A quantum walk based search algorithm, and its optical realisation

Transcription

1 A quantum walk based search algorithm, and its optical realisation Aurél Gábris FJFI, Czech Technical University in Prague with Tamás Kiss and Igor Jex Prague, Budapest Student Colloquium and School on Mathematical Physics Stará Lesná, Slovakia 23 29th August 28 / 26

2 Outline Quantum walks on graphs The quantum mechanical search problem The SKW quantum walk search algorithm Optical implementation Analysis of errors J. Kempe: Contemp. Phys (23), quant-ph/338 N. Shenvi, J. Kempe and K. B. Whaley: Phys. Rev. A 67, 5237 (23) A. Gábris, T. Kiss and I. Jex: Phys. Rev. A 76, 6235 (27) 2 / 26

3 Quantum walks on graphs Graph G = (V, E) state of system: ψ H V = Span{ v i : v i V} dynamics: according to E How to go from classical to quantum walks classical prob. distrib. stochastic Two alternative approaches discrete time quantum walk continuous time quantum walk quantum quantum state unitary (relationship: Strauch: Phys. Rev. A 74, 33 (26) Introductory review: J. Kempe: Contemp. Phys. 44, 37 (23), quant-ph/338 3 / 26

4 Discrete time quantum walk (coined QW) Focussing generalization on: Stochastic process Unitary process Simple example Galton s board (quincunx) at each pin: 5%-5% probability tossing a coin Resolution Replace classical coin by quantum coin! { j v j : j =,..., degree(v j )} additional degree of freedom 4 / 26

5 Discrete quantum walk G = (V, E) n-regular graph coin states:,..., n H C = Span { j j =,..., n} H = H C H V Dynamics coin operator: C Lin(H C ), unitary translation operator: S Lin(H), unitary step operator: U = S(C ), uniform coin Coin operators balanced: Cij 2 = /d unbalanced permutation invariant Grover coin: G = s s Grover (diffusion) operator 2 d 2 2 d d 2 2 d d 2 d d d d / 26

6 Continuous time quantum walk Discrete transition probabilities p t = (p t, pt 2,..., pt V ) p t+ = Mp t M ij transition prob. Continuous time Markov chain (classical) γ, if i j, {v i, v j } E dp(t) = Hp(t) H ij =, if i j, {v dt i, v j } E d i γ, if i = j (d i : rank of v i ) p(t) = e Ht p() Quantum mechanical generalization Choose: Ĥ = γ d i v i v i γ i Û(t) = e iĥt {v i,v j } E v i v j 6 / 26

7 Applications of QWs Designing efficient quantum algorithms is difficult QWs may be used for this task Some findings: traversal of binary trees (exp. faster hitting time) traversal of randomly connected binary trees (quadratic algorithmic speedup) evaluation of NAND formulas quantum searching... 7 / 26

8 The quantum mechanical search problem Classical specification given search space X given function f : X {, } task: find x X s. t. f (x) = Quantum mechanics Try O: O x = f (x) not unitary! Unitary choice: i.e. O x = 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

9 Quantum walk on a hypercube hypercube in n dimensions: n-regular graph coined quantum walk (n = 4) Vertices n dimensions x {,,..., 2 n } integer binary string repr.: x, n binary digits Useful definitions Hamming weight: x sum of digits/number of s Hamming distance: d( x, y) = x y Edges x, y V are connected iff d( x, y) = 9 / 26

10 Grover walk on a hypercube Definition Graph: n dimensional hypercube n-regular graph N = 2 n vertices, labels: n bit long binary strings ( x =,..., 2 n ) (9 =... ) Coin: C = G (Grover operator) Propagator: S = d, x e d d, x Symmetries x,d ( ed = 2 d, edges ) shift symmetry: x x x s permutation symmetry: P ij swap bits at positions i, j / 26

11 Spectrum of Grover walk on a hypercube Solve problem in Fourier space... Trivial eigenvectors λ = ± Non-trivial eigenvectors λ k = exp(±iω k ) = 2 k n v k, v k = x,d ( ) k x 2 n/2 ± 2i n k (n k ) / k, if k d = d, x 2 i/ n k, if k d = / 26

12 The SKW algorithm SKW quantum walk (perturbed Grover walk) C initial state: d, x, = G, nn C = d, x O marks x tg = C = C + (C C ) x tg x tg time step operator: U = SC After optimal number of steps: t f = O( N) probability of x tg : p = d, x tg ψ(t f ) 2 = 2 O ( ) n d Classical analysis and protocol Execute SKW QW 2 obtain x m 3 verify x m using O repeat until x tg found: ε certainty r ε repetitions 2 / 26

13 Proof of principle (sketch) Collapse onto a line x tg can be chosen x tg = : C = C +(C C ) C, C permutation symmetric permutation symmetric ψ permutation symmetric time evolution in an invariant subspace Basis: R, x, L, x (x =,,..., n) R, x = N R,x d, x, L, x = N L,x x =x x d = Re-formulate the QW ψ : express in R, x, L, x basis U = U 2 L, R, prob. success: p = R, ψ f 2 d, x x =x x d = ψ has property such that, R, ψ 2 = /2. 3 / 26

14 Proof of principle (sketch) Collapse onto a line x tg can be chosen x tg = : C = C +(C d, x C ) /3 C, C permutation symmetric /3permutation symmetric /3 ψ permutation symmetric time evolution in an invariant subspace Basis: R, 2/3 x, /3 L, x (x =,,..., n) L, x, R, x R, x = N R,x d, x, L, x = N L,x d, x 2 3 /3 2/3 x =x x d = Re-formulate the QW ψ : express in R, x, L, x basis U = U 2 L, R, prob. success: p = R, ψ f 2 x =x x d = ψ has property such that, R, ψ 2 = /2. 3 / 26

15 Proof of principle (sketch) 2 Treat as perturbation U = U + U permutation symmetric subspace: no trivial eigenvalues eigenvectors: v k = ( n v k k) k =k Perturbed eigenvalues QUANTUM RANDOM-WALK SEARCH ALGORITHM ω 2 n PHYSICAL Corresponding eigenvectors ψ 2 ( ω + ω ) ψ 2 ( ω ω ) = (U ) t ψ cos ω t ψ sin ω t ψ Optimal time t f = π 2 ω = π 2 2 n + O ( ) n 4 / 26

16 Linear quantum optics Basic elements beam splitter described by an SU(2) matrix R (â out, ˆb out ) = R (â in, ˆb in ) b out R phase shifter changes optical path length â out = e iϕ â in a in b in a out Optical multiport n input, n output transformation: SU(n) beam splitter: 2 2 multiport Can be assembled from beam splitters (and phase shiters) 5 / 26

17 Quantum optical realization of SKW algorithm n n multiport n n n n Single photon d one photon at port d transformation: n d= a d d n coin operator d,k= C dk a k d 6 / 26

18 Quantum optical realization of SKW algorithm n n multiport n n Scattering random walk x = t t + n n transformation: n d= a d d n coin operator d,k= C dk a k d x = x = 2 x = 3 x = 4 d, x d, x e d 6 / 26

19 Quantum optical realization of SKW algorithm n n multiport n n Scattering random walk x = t t + n n transformation: n d= a d d n coin operator d,k= C dk a k d x = x = 2 x = 3 x = 4 d, x d, x e d S(C ) a dx d, x = C dk a kx d, x e d dx dxk 6 / 26

20 Quantum optical realization of SKW algorithm n n multiport n n n With loopback x = Ansingle column: scattering quantum random walk (similar: by Bužek and Košík) x = 2 transformation: n d= a d d n coin operator d,k= C dk a k d Scattering random walk x = x = 3 x = 4 S(C ) a dx d, x = C dk a kx d, x e d dx dxk t t + d, x d, x e d 6 / 26

21 QRW search in linear optical network Column of multiports x = x = x = 2 x = 3 x = 4 Consequence Coin at x = differs! non-uniform coin QRW search 7 / 26

22 QRW search in linear optical network Column of multiports x = x = x = 2 x = 3 x = 4 Consequence Coin at x = differs! non-uniform coin QRW search Graph topology: hypercube connect: port d of x port d of x 2 d 7 / 26

23 QRW search in linear optical network Column of multiports x = x = x = 2 x = 3 x = 4 Consequence Coin at x = differs! non-uniform coin QRW search Graph topology: hypercube connect: port d of x port d of x 2 d Search problem Find the multiport different from the rest! 7 / 26

24 Losses, errors and decoherence Decoherence Basis of quantum computing: quantum coherence Losses and errors are inevitable in all physical realization: non-unitary time evolution Typical errors in optics Photon loss (absorption or scattering in media) Phase errors (difference in optical paths) 8 / 26

25 Uniform loss Loss model D(ϱ) = η 2 ϱ + ( η 2 ) after t iterations: ϱ D t (U t ϱu t ) Initial pure state ψ : (η ) ψ D ψ = η ψ Evolution operator: U = ηu 9 / 26

26 Uniform loss Loss model D(ϱ) = η 2 ϱ + ( η 2 ) after t iterations: ϱ D t (U t ϱu t ) Initial pure state ψ : (η ) ψ D ψ = η ψ Evolution operator: U = ηu log 2 x ε + n/2 /2 Success probability Maximum success probability (p max ) x = log 2 η 2 n η = 2 ε p max (η) = for large n: p max (x) 2 [ exp( 2x acot x) +x 2 2 O ( ( )] n) + x acot x O n exp( 2x acot x) +x 2 9 / 26

27 Comparison with numerical results Maximum success probability (p max ) ε = 2 ε = 8 ε = 6 ε = Rank of hypercube (n) 2 4 p max (η) = [ exp( 2x acot x) +x 2 2 O ( ( )] n) + a acot xo n 2 / 26

28 Direction dependent loss η η η n η, η,..., η n can be different Warning: original symmetry of hypercube broken! Walk cannot be collapsed onto a line 2 / 26

29 Direction dependent loss Description starting with pure state: D = n d= η d d d, {η} = {η d d =,..., n } a lower bound: p({η}, t) η 2t { pi (t) (η max / η) [ (η max / η) t ] } 2 η = η max + η min 2 2 / 26

30 Direction dependent loss Description starting with pure state: D = n d= η d d d, {η} = {η d d =,..., n } a lower bound: p({η}, t) η 2t { pi (t) (η max / η) [ (η max / η) t ] } 2 η = n n d= η = η max + η min 2 η d, δ d = η d η, Q 2 = n n d= δ 2 d, 2 / 26

31 Direction dependent loss Description starting with pure state: D = n d= η d d d, {η} = {η d d =,..., n } a lower bound: p({η}, t) η 2t { pi (t) (η max / η) [ (η max / η) t ] } 2 η = η max + η min 2 η = n n d= η d, δ d = η d η, Q 2 = n n d= δ 2 d, W3 = n n δ 3 d d= Taylor expansion around η, (using permutation symmetry) p max ({η}) = p max ( η ) + BQ 2 + CW 3 + R 2 / 26

32 Numerical results Second order Taylor coefficient (n = 8) Second order Taylor coefficient (B) (log 2 scale) Average transmission ( η ).9 Effect of higher orders: p max ({η}) p max ( η ) + 2 n Q 2 22 / 26

33 Numerical results Second order Taylor coefficient (n = 8) Second order Taylor coefficient (B) (log 2 scale) 64 6 Empirical lower bound 4 using size, and the elementary statistical properties of noise: η and Q Average transmission ( η ) Effect of higher orders: p max ({η}) p max ( η ) + 2 n Q / 26

34 Numerical results Absolute improvement (n = 7) 4 Absolute improvement (%) Second moment (Q) p max ({η}) p max (η max ) p max (η max ) increase efficiency by loss! 23 / 26

35 Phase Errors Description F({ϕ}) = d,x e iϕ dx d, x d, x U({ϕ}) = U F({ϕ}) Stable phase in one run ψ t ({ϕ}) = U({ϕ}) t ψ ϱ out = ψ t ({ϕ}) ψ t ({ϕ}) {ϕ} Gaussian: ϕ =, ϕ variance 24 / 26

36 Phase Errors Description F({ϕ}) = e iϕ dx d, x d, x d,x U({ϕ}) = U F({ϕ}) Single runs.4.35 Average probability (n = 6) Average probability of success Number of steps (t) ϕ = 3 o ϕ = 6 o ϕ = 9 o ϕ = 2 o 4 5 Probability of success Number of steps (t) Stable phase in one run ψ t ({ϕ}) = U({ϕ}) t ψ ϱ out = ψ t ({ϕ}) ψ t ({ϕ}) {ϕ} Gaussian: ϕ =, ϕ variance 24 / 26

37 Phase Errors.4 Description.35 F({ϕ}) = e iϕ dx d, x d, x.3 d,x U({ϕ}) =.25 U F({ϕ}).2 Single runs.5 Probability of success Average probability (n = 6) Average probability of success Number of steps (t) Average probability (n = 6) ϕ = 3 o.4 ϕ = 6 o.35 ϕ = 9 o.3 ϕ = 2 o Average probability of success Number of steps (t) Stable phase in one run ψ t ({ϕ}) = U({ϕ}) t ψ ϱ out = ψ t ({ϕ}) ψ t ({ϕ}) {ϕ} Number of steps (t) ϕ = 3 o ϕ = 6 o ϕ = 9 o ϕ = 2 o 4 5 Gaussian: ϕ =, ϕ variance 5 24 / 26

38 Phase Errors Numerical results Max. average success probability rank of hypercube (n) ideal ϕ = 3 o ϕ = 6 o ϕ = 5 o 9 25 / 26

39 Conclusions Quantum walks Interesting for new quantum algorithms its mathematics Quantum walk search algorithm proposal for optical realization analysis of errors and losses J. Kempe: Contemp. Phys (23), quant-ph/338 N. Shenvi, J. Kempe and K. B. Whaley: Phys. Rev. A 67, 5237 (23) A. Gábris, T. Kiss and I. Jex: Phys. Rev. A 76, 6235 (27) 26 / 26