﻿ Quantum walk
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# Quantum walk

quantum walk, quantum walk on graphs ppt
Quantum walks are quantum analogues of classical random walks In contrast to the classical random walk, where the walker occupies definite states and the randomness arises due to stochastic transitions between states, randomness arises in quantum walks through: 1 quantum superposition of states, 2 non-random, reversible unitary evolution and 3 collapse of the wavefunction due to state measurements

As with classical random walks, quantum walks admit formulations in both discrete and continuous time

## Contents

• 1 Motivation
• 2 Relation to Classical Random Walks
• 3 Continuous Time
• 31 Relation to Non-Relativistic Schrödinger Dynamics
• 4 Discrete Time
• 41 Discrete Time Quantum Walks on Z }
• 5 Dirac Equation
• 7 References

## Motivation

Quantum walks are motivated by the widespread use of classical random walks in the design of randomized algorithms, and are part of several quantum algorithms For some oracular problems, quantum walks provide an exponential speedup over any classical algorithm Quantum walks also give polynomial speedups over classical algorithms for many practical problems, such as the element distinctness problem, the triangle finding problem, and evaluating NAND trees The well-known Grover search algorithm can also be viewed as a quantum walk algorithm

## Relation to Classical Random Walks

Quantum walks exhibit very different features from classical random walks In particular, they do not converge to limiting distributions and due to the power of quantum interference they may spread significantly faster or slower than their classical equivalents

## Continuous Time

Main article: Continuous-time quantum walk

Continuous-time quantum walks arise when one replaces the continuum spatial domain in the Schrödinger equation with a discrete set That is, instead of having a quantum particle propagate in a continuum, one restricts the set of possible position states to the vertex set V of some graph G = V , E which can be either finite or countably infinite Under particular conditions, continuous-time quantum walks can provide a model for universal quantum computation 

### Relation to Non-Relativistic Schrödinger Dynamics

Consider the dynamics of a non-relativistic, spin-less free quantum particle with mass m propagating on an infinite one-dimensional spatial domain The particle's motion is completely described by its wave function ψ : R × R ≥ 0 → C \times \mathbb _\to \mathbb } which satisfies the one-dimensional, free particle Schrödinger equation

i ℏ ∂ ψ ∂ t = − ℏ 2 2 m ∂ 2 ψ ∂ x 2 }\hbar }=-}}\psi }}}}

where i = − 1 }=}} and ℏ is Planck's constant Now suppose that only the spatial part of the domain is discretized, R } being replaced with Z Δ x ≡ _\equiv \} where Δ x is the separation between the spatial sites the particle can occupy The wave function becomes the map ψ : Z Δ x × R ≥ 0 → C _\times \mathbb _\to \mathbb } and the second spatial partial derivative becomes the discrete laplacian

∂ 2 ψ ∂ x 2 → L Z ψ j Δ x , t Δ x 2 ≡ ψ j + 1 Δ x , t − 2 ψ j Δ x , t + ψ j − 1 Δ x , t Δ x 2 \psi }}}\to }\psi j\Delta x,t}}}\equiv }}}

The evolution equation for a continuous time quantum walk on Z Δ x _} is thus

i ∂ ψ ∂ t = − ω Δ x L Z ψ }}=-\omega _L_ }\psi }

where ω Δ x ≡ ℏ / 2 m Δ x 2 \equiv \hbar /2m\Delta x^} is a characteristic frequency This construction naturally generalizes to the case that the discretized spatial domain is an arbitrary graph G = V , E and the discrete laplacian L Z }} is replaced by the graph laplacian L G ≡ D G − A G \equiv D_-A_} where D G } and A G } are the degree matrix and the adjacency matrix, respectively Common choices of graphs that show up in the study of continuous time quantum walks are the d-dimensional lattices Z d ^} , cycle graphs Z / N Z /N\mathbb } , d-dimensional discrete tori Z / N Z d /N\mathbb ^} , the d-dimensional hypercube Q d ^} and random graphs

## Discrete Time

### Discrete Time Quantum Walks on Z }

Probability distribution resulting from one-dimensional discrete time random walks The quantum walk created using the Hadamard coin is plotted blue vs a classical walk red after 50 time steps

The evolution of a quantum walk in discrete time is specified by the product of two unitary operators: 1 a "coin flip" operator and 2 a conditional shift operator, which are applied repeatedly The following example is instructive here Imagine a particle with a spin-1/2 degree of freedom propagating on a linearly array of discrete sites If the number of such sites is countably infinite, we identify the state space with Z } The particle's state can then be described by a product state

| Ψ ⟩ = | s ⟩ ⊗ | ψ ⟩

consisting of an internal spin state

| s ⟩ ∈ H C = }_=\}

and a position state

| ψ ⟩ ∈ H P = }_=\left\ }\alpha _|x\rangle :\sum _ }|\alpha _|^<\infty \right\}}

where H C = C 2 }_=\mathbb ^} is the "coin space" and H P = ℓ 2 Z }_=\ell ^\mathbb } is the space of physical quantum position states The product ⊗ in this setting is the Kronecker tensor product The conditional shift operator for the quantum walk on the line is given by

S = | ↑ ⟩ ⟨ ↑ | ⊗ ∑ i | i + 1 ⟩ ⟨ i | + | ↓ ⟩ ⟨ ↓ | ⊗ ∑ i | i − 1 ⟩ ⟨ i | |i+1\rangle \langle i|+|\downarrow \rangle \langle \downarrow |\otimes \sum \limits _|i-1\rangle \langle i|} ,

ie the particle jumps right if it has spin up and left if it has spin down Explicitly, the conditional shift operator acts on product states according to

S | ↑ ⟩ ⊗ | i ⟩ = | ↑ ⟩ ⊗ | i + 1 ⟩

S | ↓ ⟩ ⊗ | i ⟩ = | ↓ ⟩ ⊗ | i − 1 ⟩

If we first rotate the spin with some unitary transformation C : H C → H C }_\to }_} and then apply S , we get a non-trivial quantum motion on Z } A popular choice for such a transformation is the Hadamard gate C = H , which, with respect to the standard z-component spin basis, has matrix representation

H = 1 2 1 1 1 − 1 }}1&\;\;1\\1&-1\\\end}}

When this choice is made for the coin flip operator, the operator itself is called the "Hadamard coin" and the resulting quantum walk is called the "Hadamard walk" If the walker is initialized at the origin and in the spin-up state, a single time step of the Hadamard walk on Z } is

| ↑ ⟩ ⊗ | 0 ⟩ ⟶ H 1 2 | ↑ ⟩ + | ↓ ⟩ ⊗ | 0 ⟩ ⟶ S 1 2 | ↑ ⟩ ⊗ | 1 ⟩ + | ↓ ⟩ ⊗ | − 1 ⟩ }}}|\uparrow \rangle +|\downarrow \rangle \otimes |0\rangle }}}|\uparrow \rangle \otimes |1\rangle +|\downarrow \rangle \otimes |-1\rangle }

Measurement of the system's state at this point would reveal an up spin at position 1 or a down spin at position -1, both with probability 1/2 Repeating the procedure would correspond to a classical simple random walk on Z } In order to observe non-classical motion, no measurement is performed on the state at this point and therefore do not force a collapse of the wave function Instead, repeat the procedure of rotating the spin with the coin flip operator and conditionally jumping with S This way, quantum correlations are preserved and different position states can interfere with one another This gives a drastically different probability distribution than the classical random walk Gaussian distribution as seen in the figure to the right Spatially one sees that the distribution is not symmetric: even though the Hadamard coin gives both up and down spin with equal probability, the distribution tends to drift to the right when the initial spin is | ↑ ⟩ This asymmetry is entirely due to the fact that the Hadamard coin treats the | ↑ ⟩ and | ↓ ⟩ state asymmetrically A symmetric probability distribution arises if the initial state is chosen to be

| Ψ 0 symm ⟩ = 1 2 | ↑ ⟩ + i | ↓ ⟩ ⊗ | 0 ⟩ ^}\rangle =}}|\uparrow \rangle +}|\downarrow \rangle \otimes |0\rangle }

## Dirac Equation

Consider what happens when we discretize a massive Dirac operator over one spatial dimension In the absence of a mass term, we have left-movers and right-movers[clarification needed] They can be characterized by an internal degree of freedom, "spin" or a "coin" When we turn on a mass term, this corresponds to a rotation in this internal "coin" space A quantum walk corresponds to iterating the shift and coin operators repeatedly

This is very much like Richard Feynman's model of an electron in 1 one spatial and 1 one time dimension He summed up the zigzagging paths, with left-moving segments corresponding to one spin or coin, and right-moving segments to the other See Feynman checkerboard for more details

The transition probability for a 1-dimensional quantum walk behaves like the Hermite functions which 1 asymptotically oscillate in the classically allowed region, 2 is approximated by the Airy function around the wall of the potential[clarification needed], and 3 exponentially decay in the classically hidden region

• Path integral formulation

## References

1. ^ A M Childs, R Cleve, E Deotto, E Farhi, S Gutmann, and D A Spielman, Exponential algorithmic speedup by quantum walk, Proc 35th ACM Symposium on Theory of Computing, pp 59–68, 2003, quant-ph/0209131
2. ^ A M Childs, L J Schulman, and U V Vazirani, Quantum algorithms for hidden nonlinear structures, Proc 48th IEEE Symposium on Foundations of Computer Science, pp 395–404, 2007, arXiv:07052784
3. ^ Andris Ambainis, Quantum walk algorithm for element distinctness, SIAM J Comput 37 2007, no 1, 210–239, arXiv:quant-ph/0311001, preliminary version in FOCS 2004
4. ^ F Magniez, M Santha, and M Szegedy, Quantum algorithms for the triangle problem, Proc 16th ACM-SIAM Symposium on Discrete Algorithms, pp 1109–1117, 2005, quant-ph/0310134
5. ^ E Farhi, J Goldstone, and S Gutmann, A quantum algorithm for the Hamiltonian NAND tree, Theory of Computing 4 2008, no 1, 169–190, quant-ph/0702144
6. ^ Andrew M Childs, "Universal Computation by Quantum Walk"
7. ^ Kempe, Julia 2003-07-01 "Quantum random walks - an introductory overview" Contemporary Physics 44 4: 307–327 arXiv:quant-ph/0303081  Bibcode:2003ConPh44307K doi:101080/00107151031000110776 ISSN 0010-7514
8. ^ T Sunada and T Tate, Asymptotic behavior of quantum walks on the line, Journal of Functional Analysis 262 2012 2608-2645

• Julia Kempe 2003 "Quantum random walks - an introductory overview" Contemporary Physics 44 4: 307–327 arXiv:quant-ph/0303081  Bibcode:2003ConPh44307K doi:101080/00107151031000110776
• Andris Ambainis 2003 "Quantum walks and their algorithmic applications" International Journal of Quantum Information 1 4: 507–518 arXiv:quant-ph/0403120  doi:101142/S0219749903000383
• Miklos Santha 2008 "Quantum walk based search algorithms" Th Theory and Applications of Models of Computation TAMC, Xian, April, LNCS 4978 5 8: 31–46 arXiv:08080059  Bibcode:2008arXiv08080059S
• Salvador E Venegas-Andraca 2012 "Quantum walks: a comprehensive review" Quantum Information Processing 11 5: 1015–1106 arXiv:12014780v2  doi:101007/s11128-012-0432-5
• Salvador E Venegas-Andraca "Quantum Walks for Computer Scientists" Retrieved 16 October 2008

• International Workshop on Mathematical and Physical Foundations of Discrete Time Quantum Walk
• Quantum walk

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29.10.2014

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