Complementarity in Quantum Walks - Jagiellonian University

Complementarity in Quantum Walks

Viv Kendon

VK, Barry Sanders (University of Calgary)[PRA 71 022307 2005, quant-ph/0404043]

W Dur, R Raussendorf, VK, H-J BriegelQuantum random walks in optical lattices

[PRA 66 052319 2002, quant-ph/0207137]

VK, Ben TregennaDecoherence can be useful in quantum walks

[PRA 67 042315 2003, quant-ph/0209005]

Quantum Information

School of Physics

& Astronomy

University of Leeds

Leeds LS2 9JT

[email protected]

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February 24, 2006 Complementarity in Quantum Walks

Overview

1. Introduce quantum walks

2. Decoherence in quantum walks

3. Quantum walks for algorithms

4. Physical systems doing quantum walks

5. What is ‘quantum’ about a quantum walk?

6. Summary

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Classical Random Walk on a Line

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Recipe:1. Start at the origin

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2. Toss a fair coin, result is HEADS or TAILS

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3. Move one unit: right for HEADS, left for TAILS

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4. Repeat steps 2. and 3. T times

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5. Measure position of walker,−T ≤ x ≤ T

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5. Measure position of walker,−T ≤ x ≤ TRepeat steps 1. to 5. many times−→ prob. dist. P (x, T ), binomial

standard deviation 〈x2〉1/2 =√

T

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Quantum Walk on a Line

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2. Toss a qubit (quantum coin) H|0〉 −→ (|0〉+ |1〉)/√2H|1〉 −→ (|0〉 − |1〉)/√2

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3. Move left and right according to qubit state S|x, 0〉 −→ |x− 1, 0〉S|x, 1〉 −→ |x + 1, 1〉

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5. measure position of walker,−T ≤ x ≤ T

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5. measure position of walker,−T ≤ x ≤ TRepeat steps 1. to 5. many times−→ prob. dist. P (x, T )...

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Quantum vs Classical on a Line

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prob

abili

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(x)

quantum spread∝ T compared with classical√

T

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Is it really a quantum walk?

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Add decoherence (measure with prob p at each step):

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x)p=0p=0.01p=0.02p=0.03p=0.05p=0.1p=0.2p=0.3p=0.4p=0.6p=0.8p=1

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Add decoherence (measure with prob p at each step):

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0.02

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obab

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dis

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n P(

x)p=0p=0.01p=0.02p=0.03p=0.05p=0.1p=0.2p=0.3p=0.4p=0.6p=0.8p=1

Top hat distribution for just the right amount of noise! [quant-ph/0209005]

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Cycles and Grids

Can also “quantum walk” on

a cycle:

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Cycles and Grids


a cycle:

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...or lattices:

(110)

(100)

(011)

(101)

(010)

(111)

(000) (001)

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Cycles and Grids


a cycle:

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...or lattices:

(110)

(100)

(011)

(101)

(010)

(111)

(000) (001)

...or grids:

(b)(a)

need larger coin: one dimension per edge

quant-ph/0304204, quant-ph/0504042

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Quantum Walk on a General Graph

...or on a very general graph:

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walker

coin

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Don’t know degree of graph: quant-ph/0306140

Know degree of graph: quant-ph/0404043

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Quantum Walk on a general graph

For a graph of N vertices, Hilbert spaceHvc = Hv ⊗Hc

• position on vertices: Hv = span{|j〉v : j ∈ ZN} and v〈j|j′〉v = δjj′

• d-dimensional coin: Hc = span{|k〉c : k ∈ Zd} and c〈k|k′〉c = δkk′

Mapping describes how coin states label edges in a consistent way:

ζ : ZN × Zd → ZN × Zd : (j, k) 7→ ζ(j, k) = (j′, k′)Discrete, unitary evolution:

• Toss the quantum coin: C : Hvc → Hvc : |j, k〉 7→∑k∈Zd

Cj

kk|j, k〉c

• Conditional shift: S : Hvc → Hvc : |j, k〉 7→ |j′, k′〉

In density matrix notation: ρ(t) = T tρ(0), T ≡ SC, SCρ ≡ SCρC†S†

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Classical Random Algorithms

Widely used for numerical simulations in physics (Monte Carlo, Markov chains):

• lattice QCD• polymer motion

• surface deposition and growth

• many body systems

Provide some of the best known classical algorithms for:

• factorisation• k-SAT• approximating the permanent (of a matrix)

• graph isomorphism

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Algorithms with quantum walks

Can do Grover’s search – find marked item in unsorted database

[Shenvi, Kempe, Whaley, quant-ph/0210064 (PRA 67 052307 2003)]

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Can do Grover’s search – find marked item in unsorted database

[Shenvi, Kempe, Whaley, quant-ph/0210064 (PRA 67 052307 2003)]

1. Start in a uniform distribution over graph with N nodes.

2. Use Grover coin everywhere except the marked node.

3. Run for approx π2

√N/2 steps.

4. Particle will now be at the marked node with high probability.

Quadratic speed up over classical

– inverse of starting at origin and trying to get uniform (top hat) distribution...

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“Glued trees” problem:

Find your way from

“Entrance” to “Exit”

Childs, Cleve,

Deotto, Farhi,

Gutmann, Spielman,

quant-ph/0209131

(STOC 2003)

Exit

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Entrance

Proof in principle that quantum walk algorithms can give exponential speed up

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Continuous Time Quantum Walk

Childs et al. give an approximate solution to the “glued trees” problem using a

continuous time walk:

A – adjacency matrix of the graph (Ajk = 1 iff ∃ an edge between sites j and k)

H = γA – Hamiltonian of the quantum walk

γ – transition rate (prob of moving to connected site per unit time)

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Continuous Time Quantum Walk

Childs et al. give an approximate solution to the “glued trees” problem using a

continuous time walk:

A – adjacency matrix of the graph (Ajk = 1 iff ∃ an edge between sites j and k)

H = γA – Hamiltonian of the quantum walk

γ – transition rate (prob of moving to connected site per unit time)

Walk is simply e−iHt followed by measurement at suitable time t

Making an algorithm involves significant detail (oracle, colouring...)

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“Glued Trees” measurement time

22

22

1011

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11

time

column

EXIT

ENTRANCE

glue

N = 10 Exit

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Entrance

Quantum walk proceeds

almost max speed to exit,

with some reflection at glue,

then returns...

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More quantum walk algorithms:

• Element distinctness [Ambainis quant-ph/0311001]

• Detecting triangles in graphs [Magniez, Santha, Szegedy quant-ph/0310134]

• Subset finding [Childs, Eisenberg quant-ph/0311038]

Generalises quantum walk version of Grover’s search to find more than one item.

Polynomial improvement over classical: O(N2/3) cf O(N )

[Short review of quantum walk algorithms: Ambainis, quant-ph/0403120]

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Physical systems doingquantum walks

Very different from algorithmic use,

coherent control of quantum systems:

• atom in trap (vibration states)Travaglione, Milburn quant-ph/0109076(PRA 65 032310 2002)

• phase of cavity field kicked byatom (walk in a cycle)Sanders, Bartlett, Tregenna, Knightquant-ph/0207028 (PRA 67 042305 2003)

• atom in optical latticeDur, Raussendorf, Kendon, Briegelquant-ph/0207137 (PRA 66 052319 2002)

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Quantum walk in Optical Lattice

latticemodulate

for shift

coin toss laser pulses (unfocused)

• some tricks to avoid heating

• detection need only pick out rough shape

• multiple entangled walkers: Omar/Paunkovic/Sheridan/Bose quant-ph/0411065O

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Wave walks

• can do quantum walk dynamics with classical light – same interference effects[Knight, Roldan, Sipe, PRA 68 020301 2003]

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Wave walks


• the experiment has been done: “Optical Galton Board”[Bouwmeester, Marzoli, Karman, Schleich, Woerdman, PRA 61 013410 1999]

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Wave walks



So what is QUANTUM about a quantum walk?

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Wave walks



So what is QUANTUM about a quantum walk?

Two contexts in which to answer this question:

1. physical systems

2. quantum walk algorithms

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Physical systems vs algorithms

Be clear about difference between physical systems and algorithms:

(run on digital computers – analogue computation is a different story)

Examples of Random Walks:

quantum classical

physical particle in optical lattice snakes and ladders (board game)

computer glued trees algorithm lattice QCD calculation

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quantum classical

physical particle in optical lattice snakes and ladders (board game)

computer glued trees algorithm lattice QCD calculation

• Can also do classical computer simulation of all of these four possibilities!

example: my own simulations of the glued trees quantum walk algorithm

...try to keep these multiple levels of abstraction clear...

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quantum classical

physical particle in optical lattice (1) snakes and ladders (board game)

←− (2)−→computer glued trees algorithm (3) lattice QCD calculation

• Can also do classical computer simulation of all of these four possibilities!

example: my own simulations of the glued trees quantum walk algorithm

...try to keep these multiple levels of abstraction clear...

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Quantum systems exhibit COMPLEMENTARITY“On the Notions of Causality and Complementarity”Bohr, Science 111 51 1950 [reprinted from Dialectica 2 312 1948]

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Young’s double slit experiment:

source

scre

en/d

etec

tor

path detectors

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Young’s double slit experiment:

source

scre

en/d

etec

tor

path detectors

quantum particles: measure path, fringes disappear

classical waves: no concept of “which path”, wave goes through both slits

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A quantum walk has many paths...

Quantum walk is just a more complicated set of paths between input and output:

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A quantum walk has many paths...

Quantum walk is just a more complicated set of paths between input and output:

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if measure which path, get classical random walk...

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Weak measurement of path

coupling strength β between ancillae and walker

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ancilla

meter

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Weak measurement of path

coupling strength β between ancillae and walker

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walker

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ancilla

meter

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1

interpolate between quantum (β = 0) and classical (β = 1)

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Classical wave walk – many quantum walkersclassical light wave is really made up of many photons

convenient to think of this as a coherent state:

indeterminate number of photons, but (fairly) sharp phase

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Classical wave walk – many quantum walkersclassical light wave is really made up of many photons

convenient to think of this as a coherent state:

indeterminate number of photons, but (fairly) sharp phase

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walker

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count number of photons and post-select:equivalent to n copies of single photon quantum walk

(each photon only interferes with itself) [quant-ph/0404043]

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Unary vs Binary Coding [Jozsa 1998]

Number Unary Binary

0 0

1 • 1

2 •• 10

3 • • • 11

4 • • •• 100

· · · · · · · · ·N N × • log2 N bits

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Number Unary Binary

0 0

1 • 1

2 •• 10

3 • • • 11

4 • • •• 100

· · · · · · · · ·N N × • log2 N bits

Read out:

Unary: distinguish between

measurements with N outcomes

Binary: log2 N measurements

with 2 outcomes each

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Number Unary Binary

0 0

1 • 1

2 •• 10

3 • • • 11

4 • • •• 100

· · · · · · · · ·N N × • log2 N bits

Read out:





−→ exponentially better accuracy

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Number Unary Binary

0 0

1 • 1

2 •• 10

3 • • • 11

4 • • •• 100

· · · · · · · · ·N N × • log2 N bits

Read out:





−→ exponentially better accuracy

binary encoding−→ exponential gain (reduction) in size of memory over unary

[does not have to be binary: Blume-Kohout, Caves, I. Deutsch Found. Phys. 32 1641-1670 quant-ph/0204157]

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Hilbert space is big...

Classical simulations of quantum algorithms and physical quantum systems

inefficient because Hilbert space is exponentially larger than

number of classical states available for same number of degrees of freedom:

Classical space

Hilbert space

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Hilbert space is big...

Classical simulations of quantum algorithms and physical quantum systems

inefficient because Hilbert space is exponentially larger than

number of classical states available for same number of degrees of freedom:

Classical space

Hilbert space

quantum parallelism−→ exponential gain (reduction) in memory over classical computer

see also:Blume-Kohout, Caves, I. Deutsch Found. Phys. 32 1641-1670

quant-ph/0204157; Jozsa, Linden quant-ph/0201143

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Quantum Simulation

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Quantum Simulation

A quantum system can simulate another quantum system efficiently

[Lloyd Science 273, 1073 1996] – map one Hilbert space directly onto the other

Has been demonstrated [Somaroo et al., 1999], using NMR quantum computers

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Quantum Simulation

A quantum system can simulate another quantum system efficiently

[Lloyd Science 273, 1073 1996] – map one Hilbert space directly onto the other

Has been demonstrated [Somaroo et al., 1999], using NMR quantum computers

However, like classical analogue computing...

accuracy is a problem

...does not scale efficiently with time needed to run simulation

[Brown et al. quant-ph/0601021]

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Summary

• Complementarity makes a quantum walk quantum in physics

Speed up and trade-off in computer simulation:

• exponential gain: binary coding over unary physical systemtrade off: simplicity of program (local interaction no longer local)

• exponential gain: quantum superposition over classical computertrade off: can’t get all the information out of a quantum state

• exponential gain: quantum system can simulate another quantum systemtrade: accuracy for analogue computation scales exponentially worse

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Acknowledgements and Funders

I have had interesting and helpful discussions of quantum walks with many, in particular:

Dorit Aharonov (Hebrew U)

Sougato Bose (UCL)

Richard Cleve, Andris Ambainis (U Waterloo/PI)

Julia Kempe (LRI Paris Orsay)

Andrew Childs (Caltech), Ed Farhi (MIT)

Mark Hillery (Hunter Col City U NY)

Peter Høyer, John Watrous (U Calgary)

Peter Knight, Ivens Carneiro, Ben Tregenna, Will

Flanagan, Rik Maile, Xibai Xu, Meng Loo (Imperial)

Cris Moore (New Mexico), Alex Russell (Connecticut)

Eugenio Roldan (U Valencia), John Sipe (U Toronto)

Mario Szegedy (Rutgers), Tino Tamon (Clarkson U)

Funding:

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Complementarity in Quantum Walks - Jagiellonian University