Recurrences in Quantum Walks

Transcription

1 Recurrences in Quantum Walks M. Štefaňák (1), I. Jex (1), T. Kiss (2) (1) Department of Physics, FJFI ČVUT in Prague, Czech Republic (2) Department of Nonlinear and Quantum Optics, RISSPO, Hungarian Academy of Sciences, Budapest, Hungary Stará Lesná, M. Štefaňák (FJFI ČVUT) Recurrences in QWs Stará Lesná, / 34

2 Outline 1 Introduction 2 Recurrence of RWs 3 Recurrence of QWs 4 Quantum walks on Z d Description of QWs Time evolution of QWs Asymptotics of QWs 5 Examples QW on a line QWs with tensor-product coins Grover walk Fourier walk 6 Summary M. Štefaňák (FJFI ČVUT) Recurrences in QWs Stará Lesná, / 34

3 Outline 1 Introduction 2 Recurrence of RWs 3 Recurrence of QWs 4 Quantum walks on Z d Description of QWs Time evolution of QWs Asymptotics of QWs 5 Examples QW on a line QWs with tensor-product coins Grover walk Fourier walk 6 Summary M. Štefaňák (FJFI ČVUT) Recurrences in QWs Stará Lesná, / 34

4 Motivation Random walks are one of the cornerstones of theoretical computer science database search, graph connectivity, 3-SAT, permanent of a matrix,... Quantum walks could solve the same problems on a quantum computer, maybe faster (talks by Vašek and Aurel) RWs Diffusion We work with probabilities Spread slowly σ 2 t QWs Wave propagation We work with probability amplitudes interference Spread fast σ 2 t 2 M. Štefaňák (FJFI ČVUT) Recurrences in QWs Stará Lesná, / 34

5 Motivation Random walks are one of the cornerstones of theoretical computer science database search, graph connectivity, 3-SAT, permanent of a matrix,... Quantum walks could solve the same problems on a quantum computer, maybe faster (talks by Vašek and Aurel) RWs Diffusion We work with probabilities Spread slowly σ 2 t QWs Wave propagation We work with probability amplitudes interference Spread fast σ 2 t 2 M. Štefaňák (FJFI ČVUT) Recurrences in QWs Stará Lesná, / 34

6 QW on a line Particle lives on 1-D lattice position space H P = l 2 (Z) = Span { m m Z} Moves in a discrete time steps on a lattice m m 1, m + 1 Does not preserve orthogonality { orthogonal } nonorthogonal To make the time evolution unitary we need an additional degree of freedom coin space H C = Span { + 1, 1 } M. Štefaňák (FJFI ČVUT) Recurrences in QWs Stará Lesná, / 34

7 QW on a line Time evolution equation ψ(t) = U t ψ(0), U = S (I P C) Initial state ψ(0) initial position + orientation of the coin Displacement operator S = m ( m + 1 m m 1 m 1 1 ) Coin flip C - rotates the coin state before the step itself e.g. Hadamard matrix H = 1 2 ( ) M. Štefaňák (FJFI ČVUT) Recurrences in QWs Stará Lesná, / 34

8 QW on a line M. Štefaňák (FJFI ČVUT) Recurrences in QWs Stará Lesná, / 34

9 Outline 1 Introduction 2 Recurrence of RWs 3 Recurrence of QWs 4 Quantum walks on Z d Description of QWs Time evolution of QWs Asymptotics of QWs 5 Examples QW on a line QWs with tensor-product coins Grover walk Fourier walk 6 Summary M. Štefaňák (FJFI ČVUT) Recurrences in QWs Stará Lesná, / 34

10 Pólya number of a RW 1 Probability that an unbiased random walk (RW) on Z d starting at the origin 0 ever returns to the origin If p 0 (t) is the probability that the walker is at the origin after t steps then the Pólya number is given by P = 1 Random walk is recurrent if P = 1 Random walk is transient if P < 1 1 p 0 (t) For classical random walks the Pólya numbers are characteristic for the dimension of the walk d t=0 1 G. Pólya, Mathematische Annalen 84, 149 (1921) M. Štefaňák (FJFI ČVUT) Recurrences in QWs Stará Lesná, / 34

11 Recurrence of RWs Criterion of recurrence RW is recurrent if and only if + t=0 p 0 (t) = + p 0 (t) t 1 or slower Recurrence of a RW is fully determined by the asymptotics of p 0 (t) For a classical RW on Z d the probability p 0 (t) scales like p 0 (t) t d 2 For d = 1, 2 the walks are recurrent = P = 1 For d 3 the walks are transient = P < 1 NB: x 2 t I can fill a plane in linear time, but not a free space M. Štefaňák (FJFI ČVUT) Recurrences in QWs Stará Lesná, / 34

12 Outline 1 Introduction 2 Recurrence of RWs 3 Recurrence of QWs 4 Quantum walks on Z d Description of QWs Time evolution of QWs Asymptotics of QWs 5 Examples QW on a line QWs with tensor-product coins Grover walk Fourier walk 6 Summary M. Štefaňák (FJFI ČVUT) Recurrences in QWs Stará Lesná, / 34

13 Pólya number of a QW 2 Problem Measurement change the state of the particle Our definition Prepare an ensemble of particles in the same initial state Take n-th particle, make n steps, measure at the origin In the n-th trial click with p 0 (n), no click with 1 p 0 (n) No click at all occurs with P = + (1 p 0 (t)) t=1 Complementary event at least one click recurrence Pólya number of a QW P = 1 + (1 p 0 (t)) t=1 2 MŠ, I. Jex, T. Kiss, Phys. Rev. Lett. 100, (2008) M. Štefaňák (FJFI ČVUT) Recurrences in QWs Stará Lesná, / 34

14 Recurrence of QWs QW is recurrent if and only if Criterion of recurrence + t=1 (1 p 0 (t)) = 0 + t=0 p 0 (t) = + As for classical RWs, recurrence of QWs is fully determined by the asymptotics of p 0 (t) QW is recurrent if p 0 (t) t 1 or slower QW is transient if p 0 (t) decays faster than t 1 NB: x 2 t 2 I can fill a line in linear time, but not a plane = QWs can be recurrent only for d = 1? No, recurrence depends on the asymptotics of p 0 (t), that can be compensated by interference M. Štefaňák (FJFI ČVUT) Recurrences in QWs Stará Lesná, / 34

15 Recurrence of QWs QW is recurrent if and only if Criterion of recurrence + t=1 (1 p 0 (t)) = 0 + t=0 p 0 (t) = + As for classical RWs, recurrence of QWs is fully determined by the asymptotics of p 0 (t) QW is recurrent if p 0 (t) t 1 or slower QW is transient if p 0 (t) decays faster than t 1 NB: x 2 t 2 I can fill a line in linear time, but not a plane = QWs can be recurrent only for d = 1? No, recurrence depends on the asymptotics of p 0 (t), that can be compensated by interference M. Štefaňák (FJFI ČVUT) Recurrences in QWs Stará Lesná, / 34

16 Recurrence of QWs QW is recurrent if and only if Criterion of recurrence + t=1 (1 p 0 (t)) = 0 + t=0 p 0 (t) = + As for classical RWs, recurrence of QWs is fully determined by the asymptotics of p 0 (t) QW is recurrent if p 0 (t) t 1 or slower QW is transient if p 0 (t) decays faster than t 1 NB: x 2 t 2 I can fill a line in linear time, but not a plane = QWs can be recurrent only for d = 1? No, recurrence depends on the asymptotics of p 0 (t), that can be compensated by interference M. Štefaňák (FJFI ČVUT) Recurrences in QWs Stará Lesná, / 34

17 Outline 1 Introduction 2 Recurrence of RWs 3 Recurrence of QWs 4 Quantum walks on Z d Description of QWs Time evolution of QWs Asymptotics of QWs 5 Examples QW on a line QWs with tensor-product coins Grover walk Fourier walk 6 Summary M. Štefaňák (FJFI ČVUT) Recurrences in QWs Stará Lesná, / 34

18 Hilbert space of QWs Given by the tensor product Position space H = H P H C H P = l 2 (Z d ) = Span { m m Z d} Coin space - determined by the set of displacements which the walker can make in a single step NB: in the examples we consider m m + e j { } H C = Span e j e j Z d, j = 1,..., c e j {1, 1} d = H C = C 2d = C 2... C 2 }{{} d M. Štefaňák (FJFI ČVUT) Recurrences in QWs Stará Lesná, / 34

19 Time evolution of QWs Time evolution is determined by ψ(t) m,j ψ j (m, t) m e j = U t ψ(0) U = S (I P C) Displacement operator S = m,j m + e j m e j e j Coin flip for unbiased QWs - Hadamard matrices CC = C C = I, C ij e i C e j = 1 c M. Štefaňák (FJFI ČVUT) Recurrences in QWs Stará Lesná, / 34

20 Time evolution of QWs Vectors of probability amplitudes ψ(m, t) (ψ 1 (m, t),..., ψ c (m, t)) T Time evolution of amplitudes set of difference equations ψ(m, t) = l C l ψ(m e l, t 1) e i C l e j = δ il e i C e j The matrices C l are independent of m M. Štefaňák (FJFI ČVUT) Recurrences in QWs Stará Lesná, / 34

21 Time evolution in the Fourier picture Fourier transformation ψ(k, t) m ψ(m, t)e im k, k ( π, π] d simplifies the time evolution equation Time evolution equation in the Fourier picture ψ(k, t) = Ũ(k) ψ(k, t 1) = Ũt (k) ψ(k, 0) Propagator in the Fourier picture Ũ(k) D(k) C D(k) D (e ie1 k,..., e iec k) M. Štefaňák (FJFI ČVUT) Recurrences in QWs Stará Lesná, / 34

22 Solution of the time evolution equations Matrix Ũ(k) is unitary Solution in the Fourier picture λ j (k) = exp ( iω j (k) ), corresponding eigenvectors v j (k) ψ(k, t) = j ( ) e iω j (k)t ψ(k, 0), vj (k) v j (k) Solution in the position representation ψ(m, t) = j ( π,π] d dk ( ) e i(m k ω (2π) d j (k)t) ψ(k, 0), vj (k) v j (k) M. Štefaňák (FJFI ČVUT) Recurrences in QWs Stará Lesná, / 34

23 Solution of the time evolution equations Matrix Ũ(k) is unitary Solution in the Fourier picture λ j (k) = exp ( iω j (k) ), corresponding eigenvectors v j (k) ψ(k, t) = j ( ) e iω j (k)t ψ(k, 0), vj (k) v j (k) Solution in the position representation ψ(m, t) = j ( π,π] d dk ( ) e i(m k ω (2π) d j (k)t) ψ(k, 0), vj (k) v j (k) M. Štefaňák (FJFI ČVUT) Recurrences in QWs Stará Lesná, / 34

24 Asymptotics of QWs and recurrence Probability that the particle is at origin at time t p 0 (t) p(0, t) = ψ(0, t) 2 Walk starts localized at the origin - FT of the initial state is identical to the initial coin state ψ(m, t) = 0 for m 0 = ψ(k, 0) = ψ(0, 0) ψ Asymptotics of the probability amplitude ψ(0, t) = dk (2π) d e iω j (k)t (ψ, v j (k) ) v j (k) j ( π,π] d determines the asymptotic of p 0 (t) M. Štefaňák (FJFI ČVUT) Recurrences in QWs Stará Lesná, / 34

25 Asymptotics of QWs and recurrence ψ(0, t) = j Amplitude at the origin dk (2π) d e iω j (k)t (ψ, v j (k) ) v j (k) ( π,π] d Asymptotics of ψ(0, t) can be analyzed e.g. by the method of stationary phase 3 Saddle points of the phases ω j (k) k 0 such that ω j (k 0 ) 0 determines the asymptotic behaviour Overlap between the initial state ψ and the eigenvector v j (k) can effectively cancel a saddle point ( ψ, vj (k 0 ) ) 0 3 R. Wong, Asymptotic Approximations of Integrals, SIAM, Philadelphia (2001) M. Štefaňák (FJFI ČVUT) Recurrences in QWs Stará Lesná, / 34

26 Outline 1 Introduction 2 Recurrence of RWs 3 Recurrence of QWs 4 Quantum walks on Z d Description of QWs Time evolution of QWs Asymptotics of QWs 5 Examples QW on a line QWs with tensor-product coins Grover walk Fourier walk 6 Summary M. Štefaňák (FJFI ČVUT) Recurrences in QWs Stará Lesná, / 34

27 QW on a line Propagator Ũ(k) = 1 ( e ik 2 e ik e ik e ik ) Eigenvalues λ 1,2 (k) = ±e ±iω(k), sin ω(k) = sin k 2 Saddle points of both phases k 0 = ± pi 2 Asymptotic behaviour p 0 (t) t 1 independent of the initial coin state (NB: the exact value of the probability at the origin is also independent) QW on a line is recurrent NB: Same applies to QWs on a line with different unbiased coins M. Štefaňák (FJFI ČVUT) Recurrences in QWs Stará Lesná, / 34

28 QWs with tensor-product coins Coin flip is given by the tensor product Propagator in the Fourier picture C = C 1 C 2... C d Ũ(k) = Ũ1(k 1 ) Ũ2(k 2 )... Ũd(k d ) Eigenvalues of Ũ(k) factorizes Ũ j (k j ) = D(e ik j, e ik j ) C j ω j (k) = l ω jl (k l ) = all follows from QW on a line Probability at the origin decays like p 0 (t) t d = QWs with TP coins are transient for d 2, Pólya number depend only d M. Štefaňák (FJFI ČVUT) Recurrences in QWs Stará Lesná, / 34

29 2-D Grover walk QW driven by the Grover coin G = 1 2 Eigenvalues of the propagator λ 1,2 = ±1, λ 3,4 (k 1, k 2 ) = e ±iω(k 1,k 2 ) cos (ω(k 1, k 2 )) = cos k 1 cos k 2 Contribution from λ 1,2 is constant from λ 3,4 decays like t 2 Probability at the origin p 0 (t) behaves like a constant except for ψ G = 1 (1, 1, 1, 1)T 2 which is orthogonal to v 1,2 (k) for any k M. Štefaňák (FJFI ČVUT) Recurrences in QWs Stará Lesná, / 34

30 Probability distribution for the Grover walk For any initial state ψ ψ G the probability at the origin behaves like a constant p 0 (t) const The walk is recurrent M. Štefaňák (FJFI ČVUT) Recurrences in QWs Stará Lesná, / 34

31 Probability distribution for the Grover walk For the initial state ψ = ψ G the probability at the origin decays fast p 0 (t) t 2 The walk is transient M. Štefaňák (FJFI ČVUT) Recurrences in QWs Stará Lesná, / 34

32 Recurrent QWs on Z d based on the Grover walk Even dimension Coin tensor product of d Grover matrices C = G... G Propagator factorizes Ũ(k) = Ũ(k 1, k 2 )... Ũ(k 2d 1, k 2d ) Eigenvalues of the propagator factorizes one-half are constant = p 0 (t) const. Odd dimension Add an extra walk on a line C = G... G H One-half of evs depend only on one momentum component = p 0 (t) t 1 Conclusion These QWs are recurrent in any dimension d M. Štefaňák (FJFI ČVUT) Recurrences in QWs Stará Lesná, / 34

33 Recurrent QWs on Z d based on the Grover walk Even dimension Coin tensor product of d Grover matrices C = G... G Propagator factorizes Ũ(k) = Ũ(k 1, k 2 )... Ũ(k 2d 1, k 2d ) Eigenvalues of the propagator factorizes one-half are constant = p 0 (t) const. Odd dimension Add an extra walk on a line C = G... G H One-half of evs depend only on one momentum component = p 0 (t) t 1 Conclusion These QWs are recurrent in any dimension d M. Štefaňák (FJFI ČVUT) Recurrences in QWs Stará Lesná, / 34

34 2-D Fourier walk QW driven by the Fourier coin F = i 1 i i 1 i Saddle points follow from the implicit equation ) Φ(k, ω) det (ŨF (k) e iω I = 0 All phases have common saddle points = contribution t 2 Two phases have saddle lines = contribution t 1 Probability at the origin p 0 (t) decays like t 1 except for ψ ψ F (a, b) = (a, b, a, b) T which are orthogonal to v 1,2 (k) for k lying at the saddle lines M. Štefaňák (FJFI ČVUT) Recurrences in QWs Stará Lesná, / 34

35 Probability distribution for the Fourier walk For any initial state which is not a member of the family ψ F the probability at the origin decays slowly p 0 (t) t 1 The walk is recurrent M. Štefaňák (FJFI ČVUT) Recurrences in QWs Stará Lesná, / 34

36 Probability distribution for the Fourier walk For the initial states belonging to the family ψ F the probability at the origin decays fast p 0 (t) t 2 The walk is transient M. Štefaňák (FJFI ČVUT) Recurrences in QWs Stará Lesná, / 34

37 Outline 1 Introduction 2 Recurrence of RWs 3 Recurrence of QWs 4 Quantum walks on Z d Description of QWs Time evolution of QWs Asymptotics of QWs 5 Examples QW on a line QWs with tensor-product coins Grover walk Fourier walk 6 Summary M. Štefaňák (FJFI ČVUT) Recurrences in QWs Stará Lesná, / 34

38 Summary Definition of the Pólya number and recurrence for QWs Recurrence of QW is determined by the coin flip C and the initial conditions For the class of QWs with tensor-product coins the Pólya number is fully determined by the dimension of the lattice Recurrent QW can be constructed for arbitrary dimension classical RW are recurrent only for d = 1, 2 The result is counter-intuitive due to the quadratically faster spreading of the QWs compared to RWs References G. Pólya, Mathematische Annalen 84, 149 (1921) MŠ, I. Jex and T. Kiss, Phys. Rev. Lett. 100, (2008) MŠ, T. Kiss and I. Jex, arxiv: , accepted for Phys. Rev. A M. Štefaňák (FJFI ČVUT) Recurrences in QWs Stará Lesná, / 34

39 Thank you for your attention M. Štefaňák (FJFI ČVUT) Recurrences in QWs Stará Lesná, / 34