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
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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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February 24, 2006 Complementarity in Quantum Walks
Classical Random Walk on a Line
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Recipe:1. Start at the origin
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February 24, 2006 Complementarity in Quantum Walks
Classical Random Walk on a Line
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Recipe:1. Start at the origin
2. Toss a fair coin, result is HEADS or TAILS
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February 24, 2006 Complementarity in Quantum Walks
Classical Random Walk on a Line
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Recipe:1. Start at the origin
2. Toss a fair coin, result is HEADS or TAILS
3. Move one unit: right for HEADS, left for TAILS
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February 24, 2006 Complementarity in Quantum Walks
Classical Random Walk on a Line
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−9 −8 −7 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 7 8 90
Recipe:1. Start at the origin
2. Toss a fair coin, result is HEADS or TAILS
3. Move one unit: right for HEADS, left for TAILS
4. Repeat steps 2. and 3. T times
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February 24, 2006 Complementarity in Quantum Walks
Classical Random Walk on a Line
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−9 −8 −7 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 7 8 90
Recipe:1. Start at the origin
2. Toss a fair coin, result is HEADS or TAILS
3. Move one unit: right for HEADS, left for TAILS
4. Repeat steps 2. and 3. T times
5. Measure position of walker,−T ≤ x ≤ T
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February 24, 2006 Complementarity in Quantum Walks
Classical Random Walk on a Line
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−9 −8 −7 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 7 8 90
Recipe:1. Start at the origin
2. Toss a fair coin, result is HEADS or TAILS
3. Move one unit: right for HEADS, left for TAILS
4. Repeat steps 2. and 3. T times
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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February 24, 2006 Complementarity in Quantum Walks
Quantum Walk on a Line
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February 24, 2006 Complementarity in Quantum Walks
Quantum Walk on a Line
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Recipe:1. Start at the origin
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February 24, 2006 Complementarity in Quantum Walks
Quantum Walk on a Line
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Recipe:1. Start at the origin
2. Toss a qubit (quantum coin) H|0〉 −→ (|0〉+ |1〉)/√2H|1〉 −→ (|0〉 − |1〉)/√2
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February 24, 2006 Complementarity in Quantum Walks
Quantum Walk on a Line
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Recipe:1. Start at the origin
2. Toss a qubit (quantum coin) H|0〉 −→ (|0〉+ |1〉)/√2H|1〉 −→ (|0〉 − |1〉)/√2
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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February 24, 2006 Complementarity in Quantum Walks
Quantum Walk on a Line
−9 −8 −7 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 8 90 7
Recipe:1. Start at the origin
2. Toss a qubit (quantum coin) H|0〉 −→ (|0〉+ |1〉)/√2H|1〉 −→ (|0〉 − |1〉)/√2
3. Move left and right according to qubit state S|x, 0〉 −→ |x− 1, 0〉S|x, 1〉 −→ |x + 1, 1〉
4. Repeat steps 2. and 3. T times
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February 24, 2006 Complementarity in Quantum Walks
Quantum Walk on a Line
−9 −8 −7 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 8 90 7
Recipe:1. Start at the origin
2. Toss a qubit (quantum coin) H|0〉 −→ (|0〉+ |1〉)/√2H|1〉 −→ (|0〉 − |1〉)/√2
3. Move left and right according to qubit state S|x, 0〉 −→ |x− 1, 0〉S|x, 1〉 −→ |x + 1, 1〉
4. Repeat steps 2. and 3. T times
5. measure position of walker,−T ≤ x ≤ T
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February 24, 2006 Complementarity in Quantum Walks
Quantum Walk on a Line
−9 −8 −7 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 8 90 7
Recipe:1. Start at the origin
2. Toss a qubit (quantum coin) H|0〉 −→ (|0〉+ |1〉)/√2H|1〉 −→ (|0〉 − |1〉)/√2
3. Move left and right according to qubit state S|x, 0〉 −→ |x− 1, 0〉S|x, 1〉 −→ |x + 1, 1〉
4. Repeat steps 2. and 3. T times
5. measure position of walker,−T ≤ x ≤ TRepeat steps 1. to 5. many times−→ prob. dist. P (x, T )...
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February 24, 2006 Complementarity in Quantum Walks
Quantum vs Classical on a Line
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0
0.02
0.04
0.06
0.08
prob
abili
ty P
(x)
quantum spread∝ T compared with classical√
T
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February 24, 2006 Complementarity in Quantum Walks
Is it really a quantum walk?
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February 24, 2006 Complementarity in Quantum Walks
Is it really a quantum walk?
Add decoherence (measure with prob p at each step):
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0
0.02
0.04
0.06
0.08pr
obab
ility
dis
trib
utio
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
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February 24, 2006 Complementarity in Quantum Walks
Is it really a quantum walk?
Add decoherence (measure with prob p at each step):
-100 -80 -60 -40 -20 0 20 40 60 80 100position (x)
0
0.02
0.04
0.06
0.08pr
obab
ility
dis
trib
utio
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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February 24, 2006 Complementarity in Quantum Walks
Cycles and Grids
Can also “quantum walk” on
a cycle:
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February 24, 2006 Complementarity in Quantum Walks
Cycles and Grids
Can also “quantum walk” on
a cycle:
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...or lattices:
(110)
(100)
(011)
(101)
(010)
(111)
(000) (001)
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February 24, 2006 Complementarity in Quantum Walks
Cycles and Grids
Can also “quantum walk” on
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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February 24, 2006 Complementarity in Quantum Walks
Quantum Walk on a General Graph
...or on a very general graph:
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coin
23
Don’t know degree of graph: quant-ph/0306140
Know degree of graph: quant-ph/0404043
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February 24, 2006 Complementarity in Quantum Walks
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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February 24, 2006 Complementarity in Quantum Walks
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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February 24, 2006 Complementarity in Quantum Walks
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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February 24, 2006 Complementarity in Quantum Walks
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)]
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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February 24, 2006 Complementarity in Quantum Walks
Algorithms with quantum walks
“Glued trees” problem:
Find your way from
“Entrance” to “Exit”
Childs, Cleve,
Deotto, Farhi,
Gutmann, Spielman,
quant-ph/0209131
(STOC 2003)
Exit
columns
1 2 3 4 5 6 7 8 90
Entrance
Proof in principle that quantum walk algorithms can give exponential speed up
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February 24, 2006 Complementarity in Quantum Walks
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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February 24, 2006 Complementarity in Quantum Walks
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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February 24, 2006 Complementarity in Quantum Walks
“Glued Trees” measurement time
22
22
1011
0
0
11
time
column
EXIT
ENTRANCE
glue
N = 10 Exit
columns
1 2 3 4 5 6 7 8 90
Entrance
Quantum walk proceeds
almost max speed to exit,
with some reflection at glue,
then returns...
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February 24, 2006 Complementarity in Quantum Walks
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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February 24, 2006 Complementarity in Quantum Walks
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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February 24, 2006 Complementarity in Quantum Walks
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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February 24, 2006 Complementarity in Quantum Walks
Wave walks
• can do quantum walk dynamics with classical light – same interference effects[Knight, Roldan, Sipe, PRA 68 020301 2003]
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February 24, 2006 Complementarity in Quantum Walks
Wave walks
• can do quantum walk dynamics with classical light – same interference effects[Knight, Roldan, Sipe, PRA 68 020301 2003]
• the experiment has been done: “Optical Galton Board”[Bouwmeester, Marzoli, Karman, Schleich, Woerdman, PRA 61 013410 1999]
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February 24, 2006 Complementarity in Quantum Walks
Wave walks
• can do quantum walk dynamics with classical light – same interference effects[Knight, Roldan, Sipe, PRA 68 020301 2003]
• the experiment has been done: “Optical Galton Board”[Bouwmeester, Marzoli, Karman, Schleich, Woerdman, PRA 61 013410 1999]
So what is QUANTUM about a quantum walk?
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February 24, 2006 Complementarity in Quantum Walks
Wave walks
• can do quantum walk dynamics with classical light – same interference effects[Knight, Roldan, Sipe, PRA 68 020301 2003]
• the experiment has been done: “Optical Galton Board”[Bouwmeester, Marzoli, Karman, Schleich, Woerdman, PRA 61 013410 1999]
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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February 24, 2006 Complementarity in Quantum Walks
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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February 24, 2006 Complementarity in Quantum Walks
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
• 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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February 24, 2006 Complementarity in Quantum Walks
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 (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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February 24, 2006 Complementarity in Quantum Walks
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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February 24, 2006 Complementarity in Quantum Walks
Quantum systems exhibit COMPLEMENTARITY“On the Notions of Causality and Complementarity”Bohr, Science 111 51 1950 [reprinted from Dialectica 2 312 1948]
Young’s double slit experiment:
source
scre
en/d
etec
tor
path detectors
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February 24, 2006 Complementarity in Quantum Walks
Quantum systems exhibit COMPLEMENTARITY“On the Notions of Causality and Complementarity”Bohr, Science 111 51 1950 [reprinted from Dialectica 2 312 1948]
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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February 24, 2006 Complementarity in Quantum Walks
A quantum walk has many paths...
Quantum walk is just a more complicated set of paths between input and output:
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February 24, 2006 Complementarity in Quantum Walks
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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February 24, 2006 Complementarity in Quantum Walks
Weak measurement of path
coupling strength β between ancillae and walker
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February 24, 2006 Complementarity in Quantum Walks
Weak measurement of path
coupling strength β between ancillae and walker
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interpolate between quantum (β = 0) and classical (β = 1)
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February 24, 2006 Complementarity in Quantum Walks
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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February 24, 2006 Complementarity in Quantum Walks
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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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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February 24, 2006 Complementarity in Quantum Walks
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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February 24, 2006 Complementarity in Quantum Walks
Unary vs Binary Coding [Jozsa 1998]
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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February 24, 2006 Complementarity in Quantum Walks
Unary vs Binary Coding [Jozsa 1998]
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
−→ exponentially better accuracy
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February 24, 2006 Complementarity in Quantum Walks
Unary vs Binary Coding [Jozsa 1998]
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
−→ 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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February 24, 2006 Complementarity in Quantum Walks
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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February 24, 2006 Complementarity in Quantum Walks
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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February 24, 2006 Complementarity in Quantum Walks
Quantum Simulation
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February 24, 2006 Complementarity in Quantum Walks
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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February 24, 2006 Complementarity in Quantum Walks
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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−9 −8 −7 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 7 8 90
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February 24, 2006 Complementarity in Quantum Walks
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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February 24, 2006 Complementarity in Quantum Walks
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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