A quantum walk based search algorithm, and its optical realisation

Proof of principle (sketch)
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⃗x tg can be chosen ⃗x tg = ⃗0:
100
110
⇒ C = C 0 ⊗ +(C|d,⃗x〉
1 − C 0 ) ⊗ |⃗0〉〈⃗0|
1/3
C 0 , C 1 permutation symmetric
010
101
000
111
|⃗0〉 1/3permutation symmetric
1/3
|ψ 0 〉 permutation symmetric
001 110
⇒ time evolution in an invariant subspace
Basis:
1
|R,
2/3
x〉,
1/3
|L,
∑
x〉 (x
∑
= 0, 1, . . . , n)
|L, x〉, |R, x〉
∑ ∑
|R, x〉 = N R,x |d,⃗x〉, |L, x〉 = N L,x |d,⃗x〉
0
1 2 3
1/3 2/3
1
|⃗x|=x x d =0
Re-formulate the QW
|ψ 0 〉 : express in |R, x〉, |L, x〉 basis
U ′ = U − 2 |L, 1〉〈R, 0|
prob. success: p 0 = | 〈R, 0|ψ f 〉 | 2
|⃗x|=x x d =1
|ψ 1 〉 has property such that, | 〈R, 0|ψ 1 〉 | 2 = 1/2.
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