Recurrences in Quantum Walks
|
|
- Patience Taylor
- 5 years ago
- Views:
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
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,
More informationSpatial correlations in quantum walks with two particles
Spatial correlations in quantum walks with two particles M. Štefaňák (1), S. M. Barnett 2, I. Jex (1) and T. Kiss (3) (1) Department of Physics, Faculty of Nuclear Sciences and Physical Engineering, Czech
More informationA quantum walk based search algorithm, and its optical realisation
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
More informationBound States and Recurrence Properties of Quantum Walks
Bound States and Recurrence Properties of Quantum Walks Autrans 18.07.2013 Albert H. Werner Joint work with: Andre Ahlbrecht, Christopher Cedzich, Volkher B. Scholz (now ETH), Reinhard F. Werner (Hannover)
More informationDynamics of Two-dimensional Quantum Walks. Dynamika dvoudimenzionálních kvantových procházek
Czech Technical University in Prague Faculty of Nuclear Sciences and Physical Engineering Dynamics of Two-dimensional Quantum Walks Dynamika dvoudimenzionálních kvantových procházek Research Project Author:
More informationLong-time Entanglement in the Quantum Walk
WECIQ 6 - Artigos Long-time Entanglement in the Quantum Walk Gonzalo Abal 1, Raul Donangelo, Hugo Fort 3 1 Instituto de Física, Facultad de Ingeniería, Universidad de la República, C.C. 3, C.P. 11, Montevideo,
More informationTrapping in Quantum Walks
Czech Technical University in Prague Faculty of Nuclear Sciences and Physical Engineering Trapping in Quantum Walks Uvěznění v kvantových procházkách Bachelor Thesis Author: Supervisor: Consultant: Tereza
More informationQuasi-Diffusion in a SUSY Hyperbolic Sigma Model
Quasi-Diffusion in a SUSY Hyperbolic Sigma Model Joint work with: M. Disertori and M. Zirnbauer December 15, 2008 Outline of Talk A) Motivation: Study time evolution of quantum particle in a random environment
More informationEntanglement and Decoherence in Coined Quantum Walks
Entanglement and Decoherence in Coined Quantum Walks Peter Knight, Viv Kendon, Ben Tregenna, Ivens Carneiro, Mathieu Girerd, Meng Loo, Xibai Xu (IC) & Barry Sanders, Steve Bartlett (Macquarie), Eugenio
More informationQuantum algorithms (CO 781, Winter 2008) Prof. Andrew Childs, University of Waterloo LECTURE 11: From random walk to quantum walk
Quantum algorithms (CO 781, Winter 2008) Prof. Andrew Childs, University of Waterloo LECTURE 11: From random walk to quantum walk We now turn to a second major topic in quantum algorithms, the concept
More informationDiscrete quantum random walks
Quantum Information and Computation: Report Edin Husić edin.husic@ens-lyon.fr Discrete quantum random walks Abstract In this report, we present the ideas behind the notion of quantum random walks. We further
More informationImplementing Quantum walks
Implementing Quantum walks P. Xue, B. C. Sanders, A. Blais, K. Lalumière, D. Leibfried IQIS, University of Calgary University of Sherbrooke NIST, Boulder 1 Reminder: quantum walk Quantum walk (discrete)
More informationLecture 4: Postulates of quantum mechanics
Lecture 4: Postulates of quantum mechanics Rajat Mittal IIT Kanpur The postulates of quantum mechanics provide us the mathematical formalism over which the physical theory is developed. For people studying
More informationTwo-Dimensional Quantum Walks with Boundaries
WECIQ 26 - Artigos Two-Dimensional Quantum Walks with Boundaries Amanda C. Oliveira 1, Renato ortugal 1, Raul Donangelo 2 1 Laboratório Nacional de Computação Científica, LNCC Caixa ostal 95113 25651-75
More informationRandom Walks and Quantum Walks
Random Walks and Quantum Walks Stephen Bartlett, Department of Physics and Centre for Advanced Computing Algorithms and Cryptography, Macquarie University Random Walks and Quantum Walks Classical random
More informationOn Alternating Quantum Walks
On Alternating Quantum Walks Jenia Rousseva, Yevgeniy Kovchegov Abstract We study an inhomogeneous quantum walk on a line that evolves according to alternating coins, each a rotation matrix. For the quantum
More informationQuantum Quenches in Extended Systems
Quantum Quenches in Extended Systems Spyros Sotiriadis 1 Pasquale Calabrese 2 John Cardy 1,3 1 Oxford University, Rudolf Peierls Centre for Theoretical Physics, Oxford, UK 2 Dipartimento di Fisica Enrico
More informationReturn Probability of the Fibonacci Quantum Walk
Commun. Theor. Phys. 58 (0 0 Vol. 58 o. August 5 0 Return Probability of the Fibonacci Quantum Walk CLEMET AMPADU 3 Carrolton Road Boston Massachusetts 03 USA (Received February 7 0; revised manuscript
More information!"#$%"&'(#)*+',$'-,+./0%0'#$1' 23$4$"3"+'5,&0'
!"#$%"&'(#)*+',$'-,+./0%0'#$1' 23$4$"3"+'5,&0' 6/010/,.*'(7'8%/#".9' -0:#/%&0$%'3;'0' Valencia 2011! Outline! Quantum Walks! DTQW & CTQW! Quantum Algorithms! Searching, Mixing,
More informationFourier mode dynamics for NLS Synchronization in fiber lasers arrays
Fourier mode dynamics for NLS Synchronization in fiber lasers arrays Nonlinear Schrodinger workshop Heraklio, May 21 2013 Jean-guy Caputo Laboratoire de Mathématiques INSA de Rouen, France A. Aceves, N.
More informationα x x 0 α x x f(x) α x x α x ( 1) f(x) x f(x) x f(x) α x = α x x 2
Quadratic speedup for unstructured search - Grover s Al- CS 94- gorithm /8/07 Spring 007 Lecture 11 01 Unstructured Search Here s the problem: You are given an efficient boolean function f : {1,,} {0,1},
More informationChapter 29. Quantum Chaos
Chapter 29 Quantum Chaos What happens to a Hamiltonian system that for classical mechanics is chaotic when we include a nonzero h? There is no problem in principle to answering this question: given a classical
More informationarxiv: v1 [quant-ph] 17 Jun 2009
Quantum search algorithms on the hypercube Birgit Hein and Gregor Tanner School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, UK e-mail: gregor.tanner@nottingham.ac.uk
More informationarxiv: v1 [quant-ph] 14 May 2007
The meeting problem in the quantum random walk arxiv:75.1985v1 [quant-ph] 14 May 7 M. Štefaňák (1), T. Kiss (), I. Jex (1) and B. Mohring (3) (1) Department of Physics, FJFI ČVUT, Břehová 7, 115 19 Praha
More informationHein, Birgit (2010) Quantum search algorithms. PhD thesis, University of Nottingham.
Hein, Birgit (00) Quantum search algorithms. PhD thesis, University of Nottingham. Access from the University of Nottingham repository: http://eprints.nottingham.ac.uk/5//phd_thesis_birgit_hein.pdf Copyright
More informationarxiv: v2 [quant-ph] 6 Feb 2018
Quantum Inf Process manuscript No. (will be inserted by the editor) Faster Search by Lackadaisical Quantum Walk Thomas G. Wong Received: date / Accepted: date arxiv:706.06939v2 [quant-ph] 6 Feb 208 Abstract
More informationDispersion relations, linearization and linearized dynamics in PDE models
Dispersion relations, linearization and linearized dynamics in PDE models 1 Dispersion relations Suppose that u(x, t) is a function with domain { < x 0}, and it satisfies a linear, constant coefficient
More informationQuantum walk algorithms
Quantum walk algorithms Andrew Childs Institute for Quantum Computing University of Waterloo 28 September 2011 Randomized algorithms Randomness is an important tool in computer science Black-box problems
More informationSOLUTIONS for Homework #2. 1. The given wave function can be normalized to the total probability equal to 1, ψ(x) = Ne λ x.
SOLUTIONS for Homework #. The given wave function can be normalized to the total probability equal to, ψ(x) = Ne λ x. () To get we choose dx ψ(x) = N dx e λx =, () 0 N = λ. (3) This state corresponds to
More informationC/CS/Phys C191 Grover s Quantum Search Algorithm 11/06/07 Fall 2007 Lecture 21
C/CS/Phys C191 Grover s Quantum Search Algorithm 11/06/07 Fall 2007 Lecture 21 1 Readings Benenti et al, Ch 310 Stolze and Suter, Quantum Computing, Ch 84 ielsen and Chuang, Quantum Computation and Quantum
More informationLecture 2: From Classical to Quantum Model of Computation
CS 880: Quantum Information Processing 9/7/10 Lecture : From Classical to Quantum Model of Computation Instructor: Dieter van Melkebeek Scribe: Tyson Williams Last class we introduced two models for deterministic
More informationTransport Properties of Quantum Walks, Deterministic and Random
Transport Properties of Quantum Walks, Deterministic and Random Alain JOYE i f INSTITUT FOURIER AQOS, Autrans, 8-19/7/2013 p.1/29 Unitary Quantum Walks Interdisciplinary Concept Quantum from Quantum Physics
More informationNew Journal of Physics
New Journal of Physics The open access journal for physics Entanglement in coined quantum walks on regular graphs Ivens Carneiro, Meng Loo, Xibai Xu, Mathieu Girerd,2, Viv Kendon,3 and Peter L Knight QOLS,
More informationFourier series on triangles and tetrahedra
Fourier series on triangles and tetrahedra Daan Huybrechs K.U.Leuven Cambridge, Sep 13, 2010 Outline Introduction The unit interval Triangles, tetrahedra and beyond Outline Introduction The topic of this
More informationNotes on Fourier Series and Integrals Fourier Series
Notes on Fourier Series and Integrals Fourier Series et f(x) be a piecewise linear function on [, ] (This means that f(x) may possess a finite number of finite discontinuities on the interval). Then f(x)
More informationReflections in Hilbert Space III: Eigen-decomposition of Szegedy s operator
Reflections in Hilbert Space III: Eigen-decomposition of Szegedy s operator James Daniel Whitfield March 30, 01 By three methods we may learn wisdom: First, by reflection, which is the noblest; second,
More informationarxiv: v1 [quant-ph] 15 Nov 2018
Lackadaisical quantum walk for spatial search Pulak Ranjan Giri International Institute of Physics, Universidade Federal do Rio Grande do orte, Campus Universitario, Lagoa ova, atal-r 59078-970, Brazil
More informationSuperposition of electromagnetic waves
Superposition of electromagnetic waves February 9, So far we have looked at properties of monochromatic plane waves. A more complete picture is found by looking at superpositions of many frequencies. Many
More information8 Quantized Interaction of Light and Matter
8 Quantized Interaction of Light and Matter 8.1 Dressed States Before we start with a fully quantized description of matter and light we would like to discuss the evolution of a two-level atom interacting
More informationLecture 3: Central Limit Theorem
Lecture 3: Central Limit Theorem Scribe: Jacy Bird (Division of Engineering and Applied Sciences, Harvard) February 8, 003 The goal of today s lecture is to investigate the asymptotic behavior of P N (εx)
More informationLecture 3: Central Limit Theorem
Lecture 3: Central Limit Theorem Scribe: Jacy Bird (Division of Engineering and Applied Sciences, Harvard) February 8, 003 The goal of today s lecture is to investigate the asymptotic behavior of P N (
More informationG : Quantum Mechanics II
G5.666: Quantum Mechanics II Notes for Lecture 5 I. REPRESENTING STATES IN THE FULL HILBERT SPACE Given a representation of the states that span the spin Hilbert space, we now need to consider the problem
More informationarxiv: v1 [quant-ph] 11 Oct 2017
Playing a true Parrondo s game with a three state coin on a quantum walk Jishnu Rajendran and Colin Benjamin School of Physical Sciences, National Institute of Science Education & Research, HBNI, Jatni-752050,
More informationA Minimal Uncertainty Product for One Dimensional Semiclassical Wave Packets
A Minimal Uncertainty Product for One Dimensional Semiclassical Wave Packets George A. Hagedorn Happy 60 th birthday, Mr. Fritz! Abstract. Although real, normalized Gaussian wave packets minimize the product
More informationLinear Algebra and Dirac Notation, Pt. 3
Linear Algebra and Dirac Notation, Pt. 3 PHYS 500 - Southern Illinois University February 1, 2017 PHYS 500 - Southern Illinois University Linear Algebra and Dirac Notation, Pt. 3 February 1, 2017 1 / 16
More information2. As we shall see, we choose to write in terms of σ x because ( X ) 2 = σ 2 x.
Section 5.1 Simple One-Dimensional Problems: The Free Particle Page 9 The Free Particle Gaussian Wave Packets The Gaussian wave packet initial state is one of the few states for which both the { x } and
More informationLikelihood in Quantum Random Walks
Likelihood in Quantum Random Walks Peter Jarvis School of Physical Sciences University of Tasmania peter.jarvis@utas.edu.au Joint work with Demosthenes Ellinas (Technical University of Crete, Chania) and
More informationQuantum algorithms (CO 781, Winter 2008) Prof. Andrew Childs, University of Waterloo LECTURE 15: Unstructured search and spatial search
Quantu algoriths (CO 781, Winter 2008) Prof Andrew Childs, University of Waterloo LECTURE 15: Unstructured search and spatial search ow we begin to discuss applications of quantu walks to search algoriths
More informationIntroduction to Neutrino Physics. TRAN Minh Tâm
Introduction to Neutrino Physics TRAN Minh Tâm LPHE/IPEP/SB/EPFL This first lecture is a phenomenological introduction to the following lessons which will go into details of the most recent experimental
More informationHamiltonian simulation with nearly optimal dependence on all parameters
Hamiltonian simulation with nearly optimal dependence on all parameters Dominic Berry + Andrew Childs obin Kothari ichard Cleve olando Somma Quantum simulation by quantum walks Dominic Berry + Andrew Childs
More informationPhonons and lattice dynamics
Chapter Phonons and lattice dynamics. Vibration modes of a cluster Consider a cluster or a molecule formed of an assembly of atoms bound due to a specific potential. First, the structure must be relaxed
More informationQUANTUM MECHANICS I PHYS 516. Solutions to Problem Set # 5
QUANTUM MECHANICS I PHYS 56 Solutions to Problem Set # 5. Crossed E and B fields: A hydrogen atom in the N 2 level is subject to crossed electric and magnetic fields. Choose your coordinate axes to make
More informationQuantum Monte Carlo Simulations in the Valence Bond Basis. Anders Sandvik, Boston University
Quantum Monte Carlo Simulations in the Valence Bond Basis Anders Sandvik, Boston University Outline The valence bond basis for S=1/2 spins Projector QMC in the valence bond basis Heisenberg model with
More informationOptimizing the discrete time quantum walk using a SU(2) coin
PHYSICAL REVIEW A 77, 336 8 Optimizing the discrete time quantum walk using a SU() coin C. M. Chandrashekar, 1 R. Srikanth,,3 and Raymond Laflamme 1,4 1 Institute for Quantum Computing, University of Waterloo,
More informationRenormalization Group for Quantum Walks
Renormalization Group for Quantum Walks CompPhys13 Stefan Falkner, Stefan Boettcher and Renato Portugal Physics Department Emory University November 29, 2013 arxiv:1311.3369 Funding: NSF-DMR Grant #1207431
More informationFigure 4.4: b < f. Figure 4.5: f < F b < f. k 1 = F b, k 2 = F + b. ( ) < F b < f k 1 = f, k 2 = F + b. where:, (see Figure 4.6).
Figure 4.4: b < f ii.) f < F b < f where: k = F b, k = F + b, (see Figure 4.5). Figure 4.5: f < F b < f. k = F b, k = F + b. ( ) < F b < f k = f, k = F + b iii.) f + b where:, (see Figure 4.6). 05 ( )
More information5.80 Small-Molecule Spectroscopy and Dynamics
MIT OpenCourseWare http://ocw.mit.edu 5.80 Small-Molecule Spectroscopy and Dynamics Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. Lectures
More informationShort Course in Quantum Information Lecture 2
Short Course in Quantum Information Lecture Formal Structure of Quantum Mechanics Course Info All materials downloadable @ website http://info.phys.unm.edu/~deutschgroup/deutschclasses.html Syllabus Lecture
More informationMP463 QUANTUM MECHANICS
MP463 QUANTUM MECHANICS Introduction Quantum theory of angular momentum Quantum theory of a particle in a central potential - Hydrogen atom - Three-dimensional isotropic harmonic oscillator (a model of
More information1 Planar rotations. Math Abstract Linear Algebra Fall 2011, section E1 Orthogonal matrices and rotations
Math 46 - Abstract Linear Algebra Fall, section E Orthogonal matrices and rotations Planar rotations Definition: A planar rotation in R n is a linear map R: R n R n such that there is a plane P R n (through
More informationQuantum Searching. Robert-Jan Slager and Thomas Beuman. 24 november 2009
Quantum Searching Robert-Jan Slager and Thomas Beuman 24 november 2009 1 Introduction Quantum computers promise a significant speed-up over classical computers, since calculations can be done simultaneously.
More informationarxiv: v1 [quant-ph] 1 Mar 2011
Quantum walk with a four dimensional coin Craig S. Hamilton 1, Aurél Gábris 1, Igor Jex 1 and Stephen M. Barnett 1, 1 Department of Physics Faculty of Nuclear Sciences and Physical Engineering, Czech Technical
More informationHigher-order exceptional points
Higher-order exceptional points Ingrid Rotter Max Planck Institute for the Physics of Complex Systems Dresden (Germany) Mathematics: Exceptional points Consider a family of operators of the form T(κ) =
More informationProblem 1: Spin 1 2. particles (10 points)
Problem 1: Spin 1 particles 1 points 1 Consider a system made up of spin 1/ particles. If one measures the spin of the particles, one can only measure spin up or spin down. The general spin state of a
More informationNumerical construction of Wannier functions
July 12, 2017 Internship Tutor: A. Levitt School Tutor: É. Cancès 1/27 Introduction Context: Describe electrical properties of crystals (insulator, conductor, semi-conductor). Applications in electronics
More informationGround State Projector QMC in the valence-bond basis
Quantum Monte Carlo Methods at Work for Novel Phases of Matter Trieste, Italy, Jan 23 - Feb 3, 2012 Ground State Projector QMC in the valence-bond basis Anders. Sandvik, Boston University Outline: The
More informationQuantum speedup of backtracking and Monte Carlo algorithms
Quantum speedup of backtracking and Monte Carlo algorithms Ashley Montanaro School of Mathematics, University of Bristol 19 February 2016 arxiv:1504.06987 and arxiv:1509.02374 Proc. R. Soc. A 2015 471
More informationMath 216 Final Exam 24 April, 2017
Math 216 Final Exam 24 April, 2017 This sample exam is provided to serve as one component of your studying for this exam in this course. Please note that it is not guaranteed to cover the material that
More informationarxiv:quant-ph/ v2 14 Jul 2005
Entanglement in coined quantum walks on regular graphs arxiv:quant-ph/5442 v2 4 Jul 25 Ivens Carneiro, Meng Loo, Xibai Xu, Mathieu Girerd, 2, Viv Kendon,, and Peter L. Knight QOLS, Blackett Laboratory,
More informationDecoherence on Szegedy s Quantum Walk
Decoherence on Szegedy s Quantum Walk Raqueline A. M. Santos, Renato Portugal. Laboratório Nacional de Computação Científica (LNCC) Av. Getúlio Vargas 333, 25651-075, Petrópolis, RJ, Brazil E-mail: raqueline@lncc.br,
More informationarxiv: v1 [quant-ph] 10 Sep 2012
Quantum Walks on Sierpinski Gaskets arxiv:1209.2095v1 [quant-ph] 10 Sep 2012 Pedro Carlos S. Lara, Renato Portugal, and Stefan Boettcher May 2, 2014 Abstract We analyze discrete-time quantum walks on Sierpinski
More informationIntegrated devices for quantum information with polarization encoded qubits
Integrated devices for quantum information with polarization encoded qubits Dottorato in Fisica XXV ciclo Linda Sansoni Supervisors: Prof. Paolo Mataloni, Dr. Fabio Sciarrino http:\\quantumoptics.phys.uniroma1.it
More information1 Time-Dependent Two-State Systems: Rabi Oscillations
Advanced kinetics Solution 7 April, 16 1 Time-Dependent Two-State Systems: Rabi Oscillations a In order to show how Ĥintt affects a bound state system in first-order time-dependent perturbation theory
More informationChemistry 365: Normal Mode Analysis David Ronis McGill University
Chemistry 365: Normal Mode Analysis David Ronis McGill University 1. Quantum Mechanical Treatment Our starting point is the Schrodinger wave equation: Σ h 2 2 2m i N i=1 r 2 i + U( r 1,..., r N ) Ψ( r
More informationarxiv:quant-ph/ v1 23 Oct 2002
Quantum Walks driven by many coins Todd A. Brun, 1, Hilary A. Carteret,, and Andris Ambainis 1, 1 Institute for Advanced Study, Einstein Drive, Princeton, NJ 08540 Department of Combinatorics and Optimization,
More informationarxiv:hep-th/ v1 2 Jul 1998
α-representation for QCD Richard Hong Tuan arxiv:hep-th/9807021v1 2 Jul 1998 Laboratoire de Physique Théorique et Hautes Energies 1 Université de Paris XI, Bâtiment 210, F-91405 Orsay Cedex, France Abstract
More informationarxiv:quant-ph/ v1 15 Apr 2005
Quantum walks on directed graphs Ashley Montanaro arxiv:quant-ph/0504116v1 15 Apr 2005 February 1, 2008 Abstract We consider the definition of quantum walks on directed graphs. Call a directed graph reversible
More informationQuantum Monte Carlo Simulations in the Valence Bond Basis
NUMERICAL APPROACHES TO QUANTUM MANY-BODY SYSTEMS, IPAM, January 29, 2009 Quantum Monte Carlo Simulations in the Valence Bond Basis Anders W. Sandvik, Boston University Collaborators Kevin Beach (U. of
More informationarxiv: v4 [quant-ph] 25 Sep 2017
Grover Search with Lackadaisical Quantum Walks Thomas G Wong Faculty of Computing, University of Latvia, Raiņa ulv. 9, Rīga, LV-586, Latvia E-mail: twong@lu.lv arxiv:50.04567v4 [quant-ph] 5 Sep 07. Introduction
More informationarxiv: v2 [quant-ph] 21 Dec 2016
The Lackadaisical Quantum Walker is NOT Lazy at all arxiv:6203370v2 [quant-ph] 2 Dec 206 Kun Wang, Nan Wu,, Ping Xu, 2 and Fangmin Song State Key Laboratory for Novel Software Technology, Department of
More informationTime dependent perturbation theory 1 D. E. Soper 2 University of Oregon 11 May 2012
Time dependent perturbation theory D. E. Soper University of Oregon May 0 offer here some background for Chapter 5 of J. J. Sakurai, Modern Quantum Mechanics. The problem Let the hamiltonian for a system
More informationLyapunov-based control of quantum systems
Lyapunov-based control of quantum systems Symeon Grivopoulos Bassam Bamieh Department of Mechanical and Environmental Engineering University of California, Santa Barbara, CA 936-57 symeon,bamieh@engineering.ucsb.edu
More informationFrom Quantum Cellular Automata to Quantum Field Theory
From Quantum Cellular Automata to Quantum Field Theory Alessandro Bisio Frontiers of Fundamental Physics Marseille, July 15-18th 2014 in collaboration with Giacomo Mauro D Ariano Paolo Perinotti Alessandro
More informationBaryon Resonance Determination using LQCD. Robert Edwards Jefferson Lab. Baryons 2013
Baryon Resonance Determination using LQCD Robert Edwards Jefferson Lab Baryons 2013 Where are the Missing Baryon Resonances? What are collective modes? Is there freezing of degrees of freedom? What is
More informationSSH Model. Alessandro David. November 3, 2016
SSH Model Alessandro David November 3, 2016 Adapted from Lecture Notes at: https://arxiv.org/abs/1509.02295 and from article: Nature Physics 9, 795 (2013) Motivations SSH = Su-Schrieffer-Heeger Polyacetylene
More informationCHAPTER 1. Polarisation
CHAPTER 1 Polarisation This report was prepared by Abhishek Dasgupta and Arijit Haldar based on notes in Dr. Ananda Dasgupta s Electromagnetism III class Topics covered in this chapter are the Jones calculus,
More informationFourier transforms, Generalised functions and Greens functions
Fourier transforms, Generalised functions and Greens functions T. Johnson 2015-01-23 Electromagnetic Processes In Dispersive Media, Lecture 2 - T. Johnson 1 Motivation A big part of this course concerns
More informationThe quantum mechanics approach to uncertainty modeling in structural dynamics
p. 1/3 The quantum mechanics approach to uncertainty modeling in structural dynamics Andreas Kyprianou Department of Mechanical and Manufacturing Engineering, University of Cyprus Outline Introduction
More informationQuestioning Quantum Mechanics? Kurt Barry SASS Talk January 25 th, 2012
Questioning Quantum Mechanics? Kurt Barry SASS Talk January 25 th, 2012 2 Model of the Universe Fundamental Theory Low-Energy Limit Effective Field Theory Quantum Mechanics Quantum Mechanics is presently
More informationExtreme Values and Positive/ Negative Definite Matrix Conditions
Extreme Values and Positive/ Negative Definite Matrix Conditions James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University November 8, 016 Outline 1
More informationECE 595, Section 10 Numerical Simulations Lecture 12: Applications of FFT. Prof. Peter Bermel February 6, 2013
ECE 595, Section 10 Numerical Simulations Lecture 12: Applications of FFT Prof. Peter Bermel February 6, 2013 Outline Recap from Friday Real FFTs Multidimensional FFTs Applications: Correlation measurements
More informationEntanglement Entropy in Extended Quantum Systems
Entanglement Entropy in Extended Quantum Systems John Cardy University of Oxford STATPHYS 23 Genoa Outline A. Universal properties of entanglement entropy near quantum critical points B. Behaviour of entanglement
More informationExotic and excited-state radiative transitions in charmonium from lattice QCD
Exotic and excited-state radiative transitions in charmonium from lattice QCD Christopher Thomas, Jefferson Lab Hadron Spectroscopy Workshop, INT, November 2009 In collaboration with: Jo Dudek, Robert
More informationAdvanced Cryptography Quantum Algorithms Christophe Petit
The threat of quantum computers Advanced Cryptography Quantum Algorithms Christophe Petit University of Oxford Christophe Petit -Advanced Cryptography 1 Christophe Petit -Advanced Cryptography 2 The threat
More informationUnderstand the existence and uniqueness theorems and what they tell you about solutions to initial value problems.
Review Outline To review for the final, look over the following outline and look at problems from the book and on the old exam s and exam reviews to find problems about each of the following topics.. Basics
More informationHomework 2. Solutions T =
Homework. s Let {e x, e y, e z } be an orthonormal basis in E. Consider the following ordered triples: a) {e x, e x + e y, 5e z }, b) {e y, e x, 5e z }, c) {e y, e x, e z }, d) {e y, e x, 5e z }, e) {
More informationarxiv:cond-mat/ v1 20 Sep 2004
One-dimensional continuous-time quantum walks DANIEL BEN-AVRAHAM ERIK M. BOLLT CHRISTINO TAMON September, 4 arxiv:cond-mat/4954 v Sep 4 Abstract We survey the equations of continuous-time quantum walks
More informationKernel Methods. Machine Learning A W VO
Kernel Methods Machine Learning A 708.063 07W VO Outline 1. Dual representation 2. The kernel concept 3. Properties of kernels 4. Examples of kernel machines Kernel PCA Support vector regression (Relevance
More informationDiscrete Quantum Theories
Discrete Quantum Theories Andrew J. Hanson 1 Gerardo Ortiz 2 Amr Sabry 1 Yu-Tsung Tai 3 (1) School of Informatics and Computing (2) Department of Physics (3) Mathematics Department Indiana University July
More information2 The Density Operator
In this chapter we introduce the density operator, which provides an alternative way to describe the state of a quantum mechanical system. So far we have only dealt with situations where the state of a
More information