A quantum walk based search algorithm, and its optical realisation

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The quantum mechanical search problem Classical specification given search space X given function f : X → {0, 1} task: find ∀x ∈ X s. t. f (x) = 1 Quantum mechanics Try O: O |x〉 = |f (x)〉 ⇒ not unitary! Unitary choice: i.e. O |x〉 |0〉 = |x〉 |f (x)〉 O |x〉 |s〉 = |x〉 |f (x) ⊕ s〉 Complexity O oracle operator Often measured as the number of oracle calls (query complexity) 8 / 26
Quantum walk on a hypercube hypercube in n dimensions: n-regular graph coined quantum walk (n = 4) Vertices n dimensions → x ∈ {0, 1, . . . , 2 n − 1} integer binary string repr.: ⃗x, n binary digits Useful definitions Hamming weight: |⃗x| sum of digits/number of 1s Hamming distance: d(⃗x,⃗y) = |⃗x ⊕ ⃗y| Edges ⃗x,⃗y ∈ V are connected iff d(⃗x,⃗y) = 1 9 / 26
Page 1 and 2: A quantum walk based search algorit
Page 3 and 4: Quantum walks on graphs Graph G = (
Page 5 and 6: Discrete quantum walk G = (V, E) n-
Page 7: Applications of QWs Designing effic
Page 11 and 12: Spectrum of Grover walk on a hyperc
Page 13 and 14: Proof of principle (sketch) Collaps
Page 15 and 16: Proof of principle (sketch) 2 Treat
Page 17 and 18: Quantum optical realization of SKW
Page 19 and 20: Quantum optical realization of SKW
Page 21 and 22: QRW search in linear optical networ
Page 23 and 24: QRW search in linear optical networ
Page 25 and 26: Uniform loss Loss model D(ϱ) = η
Page 27 and 28: Comparison with numerical results M
Page 29 and 30: Direction dependent loss Descriptio
Page 31 and 32: Direction dependent loss Descriptio
Page 33 and 34: Numerical results Second order Tayl
Page 35 and 36: Phase Errors Description F({ϕ}) =
Page 37 and 38: Phase Errors 0.4 Description 0.35 F
Page 39: Conclusions Quantum walks Interesti

A quantum walk based search algorithm, and its optical realisation

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