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The Quest for Quantum Ants
QIP Seminar, July 2007
Yair Wiener
Agenda




Classical Ants
Ants Turn Quantum
Search by Quantum Robots
Search by Quantum Random Walk
Classical Ants
“Go to the ant, thou sluggard; consider her ways, and be wise”
Biological Ants


Ants (initially) wander randomly, and upon
finding food return to their colony while laying
down pheromone trails
If other ants find such a path, they are likely
to follow the trail, returning and reinforcing it
if they eventually find food
Ant Inspired Algorithms *



Ant Colony Optimization (ACO)
Edge Ant Walk (EAW)
Vertex Ant Walk (VAW)
(*) Partial list
Typical Problems





Searching a graph (static targets)
Hunters (dynamic targets)
Combinatorial optimization (e.g traveling
salesman problem)
Finding shortest path
And much more …
Ant Colony Optimization (ACO) *

a probabilistic technique for solving
computational problems which can be
reduced to finding good paths through
graphs
In ACO pheromones attract ants
(*) Introduced by Marco Dorigo in his PhD thesis (1992)
ACO framework (quick overview)




The pheromone model induce probability distribution
over solution space
Multiple solutions (ants) are sampled and, optionally,
locally optimized
Pheromone model is updated according to solutions
(“good” solutions increase local probability)
Repeat sampling solution space and update
pheromone model
Searching a Graph



Consider memoryless agent that searches a
graph G(V,E) for food
Each agent (ant) has the ability to leave
pheromone traces on vertices and sense the
smell
Pheromone traces dissipate over time
Formalism

A vertex v at time t is marked by the pair :
( t (v), t (v))
where  t (v) is the number of marks left on v, and
 t (v) is the time of the most recent mark left
Vertex Ant Walk (VAW) *
1.
2.
3.
4.
5.
v := u’s neighbor with minimal value
of ( (.),  (.))
 (u ) :  (u )  1
 (u ) : t
t : t  1
go to v
In VAW pheromones repel ants
(*) “Efficiently Searching a Graph by a Smell-Oriented Vertex Process”, A. Wagner et al, Annals of Mathematics
and Artificial Intelligence 24 (1998) pp. 211-223
Some VAW Results

Theorem 1: Denote by d the diameter of G, and by
n the number of vertices. Then after at most nd
steps the graph G is covered.
Searching graph G is O(nd)
Comparison to random walk


Random Walk: O(n2)
VAW: O(n )
n fully connected graph
Comparison to random walk




See example graph G4
Complexity of reaching
rightmost node:
Random Walk: O(2n)
VAW: O(n2n )
(*) The graph is from “An example of the difference between quantum and classical random walks”, A.Childs et al,
Quantum Information Processing, 1:35, 2002.
Summary



Ant inspired algorithms can find
approximations to NP-hard problems
VAW can search any unknown graph in
complexity O(nd)
The introduction of pheromones can
improve search performance over random
walk
Ants Turn Quantum
How Can We Turn Ants Quantum ?

Leave pheromones in quantum states (exploit
quantum communication between ants)

Put each ant in superposition (travel different
directions at the same time)
Schrödinger's Ant


We will pursue with the second direction.
Relevant work include quantum robots and
quantum random walk.
Quantum Robots
Quantum Robot
“A quantum robot is a mobile quantum system, including an
on board quantum computer and needed ancillary systems,
that interact with an environment of quantum systems”
Paul Benioff, 1998 (*)
(*) “Quantum Robots Plus Environments”, Paul Benioff, Phys. Rev. A 58, 1998
Quantum Robot

Quantum Robot model is a generalization of
previous work on quantum computers with
interactions with the environment (noise effects,
data base searching and quantum oracle
computing)
Quantum Robot Model

Quantum Robot consist of
–
–
–
–

On board quantum computer
Memory system (m)
Output system (o)
Control qubit (c)
Task Dynamics
–
Alternating computation and action phases
Quantum Robot Model
 (n)  | j  m | d  o | x  qr | i  c ,
  ( P0c  P1c )  a  c
 (n)   n  (0)
c Pxqr  Pxqr c
a Pdo  Pdo a
i  0,1
Space Search with Quantum Robot *
X0
N
Y0
N
(*) “Space Searches with a Quantum Robot”, Paul Benioff, AMS Contemporary Math Series, Vol 305, 2002
Grover algorithm (reminder)

Function f takes the value 0 on all elements except
one, w
1
| x 
N x
Q   I I w  (1  2 P )(1  2 Pw )

1
 |x
N  1 x w
 cos  sin  
 ,
Q  

sin

cos



|  

cos   1 
2
N
m iterations of Q corresponds to a rotation by mθ in
the 2 dimensional Hilbert space spanned by the
orthogonal vectors |α> and |w>.
Can we use Grover algorithm ?



Can we directly use Grover algorithms to solve the
grid search problem in O(N) ?
The problem is that for efficient implementation of
the algorithm it is required to determine, in small
number of steps if x = w
In the grid search problem we don’t have access to
the phase oracle ( I w )
Using Quantum Robot



Initial state: | X , Y  m | 0  L | dn o | 1 c | 0,0 QR
Copy m state onto L: | X , Y  m | X , Y  L | dn o | 1 c | 0,0 QR
Computation Phase:
–
–
–
–
| X , Y  L  | X  1, Y  L ,
| ? o  |  x o ,
| 1 c  | 0 c
If X > 0:
| ? o  |  y o ,
| 1 c  | 0 c
Else if Y > 0: | 0, Y  L  | 0, Y 1  L ,
Else test presence of s at the robot location and “record” it by
changing memory state phase.
Go back to the origin following the same path and change
output state to |dn> upon arrival.
Using Quantum Robot

Action phase
–
–
The quantum robot moves one lattice site according to the
output state direction
Upon arrival back to origin transfers motion to some ballast
system
We need to preserve reversibility and
unitarily of the dynamics
Using Quantum Robot

We start with a quantum robot with the initial state:
1
m 
N

N 1
| X , Y 
X ,Y 0
m
After the quantum robot returns to the origin the state
is:

1

 f    | X , Y  m  | X 0 , Y0  m 
N  X ,Y  X 0Y0


The complexity of getting  f is O( N log N )
Using Quantum Robot
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Using quantum robot for evaluation of the phase
oracle in Grover search algorithm results in overall
complexity of O( N 2 log N )
The advantage of quantum, over classical searching
is lost for 2 dimensional regions
What about d dimensional regions ?
O( N
d 
 1 
2 
log N )
Have we missed anything ?

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Our discussion ignored the entanglement problem
Entanglement occurs because the unitary dynamics
is reversible and the number of steps needed to
complete the search task is different for different
component states of m
Grover algorithm requires the removal of this
entanglement
Benioff claimed that it is improbable that Grover
algorithm will be used to speed up spatial 2D search
Is it the end of the road ?


Scott Aaronson and Andris Ambainis have shown in
2003 that Benioff’s claim is mistaken *
Searching 2-dimensional graph can be done in
3
2
O( N log N )

And searching d-dimensional graph (d > 3) can be
done in
d
2
O( N )
(*) “Quantum search of spatial regions”, S. Aaronson and A. Ambainis, In Proc. 44 th Annual IEEE Symp. On
Foundations of Computer Science (FOCS), pages 200-209, 2003
Divide-and-conquer algorithm
Divide-and-conquer algorithm
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Partition the region into N squares
Travel from start vertex to any setsquare C: O(N )
Search C classically and return to start vertex: O(N )
Applying Grover algorithm on C’s3 results: O( N )
Overall search complexity: O( N 2 )
Divide-and-conquer algorithm
2
3

Now we can partition the region into N squares
Travel from start vertex to any setsquare C: O(N )
2 3

3
Search C using previous technique: O( N 2 )  O( N )
21
1

32
3
Applying Grover algorithm on C’s
results:
O
(
N
)

O
(
N
)
4
Overall search complexity: O( N 3 )

Applying this technique recursively we get: O(N )

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Divide-and-conquer algorithm
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The problem is that, with each additional layer of
recursion, the robot needs to repeat the search
more often to upper bound the error probability
Amplitude amplification approach is used to
overcome this issue and achieve the improved
bounds
Summary




The introduction of physical constrains to quantum
computations yields interesting results
Quantum robot: dynamic quantum system with
alternating computation and action phases
Grover algorithm can indeed speed up spatial
search
3
2D grid can be searched in O( N log 2 N ) using Grover
algorithm and quantum robots
Quantum Random Walk
Discrete Quantum Random Walk


We will start with one dimensional quantum walk
Let H p be the Hilbert space spanned by the position
of the particle
H p  {| i  : i  0 ... N  1}

Let H c be the ‘coin’-space spanned by two basis
states
H c  {| , |}

States of the total system are in the space
H  Hc  H p
Discrete Quantum Random Walk

The conditional translation of the system can be
described by the following operator
S  ||  | i  1  i |  ||  | i  1  i |
i



i
The unitary transformation C is very arbitrary
An example of coin is Hadamard coin H
Measuring the coin state after each iteration of
S  (C  I ) removes the correlation between positions
and we obtain the classical random walk
Discrete Quantum Random Walk


We will not measure the coin state between
iterations
The interference causes radically different behavior
than classical random walk
U  S  (H  I )
U
1
|  | 1   |  | 1 
| start  
2
U 1
 |  | 2  |  | | 0   |  | 2  
2
U
1
|  | 3   |  | 1  | 2 | | 1   |  | 3 

2 2
Discrete Quantum Random Walk

The asymmetry (bias to the left) comes from the
Hadamard coin




1
|  |
2
1
H | 
|  |
2
H | 

A symmetric coin *
1 1 i 


W
2  i 1
(*) “Quantum walks and their algorithmic applications”, A. Ambainis, Int. J. Quantum Inf. 1, 507–518, 2003
Discrete Quantum Random Walk
U  S  (H  I )
U  S  (W  I )
Why do we need a coin state ?

Lets look on quantum random walk on a single line
| n   a | n 1   b | n   c | n 1 

David Meyer have shown (*) that the transformation
U defined by the above equation is unitary only if
|a|  1  |b|  1  |c|  1
(*) “From quantum cellular automata to quantum lattice gases ”, D. Meyer, J. Stat. Phys. 85 (1996) 551-574
The Model




Given undirected graph G  (V , E )
Each vertex v stores a variable av {0,1}
At one step an algorithm can examine the current
vertex or move to a neighboring vertex
The algorithm is a sequence of unitary
transformations on a Hilbert space H i  H v
The Model



Query transformation U i consists of two
transformations (U i0 , U i1 )
U i0  I is applied to all H i  | v  for which av  0 and
U i1  I is applied to all H i  | v  for which av  1
Z-local transformation *
U i (|    | v )  H i  H  ( v )
(*) “Quantum search of spatial regions”, S. Aaronson and A. Ambainis, In Proc. 44 th Annual IEEE Symp. On
Foundations of Computer Science (FOCS), pages 200-209, 2003
The Model cont



The algorithm starts in a fixed starting state | start 
and applies U1 , ...., U t
The result is |  final   U tU t 1 .... U1 | start 
Then we measure the final state
Search by Quantum Random Walk *

Unperturbed “coin-flip” transformation
C0  2 | s  s |  I d ,
1
| s 
d
d
| i 
i 1
C  C0  I N

Perturbed “coin-flip” transformation
C1   I d

Final “coin-flip” transformation
C  C0  ( I  | v  v |)  C1  | v  v |  C  (C0  C1 ) | v  v |
(*) “Coins Make Quantum Walks Faster”, A. Ambainis, J. Kempe and A. Rivosh, Proc. 16th ACM-SIAM SODA, p.
1099-1108 (2005)
Search by Quantum Random Walk


S is a shift controlled by the coin register
S :| i   | x   |  (i )   | ~
x
Where  is a permutation of the d basis states of the
coin space
The “marked walk” operator
U   S  C
Quantum Walk Search Algorithm
1
|  0 
dN
d
N
 | i   | x 
i 1 x 1
Quantum Walk Search Algorithm
•
•
•
•
Initialize the quantum system in the uniform
superposition |  0 
Do T times: Apply the marked walk U 
Measure the position register
Check if the measured vertex is the marked item
Grover as a Quantum Walk

Grover search algorithm can be viewed as a random
walk search algorithm on a complete graph
S :| i   | j   | j   | i 
C1  C0

C   C0  ( I  2 | v  v |)

Lets define
I v  I  2 Pv  ( I  2 | v  v |)
I s  I  2 Ps  ( I  2 | s  s |)
C  I s  Iv
Grover as a Quantum Walk

Now the random walk based algorithm is
| 0   | s   | s 
U  |  0   S  C  |  0   S  I s | s   I v | s    I v | s   I s | s 
U 2 |  0   S  I s I v | s   I v I s | s   I v I s | s   I s I v | s 

The random walk gives exactly Grovers algorithm on
both coin space and the vertex space (at the expense
of factor 2 in the number of applications)
Searching a 2D grid


The choice of the coin transformation (or permutation
 ) is crucial for the performance of the random walk
Lets define two shift operators
S ff :
|  | x, y   |  | x  1, y 
Sm :
|  | x, y   |  | x  1, y 
|  | x, y   |  | x  1, y 
|  | x, y   |  | x  1, y 
|  | x, y   |  | x, y  1 
|  | x, y   |  | x, y  1 
|  | x, y   |  | x, y  1 
|  | x, y   |  | x, y  1 
Searching a 2D grid


The quantum walk U  S ff  C  can search N x N
grid in O ( N log N ) steps
The quantum walk U  S m  C can search N x N
grid in at least ( N 2 ) steps
Summary



Quantum random walk exhibit substantially different
behavior than classical random walk
The performance of quantum random walk as search
algorithm highly depends on the coin transformation
2D grid can be searched in O( N log N ) using
quantum random walk
Conclusion
Conclusion
Classical
Grover
(Quantum
Robots)
Quantum
Random Walk
Structured
Graph
(N x N grid)
O( N )
O( N log N )
O ( N log N )
Unknown
Graph
O(nd )
?
?
2
3
2