DVIOUT-fujin

2005 Limit Distribution of Quantum Walks and Weyl Equation 2006 3 2

1 2 2 4 2.1...................... 4 2.2......................... 5 2.3..................... 6 3 8 3.1........... 8 3.2.......................... 9 3.3 [7, 8]......................... 10 4 13 4.1..................... 13 4.2..................... 15 4.3....................... 16 5 19 5.1............... 19 5.2........................... 19 6 21 6.1 3.................... 21 6.2 3.................. 22 6.3 4......................... 22 7 25 A 27 B 29 1

1 1936 70 1985 2 [1, 2] 1994 AT&T,Bell RSA 2

[3] 3

2 2.1 1 1 Z = {, 2, 1, 0, 1, 2, } q 1 q p 1 p n x Z P n (1) (x) (P n (2) (x)) Ã! P n (1) Z Ã! (x) π P n (2) = (x) π 2π eikx W (k) n q (2.1) 1 q à e ik 0 W (k) = 0 e ik!ã 1 p p p 1 p! (2.2) i = 1 p =1/2 W (k) λ =0 λ =cosk P n (1) (x) =P n (2) (x) P n (x)/2 q Z Ã! π P n (x) = 2π eikx cos n k = 1 n 2 n (n + x)/2 π τ a n = t/τ,x x/a, k ak a, τ 0 p t (x) = 4

P t/τ (x/a)/a τ = a 2 0 (cos ak) t/τ =(1 a 2 k 2 /2+ ) t/τ e tk2 /2 Z p t (x) = 2π exp t 2 k2 + ikx = 1 2πt e x2 /2t t 1 2 p 2 x 2 t (x) =0 lim t 0 p t (x) =δ(x) t ( 2.1) 2.1: 200 2.2 (2.1) 2 2 Ã! Ψ (1) n (x) Ψ n (x) = Ψ (2) (2.3) n (x) 5

Ã! Ã! q α (2.1) 1 q β α, β C α 2 + β 2 =1 W (k) Ã! e ik 0 U(k) = A (2.4) 0 e ik A 2 2 (2.3) 1/2 σ j,j =1, 2, 3 2 2 I 2 3 p =(p 1,p 2,p 3 )= i h H(p) =σ p (2.5) σ =(σ 1, σ 2, σ 3 ) 0 1/2 i h t ˆΦ t (p) =(σ p)ˆφ t (p) 0 [4] 2.3 Ψ (1) Ψ (2) n x Ã! Ψ (1) n (x) Ψ (2) n (x) U U 6

S A U = SA Ã! Ã s + 0 S =, A = 0 s a c b d! s +,s s + Ψ (1) n (x) =Ψ (1) n (x +1) s Ψ (1) n (x) =Ψ (1) n (x 1) A AA = A A = I 2 A Ã! A = H = 1 1 1 2 1 1 Ã! Ã! Ψ (1) n+1 (x) Ψ (1) n (x) Ψ (2) n+1 (x) = U Ψ (2) n (x) Ã!Ã!Ã! s + 0 a b Ψ (1) n (x) = 0 s c d Ψ (2) n (x) Ã! aψ (1) n (x +1)+bΨ (2) n (x +1) = cψ (1) n (x 1) + dψ (2) (2.6) n (x 1) [5] [6] 7

3 lim n 3.1 (2.3) (2.6) µ (1) aψ n (x +1)+bΨ (2) n (x +1) Ψ n+1 (x) = cψ (1) n (x 1) + dψ (2) (3.1) n (x 1) µ a b A 2 2 A = U(2) c d k [ π, π) ˆΨ n (k) = µ ˆΨ(1) n (k) ˆΨ (2) n (k) Ψ (j) n (x) = Z π π 2π eikx ˆΨ(j) n (k) ˆΨ (j) n (k) = X Ψ (j) n (x)e ikx j =1, 2 x Z (3.1) ˆΨ n+1 (k) =U(k) ˆΨ n (k), n =0, 1, 2, U(k) (2.4) µ n ˆΨ α 0 (k) =, α, β β C, α 2 + β 2 =1 U(k) n ˆΨ n (k) =U(k) n ˆΨ 0 (k) (3.2) 8

n Ψ n (x) = = Z π π Z π π 2π eikx ˆΨn (k) 2π eikx U(k) n ˆΨ0 (k) n P n (x) = Ψ n (x) 2 = Ψ n(x)ψ n (x) Z π 0 Z π = 0 π 2π π 2π ei(k k )x ³ ˆΨ 0 (k0 )U (k 0 ) n ³U(k) n ˆΨ0 (k) (3.3) U(2) SU(2) e iϕ, ϕ [ π/2, π/2) U(2) SU(2) (3.3) A SU(2) determinant 1 2 2 H = 1 µ 1 1 U(2) (3.4) 2 1 1 H = e iπ/2 A, A = 1 µ i i SU(2) (3.5) 2 i i SU(2) ( Ã! ) a b SU(2) = A = ; a, b C, a 2 + b 2 =1 b a ( Ã! ue iθ 1 u2 e iφ = A = ; 1 u 2 e iφ ue iθ o u [0, 1], θ, φ [ π, π) (3.6) SU(2) u, θ, φ ( ) 3.2 X n n n =0, 1, 2, x Z f f(x n ) D E f(x n ) = X f(x)p n (x) x Z = X Z π 0 Z π f(x) 0 x Z π 2π e ik x ˆΨ n (k 0 ) π 2π eikx ˆΨn (k) 9

f(x) =x r,r =0, 1, 2, D E Xn r = X x Z Z π π 0 2π e ik0 x ˆΨ n (k 0 ) Z π π ½µ i d r ¾ e ikx ˆΨ n (k) 2π ˆΨ n (k) k [ π, π) Z π π 2π ½µ i d r ¾ Z π µ e ikx ˆΨ n (k) = π 2π eikx i d r ˆΨn (k) X r n D E Xn r = X e ixk =2πδ(k) x Z Z π π µ 2π ˆΨ n(k) i d r ˆΨn (k) f(x) x =0 f(x) = P j=0 a jx j D E Z π µ f(x n ) = π 2π ˆΨ n(k)f i d ˆΨ n (k) (3.7) 3.3 [7, 8] U(k) n x Z (3.3) n f (3.7) U(k) U(k) λ λ =1 Ψ n (x) P n (x) n 2.1 X n /n n [7, 8] r =0, 1, 2, h(x n /n) r i Z dy y r ν(y) n 10

ν(y) µ(y; a ) = p 1 a 2 π(1 y 2 ) p a 2 y 2 (3.8) µ I(y; a, b; α, β) = 1 α 2 β 2 + αβ ab + α βa b y (3.9) a 2 ν(y) =µ(y; a )I(y; a, b; α, β)1 { y < a } (3.10) 1 {ω} ω ω 1 {ω} =1 1 {ω} =0 A (3.4) (2.1) (2.2) p =1/2 I 6= 1 ( 3.1) α = β αβ ab + α βa b =0(a, b A ) µ(y; a )1 { y < a } [7, 8]( 3.2 ) 11

3.1: 200 A α =1, β =0 3.2: α =1/ 2, β = i/ 2 3.1 12

4 4.1 µ µ µ 0 1 0 i 1 0 σ 1 =, σ 2 =, σ 3 = 1 0 i 0 0 1 3 q =(q 1,q 2,q 3 ) q = q (σ q) 2 = q 2 I 2 e iσ q X 1 = ( iσ q)n n! n=0 Ã! = I 2 iσ ˆq tan q cos q (4.1) ˆq ˆq = q/q A SU(2) (2.4) µ ue i(k+θ) 1 u2 e i(k+φ) U(k) = 1 u 2 e i(k+φ) ue i(k+θ) = u cos(k + θ) " Ã!# 1 u 2 sin(k + φ) 1 u I 2 + i u cos(k + θ) σ 2 cos(k + φ) 1 + u cos(k + θ) σ 2 + tan(k + θ) σ 3 U(k) (4.1) q 1 u 2 u 1 u 2 u u cos(k + θ) =cosq sin(k + φ) cos(k + θ) = ˆq 1 tan q cos(k + φ) cos(k + θ) = ˆq 2 tan q tan(k + θ) = ˆq 3 tan q (4.2) q(k) " # q(k) = arccos u cos(k + θ) = arctan " r # 1 1 cos(k + θ) u 2 cos2 (k + θ) (4.3) (4.4) 13

arccos x arctan x q(k) k [ π, π) 1 u2 /u ˆq 1 (k) = p sin(k + φ) 1/u2 cos 2 (k + θ) 1 u2 /u ˆq 2 (k) = p cos(k + φ) 1/u2 cos 2 (k + θ) 1 ˆq 3 (k) = p sin(k + θ) (4.5) 1/u2 cos 2 (k + θ) 3 ³ q(k) = q(k)ˆq 1 (k),q(k)ˆq 2 (k),q(k)ˆq 3 (k) (4.6) (4.1) U(k) U(k) =e iσ q(k) (4.7) n =0, 1, 2, t [0, ) ˆΨ t (k) =e itσ q(k) ˆΨ0 (k), k [ π, π) (4.8) h =1 i t ˆΨ t (k) =H(q(k)) ˆΨ t (k) (4.9) (2.5) (4.3)- (4.6) k [ π, π) 7 q p- (3.2) (4.9) p (2.5) [4] λ = ±p p = p p ˆp = p/p µ (1) µ ψ + (ˆp) cos(θp /2) ψ + (ˆp) = ψ (2) = + (ˆp) sin(θ p /2)e iϕ p µ (1) µ ψ (ˆp) sin(θp /2)e iϕ p ψ (ˆp) = ψ (2) = (4.10) (ˆp) cos(θ p /2) p 1 = p sin θ p cos ϕ p,p 2 = p sin θ p sin ϕ p,p 3 = p cos θ p, ψ + (ˆp) 2 = ψ + (ˆp)ψ +(ˆp) =1 ψ (ˆp) 2 = ψ (ˆp)ψ (ˆp) =1 ψ +(ˆp)ψ (ˆp) =ψ (ˆp)ψ + (ˆp) = 0 (4.11) 14

(4.3)-(4.6) k 7 q p- 4.2 (4.3)-(4.5) q j (k),j =1, 2, 3 k [ π, π) q(k) p- k [ π, π) ³ ê 3 = u cos(φ θ),usin(φ θ) p 2 1 u (4.12) q(k) ê 3 =0 k [ π, π) q(k) ê 3 ê 3 Π(u, θ, φ) (4.3) arccos x 0 u 1 u 0 arccos u π/2 u =0 arccos u = π/2 u =1 arccos u =0 A q(k) = q(k) k = θ(mod 2π) q min = arccos u k = π θ (mod 2π) q max = π q min k = π/2 θ k = π/2 θ (mod 2π) q(k) =π/2 ³ ê 1 = sin(φ θ), cos(φ θ), 0 ³p ê 2 = 1 u2 cos(φ θ), p 1 u 2 sin(φ θ),u (4.13) (ê 1, ê 2, ê 3 ) p- Π(u, θ, φ) (ê 1, ê 2 )- ê 1 k 0 = θ (mod 2π) q(k 0 ) γ ˆq(k) =q(k)/q(k) cos γ = ˆq(k) ê 1 cos γ = ( 1 u 2 /u) cos(k + θ) p 1/u2 cos 2 (k + θ) sin γ = (1/u)sin(k + θ) p 1/u2 cos 2 (k + θ) (4.14) (4.15) (4.4) (4.14) Π(u, θ, φ) (q, γ), 0 q<, γ [ π, π) tan q = 1 u 2 u 1 cos γ (4.16) 15

(4.3)-(4.6) q γ q 1 = q p 1 u 2 cos(φ θ)sinγ q sin(φ θ)cosγ q 2 = q p 1 u 2 sin(φ θ)sinγ q cos(φ θ)cosγ q 3 = qu sin γ (4.17) q Π(u, θ, φ) (q, γ) {q(k); k [ π, π)} SU(2) 3 u, θ, φ ê 3 (4.12) 4.1 u =0 z u 0 1 ( cos(φ θ), sin(φ θ), 0) (x, y) arctan x x 0 (0 arctan x<π/2) x <0 (π/2, π) (4.16) " # 1 u 2 1 q =arctan u cos γ 4.1 u =0 q π/2 π/2 π/2 γ π/2 cos γ 0 u 1 q 0 π γ < π/2 π/2 < γ < π q π u (0, 1) 4.2 u 4.3 (3.7) [ π, π) k γ [ π, π) k γ J = /dγ A) 1 u 2 J = 1 u 2 sin 2 γ (4.18) k Z π π Z π 2π f(k) = dγ π 2π 1 u 2 1 u 2 sin 2 f(k(γ)) (4.19) γ k(γ) (A.2) (A.3) γ y ; y = u sin γ (4.20) 16

4.1: u φ θ =0 u =0,u =1/ 2( ),u =0.9 4.2: u u ' 0, u =1/ 2( ), u ' 1 17

(4.19) Z π π Z u dy 1 1 u 2 f(k) =2 p 2π u 2π u2 y 2 1 y 2 f(k(y)) (3.8) Z π Z u π 2π f(k) = dy µ(y; u)f(k(y)) u 18

5 5.1 x =0 Ã! Ã! α Ψ 0 (x) =δ(x) ˆΨ α 0 (k) = β β α, β C α 2 + β 2 =1 ˆΨ 0 (k) p = q(k) H(p) (4.10) ˆΨ 0 (k) =C + (ˆq(k))ψ + (ˆq(k)) + C (ˆq(k))ψ (ˆq(k)) (5.1) C + (ˆp) C (ˆp) ˆp = p/ p ( B (B.1) ) n =0, 1, 2, (4.8) ) ˆΨ n (k) = e (C ih(q(k))n + (ˆq(k))ψ + (ˆq(k)) + C (ˆq(k))ψ (ˆq(k)) = e iq(k)n C + (ˆq(k))ψ + (ˆq(k)) + e iq(k)n C (ˆq(k))ψ (ˆq(k))(5.2) ψ ± (ˆp) λ = ±p H(p) q(k) q(k) = q(k) 5.2 (3.7) f(x) =x r,r =0, 1, 2,, [9] (5.2) µ i d r ˆΨn (k) µ r dq(k) = e iq(k)n C + (ˆq(k))ψ + (ˆq(k))n r µ + dq(k) r e iq(k)n C (ˆq(k))ψ (ˆq(k))n r + O(n r 1 ) (5.3) 19

(3.7) (5.2) (5.3) (4.11) n [9] (A.1) = lim h(x n/n) r i n Z )Ã π ( C + (ˆq(k)) 2 +( 1) r C (ˆq(k)) 2 sin(k + θ) p 2π 1/u2 cos 2 (k + θ) π! r (5.4) k γ (4.15), (4.19) (B.2) Z π lim h(x n/n) 2m dγ 1 u 2 i = n π 2π 1 u 2 sin 2 (u sin γ)2m γ ( ) 1 u lim h(x n/n) 2m+1 i = ( α 2 β 2 2 )+ (αβ e i(φ θ) + α βe i(φ θ) ) n Z π dγ π 2π u 1 u 2 1 u 2 sin 2 (u sin γ)2m+2 γ m =0, 1, 2, (4.20) (3.8) lim h(x n/n) 2m i = n Z u lim h(x n/n) 2m+1 i = n dy µ(y; u)y 2m u ( ) 1 u ( α 2 β 2 2 )+ (αβ e i(φ θ) + α βe i(φ θ) ) u Z u dy µ(y; u)y 2m+2 u [7, 8] X n n Ã! a b A = b a a, b C, a 2 + b 2 =1 µ α Ψ 0 (x) =δ(x), α 2 + β 2 =1, α, β C β Z f(x) (5.5) hf(x n /n)i Z dy f(y)ν(y) ν(y) (3.10)-(3.9) n 20

6 6.1 3 2 3 Ψ (1) n (x) Ψ (2) n (x) Ψ (3) n (x) 3 U = SA s + 0 0 S = 0 s 0 0 0 0 s, A = a b c d e f g h i s 0 s 0 Ψ n (x) =Ψ n (x) 2 3 G = 1 1 2 2 2 1 2 3 2 2 1 3 Ψ (1) n+1 (x) aψ (1) n (x +1)+bΨ (2) n (x +1)+cΨ (3) n (x +1) Ψ (2) n+1 (x) = dψ (1) n (x)+eψ (2) n (x)+fψ (3) n (x) Ψ (3) n+1 (x) gψ (1) n (x 1) + hψ (2) n (x 1) + iψ (3) n (x 1) (6.1) 21

6.2 3 [10] [9] ( 6.1, 6.2) 6.3 4 4 2,3 4 ( 6.3, 6.4 ) Ψ (1) n+1 (x) Ψ (2) n+1 (x) Ψ (3) n+1 (x) Ψ (4) n+1 (x) = 1 2 Ψ (1) n (x +2)+Ψ (2) n (x +2)+Ψ (3) n (x +2)+Ψ (4) n (x +2) Ψ (1) n (x +1) Ψ (2) n (x +1)+Ψ (3) n (x +1)+Ψ (4) n (x +1) Ψ (1) n (x 1) + Ψ (2) n (x 1) Ψ (3) n (x 1) + Ψ (4) n (x 1) Ψ (1) n (x 2) + Ψ (2) n (x 2) + Ψ (3) n (x 2) Ψ (4) n (x 2) 22

6.1: = t 11 2 / 6 6.2: = t (1 i 1)/ 3 23

6.3: = t (1 ii1)/2 6.4: = t (1 1 1 i)/2 24

7 3 3 [11] - 1 [12] 25

26

A q(k) (4.3) (i) q(k) =π arccos u & π 2 & arccos u k + θ = π % π 2 % 0 (ii) q(k) = arccos u % π 2 % π arccos u k + θ =0% π 2 % π x = a % b x a b x = a & b x a b π k + θ 0 dq(k)/ 0 0 k + θ < π dq(k)/ 0 (d/dx) arccos x = 1/ 1 x 2 dq(k) = sin(k + θ) p 1/u2 cos 2 (k + θ) (A.1) (4.3)-(4.5) ³ q( π θ) = q max sin(φ θ), cos(φ θ), 0 q( π/2 θ) = π ³p 1 u2 cos(φ θ), p 1 u 2 2 sin(φ θ),u ³ q( θ) = q min sin(φ θ), cos(φ θ), 0 q(π/2 θ) = π ³ p 1 u 2 2 cos(φ θ), p 1 u 2 sin(φ θ), u q min =arccosu q max = π q min (4.14) (4.15) k + θ = π γ = π k + θ = π/2 γ = π/2 k + θ =0 γ =0 k + θ = π/2 γ = π/2 sin(k + θ) = ( 1 u 2 /u)sinγ p (1 u2 )/u 2 +cos 2 γ (A.2) cos(k + θ) = (1/u)cosγ p (1 u2 )/u 2 +cos 2 γ (A.3) 27

(A.3) " # sin(k + θ) = d (1/u)cosγ p dγ dγ (1 u2 )/u 2 +cos 2 γ (A.2) " # d (1/u)cosγ p dγ (1 u2 )/u 2 +cos 2 γ 1 u 2 =sin(k + θ) 1 u 2 sin 2 γ J = /dγ (4.18) 28

B µ α φ 0 =, α, β C, α 2 + β 2 =1 β H(p) =σ p (4.10) (4.11) φ 0 = C + (ˆp)ψ + (ˆp)+C (ˆp)ψ (ˆp) C + (ˆp) = ψ + (ˆp)φ 0 = α cos θ p 2 + β sin θ p 2 e iϕp C (ˆp) = ψ (ˆp)φ 0 = α sin θ p + β cos θ p 2 eiϕp 2 (B.1) (4.17) ˆp ˆq(γ) ³p ˆp 1 ± iˆp 2 ˆq 1 ± iˆq 2 = 1 u2 sin γ i cos γ e i(φ θ) C ± (ˆq(γ)) 2 = 1 2 ± 1 n 1 u ( α 2 β 2 2 )+ (αβ e i(φ θ) + α βe ou i(φ θ) sin γ 2 u 1 2 i(αβ e i(φ θ) α βe i(φ θ) )cosγ (B.2) 29

[1],, (1997). [2],, (2002). [3] M.Katori, S.Fujino, and N.Konno, Phys. Rev. A 72, 012316 (2005) [4],, (1973). [5], No. 492, PP. 37-44, (2004). [6] J.Kempe, Contemp. Phys. 44, 307 (2003). [7] N.Konno, Quantum Inf. Process. 1, 345 (2002). [8] N.Konno, J.Math. Soc. Jpn, 57, 1179 (2005). [9] G.Grimmett, S.Janson, and P.F.Scudo, Phys. Rev. E 69, 026119 (2004). [10] N.Inui, N.Konno and E.Segawa, Phys. Rev. E 72, 056112 (2005). [11] D.Bouwmeester, I.Marzoli, G.P.Karman, W.Schleich, and J.P.Woerdman, Phys. Rev. A 61, 013410 (1999). [12] T.Oka, N.Konno, R.Arita, and H.Aoki, Phys. Rev. Lett. 94, 100602 (2005). 30