup Xn ω p ψ0 < C ω, a.s. n Z First RQW: 09 Konno: ( re iω k t ) C ω (k) = t re iω k ballistic behaviour! AQOS, Autrans, 8-19/7/2013 p.16/29

33 Random Quantum Walk: Static Disorder Generalization: RQW on Z d H = l 2 (Z d ) C 2d J. 12 ONB: { τ } τ I±, I ± {±1,±2,...,±d} for C 2d Jump Function r : I ± Z d s.t. τ r(τ) r(+2) Z/ - -2 S = τ I ± k Z d k +r(τ) k P τ r(+1) r(-1) r(-2) AQOS, Autrans, 8-19/7/2013 p.17/29

34 Random Quantum Walk: Static Disorder Generalization: RQW on Z d H = l 2 (Z d ) C 2d J. 12 ONB: { τ } τ I±, I ± {±1,±2,...,±d} for C 2d Jump Function r : I ± Z d s.t. τ r(τ) r(+2) Z/ - -2 S = τ I ± k Z d k +r(τ) k P τ C U(2d), U(C) = S(I C), U ω (C) = D(ω)U(C) on l 2 (Z d ) C 2d r(-1) r(-2) r(+1) AQOS, Autrans, 8-19/7/2013 p.17/29

35 Random Quantum Walk: Static Disorder Generalization: RQW on Z d H = l 2 (Z d ) C 2d J. 12 ONB: { τ } τ I±, I ± {±1,±2,...,±d} for C 2d Jump Function r : I ± Z d s.t. τ r(τ) r(+2) Z/ - -2 S = τ I ± k Z k +r(τ) k P d τ C U(2d), U(C) = S(I C), U ω (C) = D(ω)U(C) on l 2 (Z d ) C 2d "Large Disorder Regime" Anderson Localization If C close to certain Permutation matrices r(-1) r(-2) r(+1) + similar hyp. sup Xn ω p ψ0 < C ω, a.s. n Z Rem: Permut. matrices Off-Diag C when d = 1 Stuck Quantum Walker AQOS, Autrans, 8-19/7/2013 p.17/29

36 Random Quantum Walk: Temporal Disorder Temporal disorder New coin C ω at each time step: Kosk et al 06 U ω (n,0) = S(I C ω (n)) S(I C ω (2))S(I C ω (1)) Random Coin op s {C ω (k)} k N, i.i.d. with common distribution µ on U(2d) r(-1) Deterministic Jump Function r : I ± Z d s.t. τ r(τ) r(+2) Z/ - -2 _ r r(+1) r(-2) AQOS, Autrans, 8-19/7/2013 p.18/29

37 Random Quantum Walk: Temporal Disorder Temporal disorder New coin C ω at each time step: Kosk et al 06 U ω (n,0) = S(I C ω (n)) S(I C ω (2))S(I C ω (1)) Random Coin op s {C ω (k)} k N, i.i.d. with common distribution µ on U(2d) r(-1) Deterministic Jump Function r : I ± Z d s.t. τ r(τ) r(+2) Averaging over randomness of QM expectations: X n Ψ0 = E ω ( Xn ω Ψ0 ) w n ( ) = E ω ( ψn( ) ω 2 ) Z/ - -2 _ r r(+1) r(-2) AQOS, Autrans, 8-19/7/2013 p.18/29

38 Random Quantum Walk: Temporal Disorder Temporal disorder New coin C ω at each time step: Kosk et al 06 U ω (n,0) = S(I C ω (n)) S(I C ω (2))S(I C ω (1)) Random Coin op s {C ω (k)} k N, i.i.d. with common distribution µ on U(2d) r(-1) Deterministic Jump Function r : I ± Z d s.t. τ r(τ) r(+2) Averaging over randomness of QM expectations: X n Ψ0 = E ω ( Xn ω Ψ0 ) w n ( ) = E ω ( ψn( ) ω 2 ) Z/ - -2 _ r r(+1) r(-2) Thm "Diffusive Behaviour" Ahlbrecht et al 11, J. 11, Hamza-J. 12 lim n X n Ψ0 n = µ Drift lim n (X n nµ) 2 Ψ0 n = D Diffusion Const. AQOS, Autrans, 8-19/7/2013 p.18/29

39 Summary Coined Random Quantum Walks on Z d C uniform ballistic behaviour C ω random in space a.s. localization C ω random in time average diffusive behaviour Generic situation AQOS, Autrans, 8-19/7/2013 p.19/29

40 Summary Coined Random Quantum Walks on Z d C uniform ballistic behaviour C ω random in space a.s. localization C ω random in time average diffusive behaviour Generic situation Remarks C ω random in time and space average diffusive behaviour Hamza-J. 12, Ahlbrecht et al 13 (R)QW on Z and CMV matrices: special cases of discrete dyn. systems generated by 5-diag. unitary operators on l 2 (Z) Bourget-Howland-J. 03 Warning a notion of localization for QW point spectrum in U(C). AQOS, Autrans, 8-19/7/2013 p.19/29

41 More on Random Quantum Walk II Random evol. U ω (n,0) = S(I C ω (n)) S(I C ω (2)) S(I C ω (1)) AQOS, Autrans, 8-19/7/2013 p.20/29

42 More on Random Quantum Walk II Random evol. U ω (n,0) = S(I C ω (n)) S(I C ω (2)) S(I C ω (1)) = where J ω k (n) = τn,,τ 1 τ j I ± x Z d x Z d k ρn τn,,τ 1 nj=1 r(τ j )=k n x+ j=1 x+k x J ω k (n), r(τ j ) x Pτn C ω (n) P τ1 C ω (1) ρ = max τ I ± r(τ) P τn C ω (n) P τ1 C ω (1) M 2d (C) AQOS, Autrans, 8-19/7/2013 p.20/29

43 More on Random Quantum Walk II Random evol. U ω (n,0) = S(I C ω (n)) S(I C ω (2)) S(I C ω (1)) = where J ω k (n) = τn,,τ 1 τ j I ± x Z d x Z d k ρn τn,,τ 1 nj=1 r(τ j )=k n x+ j=1 x+k x J ω k (n), r(τ j ) x Pτn C ω (n) P τ1 C ω (1) ρ = max τ I ± r(τ) P τn C ω (n) P τ1 C ω (1) }{{} M 2d(C) τ n τ n C ω (n) τ n 1 τ 1 τ 1 C ω (1) Note Ψ ω n = U ω (n,0)( 0 ϕ 0 ) = k Jk ω (n)ϕ 0 k ψn(k) ω k ρn k ρn AQOS, Autrans, 8-19/7/2013 p.21/29

44 QM Probability distribution Lattice observables If f : Z d C, let F := I f on D(F) H s.t. F(ϕ 0 k ) = f(k)(ϕ 0 k ), ϕ 0 C 2d. AQOS, Autrans, 8-19/7/2013 p.22/29

45 QM Probability distribution Lattice observables If f : Z d C, let F := I f on D(F) H s.t. F(ϕ 0 k ) = f(k)(ϕ 0 k ), ϕ 0 C 2d. QM Expectation Value Let F, Ψ 0 = ϕ 0 0, Ψ ω n = U ω (n,0)ψ 0 F ω Ψ 0 (n) = k Z d f(k) ψ ω n(k) 2, with ψn(k) ω 2 = Jk ω (n)ϕ 0 2 C 2d s.t. k Z ψn(k) ω 2 = Ψ d 0 2 H = 1. AQOS, Autrans, 8-19/7/2013 p.22/29

46 QM Probability distribution Lattice observables If f : Z d C, let F := I f on D(F) H s.t. F(ϕ 0 k ) = f(k)(ϕ 0 k ), ϕ 0 C 2d. QM Expectation Value Let F, Ψ 0 = ϕ 0 0, Ψ ω n = U ω (n,0)ψ 0 F ω Ψ 0 (n) = k Z d f(k) ψ ω n(k) 2, with ψn(k) ω 2 = Jk ω (n)ϕ 0 2 C 2d s.t. k Z ψn(k) ω 2 = Ψ d 0 2 H = 1. Random Distribution Let X ω n be a RV on Z d s.t. Prob(X ω n = k) = W ω k (n) = J ω k (n)ϕ 0 2 C 2d, F ω Ψ 0 (n) = E W ω k (n)(f(x ω n)). AQOS, Autrans, 8-19/7/2013 p.22/29

47 Averaged QM Probability distribution Lattice observables If f : Z d C, let F := I f on D(F) H s.t. F(ϕ 0 k ) = f(k)(ϕ 0 k ), ϕ 0 C 2d. Averaged QM Expectation Value Let F, Ψ 0 = ϕ 0 0, Ψ ω n = U ω (n,0)ψ 0 F Ψ0 (n) = E ω F ω Ψ 0 (n) = k Z d f(k)e ω ψ ω n(k) 2, with E ω ψn(k) ω 2 = E ω Jk ω (n)ϕ 0 2 C 2d s.t. k Z E d ω ψn(k) ω 2 = Ψ 0 2 H = 1. Determin. Distribution Let X n be a RV on Z d s.t. Prob(X n = k) = w k (n) = E ω J ω k (n)ϕ 0 2 C 2d, F Ψ0 (n) = E wk (n)(f(x n )). Goal Understand {w k (n)} k Z d as n AQOS, Autrans, 8-19/7/2013 p.23/29

48 Ballistic vs. Diffusive Scaling Expected r -dependent Drift: Diffusive behaviour: X n vn (X n vn) n AQOS, Autrans, 8-19/7/2013 p.24/29

49 Ballistic vs. Diffusive Scaling Expected r -dependent Drift: Diffusive behaviour: X n vn (X n vn) n Tool Given X n {w k (n)} k Z d, and y T d let Φ n (y) = E w(n) (e iyx n ) = k Z d w k (n)e iyk s.t. i yj Φ n (y/n α ) y=0 = k Z d w k (n)k j /n α = E w(n) ((X n ) j /n α ) α = 1 ballistic scaling α = 1/2 diffusive scaling AQOS, Autrans, 8-19/7/2013 p.24/29

50 Ballistic vs. Diffusive Scaling Expected r -dependent Drift: Diffusive behaviour: X n vn (X n vn) n Tool Given X n {w k (n)} k Z d, and y T d let Φ n (y) = E w(n) (e iyx n ) = k Z d w k (n)e iyk s.t. i yj Φ n (y/n α ) y=0 = k Z d w k (n)k j /n α = E w(n) ((X n ) j /n α ) α = 1 ballistic scaling α = 1/2 diffusive scaling Rem: Charact. Funct. Φ n (y) is analytic in C d. AQOS, Autrans, 8-19/7/2013 p.24/29

51 Black Box Reformulation Steps Expansion of w k (n) = E ω Jk ω (n)ϕ 0 2 C sums over weighted paths 2d Extended space C d C d C d i.i.d. coin matrices Periodicity of Φ n (y) T d AQOS, Autrans, 8-19/7/2013 p.25/29

52 Black Box Reformulation Steps Expansion of w k (n) = E ω Jk ω (n)ϕ 0 2 C sums over weighted paths 2d Extended space C d C d C d i.i.d. coin matrices Periodicity of Φ n (y) T d Set for any (y,y ) T d T d D(y,y ) := d(y) d(y ), where d(y) = τ I ± e iyr(τ) τ τ, E := E ω (C ω C ω ), M(y,y ) := D(y,y )E Ψ 1 = τ I ± τ τ, Φ 0 = ϕ 0 ϕ 0 Char. Funct. For y T d, Φ n (y) = Ψ 1 M n (y v,v)φ 0 dṽ T d AQOS, Autrans, 8-19/7/2013 p.25/29

53 Spectral Analysis Φ n (y) = T d Ψ 1 M n (y v,v)φ 0 dṽ Properties For all v T d Ψ 1 = τ I ± τ τ invar. under M( v,v) and its adjoint Spr M( v,v) = M( v,v) = 1 AQOS, Autrans, 8-19/7/2013 p.26/29

54 Spectral Analysis Φ n (y) = T d Ψ 1 M n (y v,v)φ 0 dṽ Properties For all v T d Ψ 1 = τ I ± τ τ invar. under M( v,v) and its adjoint Spr M( v,v) = M( v,v) = 1 Assumption S For all v T d, σ(m( v,v)) D(0,1) = {1} and 1 is simple AQOS, Autrans, 8-19/7/2013 p.26/29

55 Spectral Analysis Φ n (y) = T d Ψ 1 M n (y v,v)φ 0 dṽ Properties For all v T d Ψ 1 = τ I ± τ τ invar. under M( v,v) and its adjoint Spr M( v,v) = M( v,v) = 1 Assumption S For all v T d, σ(m( v,v)) D(0,1) = {1} and 1 is simple Consequently 0 < δ < 1, a complex ngbhd. N of {0} T d s.t. (y,v) N σ(m(y v,v)) D(1,δ) = {λ 1 (y,v)} σ(m(y v,v))\{λ 1 (y,v)} D(0,1 δ) M(y v,v) = λ 1 (y,v)p(y,v)+r(y,v), Spr(R(y,v)) < 1 δ where λ 1 (y,v), P(y,v), R(y,v) are analytic in N and λ 1 (0,v) 1 and P(0,v) Ψ 1 Ψ 1 /(2d). AQOS, Autrans, 8-19/7/2013 p.26/29

56 Perturbation Theory Φ n (y) = T d Ψ 1 M n (y v,v)φ 0 dṽ Diffusive/Ballistic Scaling y y/n α << 1 High Powers M(y/n α v,v) n = λ 1 (y/n α,v) n P(y/n α,v)+o((1 δ) n ) AQOS, Autrans, 8-19/7/2013 p.27/29

57 Perturbation Theory Φ n (y) = T d Ψ 1 M n (y v,v)φ 0 dṽ Diffusive/Ballistic Scaling y y/n α << 1 High Powers M(y/n α v,v) n = λ 1 (y/n α,v) n P(y/n α,v)+o((1 δ) n ) Analytic Perturbation Theory For (0,v) N, λ 1 (y,v) 1+y i 2d τ I ± r(τ) 1 2 y D(v)y +O v( y 3 ) The map v D(v) M d (C) is real analytic in T d and D(v) j,k = 2 y j y k λ(0,v) For v T d, D(v) 0 and O v ( y 3 ) is unif. in v in a complex nbhd. of T d. AQOS, Autrans, 8-19/7/2013 p.27/29

58 Results in Average Assumption S a real analytic map T d v D(v) M d (R), s.t. D(v) 0. AQOS, Autrans, 8-19/7/2013 p.28/29

59 Results in Average Assumption S a real analytic map T d v D(v) M d (R), s.t. D(v) 0. Theorem Under S, unif. in y in cpct sets of C d, lim Φ n(y/n) = e iyr, r = 1 r(τ) n 2d τ I ± ry n Φ n (y/ n) = lim n e in T d e 1 2 y D(v)y dṽ. AQOS, Autrans, 8-19/7/2013 p.28/29

60 Results in Average Assumption S a real analytic map T d v D(v) M d (R), s.t. D(v) 0. Theorem Under S, unif. in y in cpct sets of C d, lim Φ n(y/n) = e iyr, r = 1 r(τ) n 2d τ I ± ry n Φ n (y/ n) = lim n e in lim n lim n Consequently T d e 1 2 y D(v)y dṽ. X i ψ0 (n) = r i Drift n (X nr) i (X nr) j ψ0 (n) = D ij (v)dṽ Diffusion Matrix. n T d + Moderate/Large Deviations Results AQOS, Autrans, 8-19/7/2013 p.28/29

61 Variants / Perspectives Variations Open Quantum Walks Interacting Quantum Walkers Localization for the Chalker-Coddington model of electron transport localization-delocalization spectral transitions for RQW defined on trees AQOS, Autrans, 8-19/7/2013 p.29/29

62 Variants / Perspectives Variations Open Quantum Walks Interacting Quantum Walkers Localization for the Chalker-Coddington model of electron transport localization-delocalization spectral transitions for RQW defined on trees Perspectives Delocalization regime for the Chalker-Coddington model? Statistics of eigenvalues for RQW on Z d? (De-)localization properties of Random Open Quantum Walks?... AQOS, Autrans, 8-19/7/2013 p.29/29