Quantum walk
quantum walk, quantum walk on graphs pptQuantum 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 nonrandom, 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 NonRelativistic Schrödinger Dynamics
 4 Discrete Time
 41 Discrete Time Quantum Walks on Z }
 5 Dirac Equation
 6 See also
 7 References
 8 Further reading
 9 External links
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[1][2] Quantum walks also give polynomial speedups over classical algorithms for many practical problems, such as the element distinctness problem,[3] the triangle finding problem,[4] and evaluating NAND trees[5] The wellknown 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: Continuoustime quantum walkContinuoustime 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, continuoustime quantum walks can provide a model for universal quantum computation [6]
Relation to NonRelativistic Schrödinger Dynamics
Consider the dynamics of a nonrelativistic, spinless free quantum particle with mass m propagating on an infinite onedimensional 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 onedimensional, 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 ddimensional lattices Z d ^} , cycle graphs Z / N Z /N\mathbb } , ddimensional discrete tori Z / N Z d /N\mathbb ^} , the ddimensional hypercube Q d ^} and random graphs
Discrete Time
This section needs expansion You can help by adding to it December 2009 
Discrete Time Quantum Walks on Z }
Probability distribution resulting from onedimensional discrete time random walks The quantum walk created using the Hadamard coin is plotted blue vs a classical walk red after 50 time stepsThe 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[7] Imagine a particle with a spin1/2 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 _i1\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 nontrivial quantum motion on Z } A popular choice for such a transformation is the Hadamard gate C = H , which, with respect to the standard zcomponent 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 spinup 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 nonclassical 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 leftmovers and rightmovers[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 leftmoving segments corresponding to one spin or coin, and rightmoving segments to the other See Feynman checkerboard for more details
The transition probability for a 1dimensional 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[8]
See also
 Path integral formulation
References
 ^ 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, quantph/0209131
 ^ 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
 ^ Andris Ambainis, Quantum walk algorithm for element distinctness, SIAM J Comput 37 2007, no 1, 210–239, arXiv:quantph/0311001, preliminary version in FOCS 2004
 ^ F Magniez, M Santha, and M Szegedy, Quantum algorithms for the triangle problem, Proc 16th ACMSIAM Symposium on Discrete Algorithms, pp 1109–1117, 2005, quantph/0310134
 ^ E Farhi, J Goldstone, and S Gutmann, A quantum algorithm for the Hamiltonian NAND tree, Theory of Computing 4 2008, no 1, 169–190, quantph/0702144
 ^ Andrew M Childs, "Universal Computation by Quantum Walk"
 ^ Kempe, Julia 20030701 "Quantum random walks  an introductory overview" Contemporary Physics 44 4: 307–327 arXiv:quantph/0303081 Bibcode:2003ConPh44307K doi:101080/00107151031000110776 ISSN 00107514
 ^ T Sunada and T Tate, Asymptotic behavior of quantum walks on the line, Journal of Functional Analysis 262 2012 26082645
Further reading
 Julia Kempe 2003 "Quantum random walks  an introductory overview" Contemporary Physics 44 4: 307–327 arXiv:quantph/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:quantph/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 VenegasAndraca 2012 "Quantum walks: a comprehensive review" Quantum Information Processing 11 5: 1015–1106 arXiv:12014780v2 doi:101007/s1112801204325
 Salvador E VenegasAndraca "Quantum Walks for Computer Scientists" Retrieved 16 October 2008
External links
 International Workshop on Mathematical and Physical Foundations of Discrete Time Quantum Walk
 Quantum walk
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