Recurrences and Full-revivals in Quantum Walks

Recurrences and Full-revivals in Quantum Walks M. Štefaňák (1), I. Jex (1), T. Kiss (2) (1) Department of Physics, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Czech Republic (2) Department of Nonlinear and Quantum Optics, RISSPO, Hungarian Academy of Sciences, Budapest, Hungary Stará Lesná, August 2009 M. Štefaňák (FJFI ČVUT) Recurrences in QWs 25. 8. 2009 1 / 40

Outline 1 Introduction 2 Recurrence of Classical Random Walks 3 Recurrence of Quantum Walks 4 Quantum Walks on Z d 5 Unbiased Quantum Walks 6 Biased Quantum Walk on a Line 7 Full-revivals in Quantum Walks 8 Summary M. Štefaňák (FJFI ČVUT) Recurrences in QWs 25. 8. 2009 2 / 40

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 RWs Diffusion We work with probabilities Spread slowly σ t QWs Wave propagation We work with probability amplitudes interference Spread fast σ t M. Štefaňák (FJFI ČVUT) Recurrences in QWs 25. 8. 2009 4 / 40

Quantum Walk 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 RW : m m 1, m + 1 = QW : m m 1 + m + 1 Does not preserve orthogonality { 0 1 + 1 orthogonal 2 1 + 3 } nonorthogonal To make the time evolution unitary we need an additional degree of freedom coin space H C = Span { L, R } M. Štefaňák (FJFI ČVUT) Recurrences in QWs 25. 8. 2009 5 / 40

Quantum Walk 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 R R + m 1 m L L Coin flip C - rotates the coin state before the step itself e.g. Hadamard matrix H = 1 2 ( 1 1 1 1 ) M. Štefaňák (FJFI ČVUT) Recurrences in QWs 25. 8. 2009 6 / 40

Quantum Walk on a line M. Štefaňák (FJFI ČVUT) Recurrences in QWs 25. 8. 2009 7 / 40

Pólya Number of a Random Walk 1 Probability that a 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 1 p 0 (t) t=0 Random walk is recurrent if P = 1 Random walk is transient if P < 1 - non-zero escape probability Unbiased random walks are recurrent for the dimensions d = 1, 2, transient for d 3 Biased random walks are transient for any dimension d 1 G. Pólya, Mathematische Annalen 84, 149 (1921) M. Štefaňák (FJFI ČVUT) Recurrences in QWs 25. 8. 2009 9 / 40

Recurrence of Random Walks 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 unbiased 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 For a biased RW p 0 (t) decays exponentially = P < 1 M. Štefaňák (FJFI ČVUT) Recurrences in QWs 25. 8. 2009 10 / 40

Pólya Number of a Quantum Walk 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, 020501 (2008) M. Štefaňák (FJFI ČVUT) Recurrences in QWs 25. 8. 2009 12 / 40

Recurrence of Quantum Walks 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 Probability at the origin is influenced by the additional degrees of freedom - coin operator C and the initial state ψ M. Štefaňák (FJFI ČVUT) Recurrences in QWs 25. 8. 2009 13 / 40

Hilbert Space of Quantum Walks 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 m m + e j { } H C = Span e j e j Z d, j = 1,..., n M. Štefaňák (FJFI ČVUT) Recurrences in QWs 25. 8. 2009 15 / 40

Time Evolution of Quantum Walks Time evolution is determined by ψ(t) m,j ψ j (m, t) m e j = U t ψ(0) Displacement operator U = S (I P C) S = m,j m + e j m e j e j Coin flip C unitary operator on H C M. Štefaňák (FJFI ČVUT) Recurrences in QWs 25. 8. 2009 16 / 40

Time Evolution of Quantum Walks Vectors of probability amplitudes ψ(m, t) (ψ 1 (m, t),..., ψ n (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 Translational invariance the matrices C l are independent of m Fourier transformation ψ(k, t) m ψ(m, t)e im k, k ( π, π] d simplifies the time evolution equation M. Štefaňák (FJFI ČVUT) Recurrences in QWs 25. 8. 2009 17 / 40

Time Evolution in the Fourier Picture 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 ien k) Walk starts localized at the origin - FT of the initial state is identical to the initial coin state ψ(m, 0) = 0 for m 0 = ψ(k, 0) = ψ(0, 0) ψ M. Štefaňák (FJFI ČVUT) Recurrences in QWs 25. 8. 2009 18 / 40

Solution of the Time Evolution Equations Solution in the momentum picture Matrix Ũ(k) is unitary λ j (k) = exp ( iω j (k) ), corresponding eigenvectors v j (k) ψ(k, t) = j e iω j (k)t ( v j (k), ψ ) v j (k) Solution in the position representation ψ(m, t) = dk e i(m k ω (2π) d j (k)t) ( v j (k), ψ ) v j (k) j ( π,π] d M. Štefaňák (FJFI ČVUT) Recurrences in QWs 25. 8. 2009 19 / 40

Asymptotics of QWs and Recurrence ψ(0, t) = j Amplitude at the origin dk (2π) d e iω j (k)t (v j (k), ψ ) v j (k) ( π,π] d Asymptotics of ψ(0, t) can be analyzed e.g. by the method of stationary phase 3 Stationary 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 stationary point ( vj (k 0 ), ψ ) 0 3 R. Wong, Asymptotic Approximations of Integrals, SIAM, Philadelphia (2001) M. Štefaňák (FJFI ČVUT) Recurrences in QWs 25. 8. 2009 20 / 40

Quantum Walk on a Line Propagator Ũ(k) = 1 ( e ik e ik 2 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 Quantum walk on a line is recurrent M. Štefaňák (FJFI ČVUT) Recurrences in QWs 25. 8. 2009 22 / 40

2-D Grover Walk 4 QW driven by the Grover coin G = 1 2 Eigenvalues of the propagator 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 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 2 (1, 1, 1, 1)T which is orthogonal to v 1,2 (k) for any k 4 MŠ, T. Kiss and I. Jex, Phys. Rev. A 78, 032306 (2008) M. Štefaňák (FJFI ČVUT) Recurrences in QWs 25. 8. 2009 23 / 40

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 25. 8. 2009 24 / 40

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 25. 8. 2009 25 / 40

2-D Fourier Walk 4 QW driven by the Fourier coin F = 1 2 1 1 1 1 1 i 1 i 1 1 1 1 1 i 1 i Stationary points follow from the implicit equation Φ(k, ω) det (ŨF (k) e iω I) = 0 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 4 MŠ, T. Kiss and I. Jex, Phys. Rev. A 78, 032306 (2008) M. Štefaňák (FJFI ČVUT) Recurrences in QWs 25. 8. 2009 26 / 40

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 25. 8. 2009 27 / 40

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 25. 8. 2009 28 / 40

Classical Biased Walk on a Line Step to the right by r units with probability p Spreading is diffusive σ t Mean varies linearly x = [p(r + 1) 1] t Recurrence vanishing mean p = 1 r + 1 M. Štefaňák (FJFI ČVUT) Recurrences in QWs 25. 8. 2009 30 / 40

Biased Quantum Walk on a Line 5 Step to the right by r units Coin operator ( ) ρ 1 ρ C = 1 ρ ρ Both mean and variance grows linearly Two peaks propagating with constant velocities v L and v R Probability in between the two peaks behave like P(m, t) t 1 Recurrence v L 0 and v R 0 5 MŠ, T. Kiss and I. Jex, New J. Phys. 11, 043027 (2009) M. Štefaňák (FJFI ČVUT) Recurrences in QWs 25. 8. 2009 31 / 40

Recurrence and Velocities of the Peaks Velocities can be found via the method of stationary phase ψ(m, t) = 2 j=1 π π dk 2π ei(ω j (k) αk)t ( vj (k), ψ ) v j (k), α = m t Peaks slower decay of the probability modified phase has higher order stationary point v L = r 1 2 ω j (k 0 ) = 0, ω j (k 0) = v L,R r + 1 ρ, vr = r 1 2 2 Condition of Recurrence ( ) r 1 2 ρ R r + 1 + r + 1 ρ 2 M. Štefaňák (FJFI ČVUT) Recurrences in QWs 25. 8. 2009 32 / 40

Stationary States and Full-revivals Some eigenvalues of Ũ(k) may be independent of the momenta k (typically single λ 1 = +1 or two λ 1,2 = ±1) The propagator U has a point spectrum Corresponding eigenvectors - stationary states Linear combinations - oscillating states The eigenvectors of Ũ(k) corresponding to constant eigenvalues depend on the momenta k Stationary and oscillating states are not localized M. Štefaňák (FJFI ČVUT) Recurrences in QWs 25. 8. 2009 34 / 40

Three-state Grover Walk on a Line The particle can move to the left, right or stay Coin operator - 3x3 Grover matrix G 3 = 1 3 1 2 2 2 1 2 2 2 1 One of the eigenvalues is constant - λ 1 = 1 Corresponding eigenvector v (3) 1 (e (k) = ik, 1 ) T 2 (1 + e ik ), 1 Fourier transformed vector - stationary state ψ (3) 2 1 [ = 1 L + 12 ] 5 ( 1 + 0 ) S + 0 R M. Štefaňák (FJFI ČVUT) Recurrences in QWs 25. 8. 2009 35 / 40

Four-state Grover Walk on a Line The particle can move to the left or right by one or two steps Coin operator - 4x4 Grover matrix G 4 = 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Two eigenvalues are constant - λ 1 = 1, λ 2 = 1 Corresponding eigenvectors v (4) 1 (k) = ( e ik + e 2ik, 1 + e 2ik, e ik + e ik, 1 + e ik) T v (4) 2 (k) = ( e 2ik, e 2ik (1 + e ik ), 1 + e ik, 1 ) T M. Štefaňák (FJFI ČVUT) Recurrences in QWs 25. 8. 2009 36 / 40

Four-state Grover Walk on a Line Fourier transformed vectors - stationary states ψ (4) 1 = 1 [ ( 2 + 1 ) 2L + ( 2 + 0 ) L + 8 ] +( 1 + 1 ) R + ( 0 + 1 ) 2R ψ (4) 2 = 1 [ 2 2L ( 2 + 1 ) L + 6 ] +( 1 + 0 ) R + 0 2R Linear combinations - oscillating states M. Štefaňák (FJFI ČVUT) Recurrences in QWs 25. 8. 2009 37 / 40

Four-state Grover Walk on a Line Initial state ψ(0) = 1 ( ) ψ (4) 2 1 + ψ(4) 2 State of the particle oscillates between two orthogonal states Autocorrelation function A(t) = ψ(0) ψ(t) 2 oscillates between one and zero M. Štefaňák (FJFI ČVUT) Recurrences in QWs 25. 8. 2009 38 / 40

Summary Main Results Extension of the concept of recurrence and Pólya number to quantum walks Recurrence of a quantum walk is determined by the coin operator C and the initial state ψ Recurrence is more stable against bias Full-revivals of a quantum state are possible References MŠ, I. Jex and T. Kiss, Phys. Rev. Lett. 100, 020501 (2008) MŠ, T. Kiss and I. Jex, Phys. Rev. A 78, 032306 (2008) MŠ, T. Kiss and I. Jex, New J. Phys. 11, 043027 (2009) Thank you for your attention M. Štefaňák (FJFI ČVUT) Recurrences in QWs 25. 8. 2009 40 / 40