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Title: Quantum Computational Geometry

1
Quantum Computational Geometry
• Dr. Marco Lanzagorta
• Center for Computational Science
• US Naval Research Laboratory
• Email marco.lanzagorta_at_nrl.navy.mil

2
Objective
• To investigate how a quantum computer could be
used as a fully functional computational device
to solve real problems found in a wide variety of
scientific, industrial and military software
systems.
• Explore the applications of QC beyond its use as
a dedicated cryptographic device or a quantum
physics simulator.

3
Presentation Outline
• Introduction
• Grovers Quantum Search Algorithm
• Quantum Computational Geometry
• Quantum Computer Graphics
• Conclusions

4
Introduction
• From bits to qubitsinto the realm of quantum
information

5
From Bits to Qubits
• Qubits
• A qubit is the quantum unit of information.
• Represented by a quantum state, it has a value of
0, 1, or any superposition
• qgt a0gt b1gt
• A quantum computer is in a state determined by a
quantum superposition
• Bits
• A bit is the classical unit of information.
• Represented by an electric/electronic on/off
switch, it has a value of either 0 or 1.
• A classical computer is in a state determined by
a set of bits
• S 0100111001

6
Probabilistic Computing
• If a classical computer is in the state 01,
when we check the output we will always read 01
with probability 1.
• If a quantum computer is in the state
• Pgt a00gt b01gt c10gt d11gt
• after a measurement we will observe the state
00 with probability a2, 01 with probability
b2,
• One of the goals of quantum information
processing systems is to transform the quantum
state in such a way that we will have a
meaningful result with probability close to 1.

7
Large Computational Space
• In a classical computer, we require O(N Log(N))
bits to store N records.
• Example 24 bits can store 8 records.
• A quantum system encodes exponentially more
information than a classical one. We only require
O(Log(N)) qubits to store N records.
• Example 24 qubits can store more than 16 million
records!
• Of course, after a measurement of the state, we
can only access 24 bits of logical information.
• Although most of the quantum information is
inaccessible, we can still exploit this
information processing capability.

8
Quantum Parallel Computation
• Let U be a linear operator such that
• Any function f can be represented with a quantum
array that implements the U linear operator.
• If U is applied to a superposition then
• Therefore, U is being applied simultaneously to
all the states in the superposition, and leads to
an entangled state.

9
Quantum Information
• Quantum information is radically different from
classical information.
• Therefore, the processing, manipulation,
transmission, encryption, distribution and
observation protocols for quantum information are
vastly different than those in classical
computing.
• Besides the theoretical foundations of quantum
information, special emphasis will be made on
fault-tolerant quantum computing, quantum
cryptography, quantum communications and quantum
networks.

10
Grovers Quantum Search Algorithm
• Searching a database using a quantum computer

11
Grovers Algorithm
• Quantum algorithm developed by Grover to
perform a search of an item from an unsorted,
unstructured list of N records.
• Performs the search in
• Instead of the O(N) required by brute force
methods in classical computing.
• This algorithm can be formalized into a theorem
about the general solution of an NP problem.

12
Quantum Oracles (1)
• A quantum oracle is a black box which computes
the value of a function.
• Let us consider a function f such that f(x)1 if
x y, and f(x) 0 for any other case.
• Then, an oracle marks the solution of a search
problem by shifting the phase of the solution.

13
Quantum Oracles (2)
• We define I as the operator such that
• Ixgt -xgt if xy, and Ixgtxgt in any other
case

I
14
Quantum Search Problem (1)
• Let us suppose we have an unsorted unstructured
list of N 2n records and one of these records y
satisfies a given property.
• Our objective is to use a quantum oracle to
search for y.
• We define a function such that f(x)1 if x y,
and f(x) 0 for any other case.
• We create a n-qubit state which is the uniform
superposition of N independent quantum states
• y gt a S xgt where a 1/Sqrt(N)
• Each state xigt represents (points at) an element
of the list of records that need to be searched.

15
Quantum Search Problem (2)
• When we measure the quantum state y gt, the
probability that the result will be the solution
to the search problem is equal to 1/N.
• lty y gt a S ltyxgt a 1/sqrt(N)
• P rr1/N
• Our aim is to modify y gt to concentrate the
amplitudes at xy.
• This can be done by using quantum operators.

16
Grovers Algorithm (1)
• We use the f function to build a quantum oracle
and a quantum operator I such that Ixgt -xgt if
xy, and Ixgtxgt in any other case.
• We define the operator D by
• Off diagonal elements 2/N
• Diagonal elements (-12/N)
• D performs an inversion about the mean

17
Grovers Algorithm (2)
• We consecutively apply k times the DI operator to
the state y gt
• Which has as solution
• where

18
Grovers Algorithm (3)
• To find a solution to the search problem we need
b(k) 1 and a(k) 0.
• Then, for large N
• k (p/4) Sqrt (N) (1/2)
• Therefore, by using Grovers algorithm, we can
find the solution of the searching problem in
O(Sqrt(N)) time!
• Note If we use a single-qubit Hadamard gate,
then we have O(Sqrt(N) Log(N)) instead. This
result depends on the specific quantum computers
architecture.

19
Visualization of the Effect of a Single Grover
Iteration
b
3
DI ygt
y
Q
1
Q/2
a
Q/2
I ygt
2
20
Grovers Algorithm is Optimal
• It can be shown that Grovers algorithm is
optimal for an unsorted unstructured search
space.
• This means that no other quantum algorithm can
solve the searching problem in less than O(N1/2).
• However, it may be possible to exploit the search
space structure to increase performance.
• For example, Shors algorithm reduces factoring
to a periodicity problem of O(log(N)).

21
Formalization (1)
• Theorem 1
• By using a quantum circuit making O(N1/2) queries
to a black-box function f, one can decide with
non-vanishing correctness probability if there is
an element in the set such that f(x)1.
• In other words, it states that Grovers algorithm
provides the right answer in O(N1/2).

22
Formalization (2)
• Theorem 2
• By using a quantum circuit, any problem in NP can
be solved with non vanishing correctness
probability in time O(N1/2P(log(N))), where P is
a polynomial depending on the particular problem.
• In other words, it generalizes Grovers algorithm
to any problem in NP. However, the number of
elements to search may grow exponentially
• N O(2M)
• In such case, the asymptotic complexity of the NP
problem remains exponential.

23
Some Generalization of Grovers Algorithm
• Grovers algorithm can easily be generalized for
the case where the list of N records has k
solutions to the search problem.
• Grovers algorithm can be generalized to the case
where the number of solutions is unknown.
• Grovers algorithm can be generalized to
calculate the mean and median of N elements.
• Grovers algorithm works with almost any
transformation, it is not restricted to the use

24
Unsorted Unstructured Datasets
• If there is no way to sort and/or structure the
dataset, then Grovers algorithm is unbeatable.
• Shapes, colors, images, graphs, semantic
memories
• However, most scientific, industrial, military
and financial datasets are alphanumerical strings
that can be sorted, structured and ordered.

25
What Grovers Algorithm Isn't Good For (1)
• Grover's algorithm has been hyped and
misrepresented as a tool to search a commercial
database.
• QC literature usually ignores the existence of
classical data structures.
• Speed up classical computational tasks
• Reorganize the original format of the data set in
a way that increases efficiency, abstraction and
reusability
• Caveats Require a non-constant time process to
store the data, and it may increase the
space/storage complexity of the original data
set.

26
What Grovers Algorithm Isn't Good For (2)
• For 1-dimensional data queries, if a classical
algorithm is permitted to spend O(N Log(N)) time
to structure the database
• A variety of searches can be performed in
O(Log(N)) time or better.
• A hash table can be created in O(N) and it can
find an item in a list in O(1).
• Therefore, classical data structures seem to be
superior to any quantum algorithm in terms of
asymptotic query-time complexity.

27
What Grovers Algorithm Isn't Good For (3)
• Grover's Algorithm has been hyped to offer a
solution to the "unsorted database problem". For
this alleged problem, the O(N log N)
preprocessing is not permitted.
• However, Grover's algorithm still requires O(N)
preprocessing to move records from classical disk
space to quantum memory, and newer classical
methods can sort integers with complexity
approaching O(N).

28
Summary
• Grover's algorithm offers a general mechanism for
satisfying search queries on a set of N records
in O(N1/2) time.
• Exact-match retrieval of records from a
1-dimensional datasets do not really exploit the
potential benefits of Grover's algorithm.
• Grovers algorithm is most appropriate for some
multidimensional spatial search problems found in
the realm of computational geometry.

29
Quantum Computational Geometry
• What Grovers Algorithm is good for

30
Computational Geometry
• Computational Geometry is concerned with the
computational complexity of geometric problems
that arise in a variety of disciplines.
• Computer Graphics
• Computer Vision
• Virtual Reality
• Multi-Object Simulation and Visualization
• Multi-Target Tracking.

31
Some Computational Geometry Problems
• Many of the most fundamental problems in
computational geometry involve
• Multidimensional searches
• Search for those objects in space that satisfy a
certain query criteria.
• Representation of spatial information
• Determination of the convex hull of a set of
points.
• Determination of object-object intersections.

32
Relevance to Naval Systems
• These computational geometry problems arise in
a wide variety of defense systems of interest to
the US Navy.
• Modeling Simulation of Combat Platforms
• VR Training Systems
• Command and Control Systems
• Missile Defense Systems
• Missile and Unmanned Aerial Vehicles Guidance
Systems
• Data Fusion in Network Centric Warfare Systems
• Robotics

33
Example Graphics and Simulation
• Virtually all of the computational bottlenecks
in graphics and simulation applications involve
some variant of spatial search
• Collision detection involves the identification
of intersections among spatial objects.
• Ray tracing involves the identification of the
spatial object that is intersected first along a
line of sight.

34
Theoretical Limitations of Classical Computing
• Many optimality results have been established for
the application of classical algorithms to
various classes of problems.
• Machine independent function defined by the
maximum number of steps taken on any instance of
size N.
• An optimal O(f(N)) gives the best worst case
scenario.
• When an optimality result is established, it
imposes a strict performance bound that we cannot
hope to exceed.
• Many of the optimality results for
multi-dimensional search algorithms imply very
pessimistic prospects for the development of very
large systems.

35
Quantum Computational Geometry
• Grovers algorithm may be used to speed up
some of the most important computational geometry
algorithms
• Quantum Multidimensional Range Searches
• Quantum Determination of the Complex Hull
• Quantum Determination of Object-Object
intersections

36
Quantum Computational Geometry Algorithms 1.
Quantum Multidimensional Range Searches
37
Introduction
• Multidimensional search algorithms are used to
find a number of specific elements from a
multidimensional set.
• Multidimensional queries are widely used
• Missile Defense Systems.
• Robots, unmanned aerial vehicles, cruise
missiles
• Visualization, Modeling and Simulation Systems.
• Combat platforms (submarines, tanks).
• Physical, atmospheric and oceanic phenomena.
• Scientific and Battlespace Visualization.
• Battlefield Command and Control Systems.

38
Multidimensional Range Searches
• Multidimensional search problems are usually cast
• Given a collection of points in space, one is to
find those that satisfy a certain query criteria.
• Range queries require the identification of all
points within a d-dimensional coordinate aligned
box.
• Range queries are the most general
multidimensional queries and special cases of
general region queries.
• Optimality results for range queries provide
lower bounds for more sophisticated queries.

39
Classical Range Queries Linear Space
• Given a dataset of N points, a classical data
structure of size O(N) can be used to satisfy
range queries in O(N1-1/d k) time, where k is
the number of points satisfying the query and d
is the number of dimensions. This is optimal.

40
Classical Range Queries Non-Linear Space
• In many applications it is possible within the CC
framework to optimize a tradeoff between
execution time and storage.
• In particular, range queries can be satisfied in
O(log d-1 N k) time using O(N log d-1 N)
storage.
• The storage complexity becomes problematic for
large N, e.g., if N is 1 million, the storage in
3D is multiplied by a factor of 400.

41
Quantum Range Queries
• It can be shown that Grovers quantum search
algorithm permits general spatial search queries
to be performed with O((N/k)1/2 k log k)
complexity.
• If k is essentially a constant, then quantum
range queries can be satisfied in O(N1/2) time,
where the exponent is independent of the
dimensionality.

42
Spatial Search Comparisons
d space dimensions
43
Practical Considerations (1)
• The observed performance of linear-space
classical data structures is often much better
than the theoretical worst case O(N1-1/d)
complexity, at least for small query ranges.
• The O((log N)d) query complexity of sophisticated
classical range query structures is also the
best-case complexity.
• It is conjectured that the run-time coefficients
associated with QC algorithms may be much smaller
than what is possible for classical computers.

44
Practical Considerations (2)
• By far the greatest practical advantage offered
by quantum computing is the ability to store
pointers to N data items using only log(N)
qubits.
• QC offers the opportunity to handle very large
datasets, far beyond of what could be stored in a
classical computers memory.
• The quantum O(N1/2) complexity may prove to be
problematic for large N if the QC run-time
coefficients are not extremely small.

45
Potential Applications
• Multidimensional Quantum Searches could be
important in several areas of interest
• Multi-target tracking and multi-sensor data
fusion
• Missile Defense
• Robotics
• Missile Guidance and Navigation systems
• Network Centric Warfare Systems
• Computer Graphics, modeling and simulation

46
Quantum Computational Geometry Algorithms 2.
Quantum Determination of the Convex Hull
47
The Convex Hull
• The convex hull of a set of points S is the
smallest convex set that contains S.
• The determination of the convex hull is a
computational geometry problem that emphasizes
the representation of spatial information.
• The convex hull of a set of points is used to
represent its spatial extent.

48
CC Convex Hull Algorithms
• The most efficient classical algorithm (known to
date) to compute the convex hull requires
O(N log(h)) time for N objects with h points
forming the convex hull.
• h is usually a constant.
• The Jarvis-March algorithm calculates the convex
hull in O(N h), but it is a good candidate to be
ported to a quantum computer using Grovers
algorithm.

49
Jarvis-March Algorithm
• Identify a point in the convex hull (the one with
the minimum x-coordinate), then, for each point
in the dataset
• Compute the angles between the line x0 and every
point in the dataset. The line with the smallest
clockwise angle goes through the next point in
the convex hull.
• Overall complexity is O(N h).

50
Quantum Jarvis- March Algorithm
• Each successive point can be determined after the
application of a simple calculation for each of
the points in the dataset.
• The angles for the points in each step can be
computed, and the minimum point retrieved in
O(N1/2) using Grovers algorithm.
• Overall complexity of O(N1/2 h).

51
Comparison of Convex Hull Algorithms
N Total number of points. h Number of points
comprising the hull
52
Quantum Computational Geometry Algorithms 3.
Quantum Determination of Object-Object
Intersections
53
Classical Algorithms
• Given a set of N objects, in general it is
impossible to avoid spending O(N2) time checking
whether or not each pair of objects intersect.
• For coordinated-aligned orthogonal boxes, it is
possible to determine the intersections in O(N
logd-2(N) m) time, where m is the total number
of intersections.

54
A Grover-Based Quantum Algorithm
• Assume that O(1) time is sufficient to determine
whether a pair of objects intersect.
• Construct a quantum register that enumerates all
the possible N2 pairs of objects in O(log N)
time.
• Use Grovers algorithm to determine which objects
intersect and retrieve them.
• This algorithm has O(N/m1/2 m log m) time
complexity.

55
Comparison of Intersection Detection Algorithms
N Total number of objects
m Total number of intersections
56
• The quantum algorithm is attractive because of
its generality.
• The complexity of the quantum algorithm holds for
any class of objects for which comparisons take
O(1) time.
• Annuli with arbitrary radii for sonar
applications.
• Nurbs and surface patches in computer graphics.
• No classical method exists for efficiently
identifying intersections among general objects
with curved surfaces.

57
Quantum Computer Graphics
• Using Quantum Computational Geometry in Computer
Graphics

58
Computer Graphics
• Computer Graphics are widely used in todays
world
• Digital film and visual effects
• Video games
• Scientific visualization
• Battlefield visualization
• Flight simulators
• Architectural applications
• Medical Applications
• Computer Aided Design

59
Rendering
• In few words, rendering means to draw an object
on the computer screen.
• In computer graphics, an object is decomposed
into several thousands of polygons or surfaces.
• The rendering process has to render each
individual element, one at the time.
• Rendering is a simple but extremely time
consuming process.

60
Rendering Example
61
Rendering Speed
• Rendering speed is an important issue. The
faster the rendering, the better
• Efficient generation of high quality
photo-realistic images.
• Cheaper production costs for film companies.
• Ability to interactively visualize and explore
larger scientific data-sets.
• Trainers and trainees can also benefit from more
accurate, faster and photo-realistic simulations.
• Battlefield commanders could rely on higher
quality Combat Planning, Decision Support and
Command Control systems.

62
Rendering as a search problem
• Most of the currently known rendering
algorithms have to perform several spatial
searches over the entire database of
objects/polygons in a scene.
• These searches tend to be a BIG bottleneck in the
rendering process as the number of
objects/polygons grows into the order of millions.

63
Improving Classical Rendering
• Research is currently being done to speedup the
rendering of complex scenes.
• New data structures and rendering algorithms.
• New hardware (graphic engines graphic cards).
• However, as we discussed in the previous section,
there are theoretical limits on the performance
of classical algorithms used in graphics and
simulation.
• Note these are algorithmic limitations. The
algorithms asymptotic behavior is independent of
the computer hardwares speed and architecture.

64
Quantum Rendering
• Brute Force Approach Apply quantum search
algorithms to speed up some of the most important
rendering techniques
• Z-Buffering
• Ray casting
• Level of Detail
• Image compression

65
Classical Ray Casting (1)
• Determines the visibility of surfaces by
tracing imaginary rays from the viewers eye to
the objects in the scene.

Eye
Scene
Screen
66
Classical Ray Casting (2)
• Ray Casting Algorithm
• For each pixel in the screen
• Calculate ray from pixel position and center of
projection.
• Shoot the ray and find p, the closest polygon
intersected.
• Shade pixel according to p.
• Ray casting technique requires O(N) operations
per pixel for a single ray.

67
Quantum Ray Casting (1)
• The most obvious application of quantum computing
to ray casting is for the determination of
object-ray intersections.
• In the sake of simplicity, we use bounding boxes
(bb).
• For each object (polygon or surface) in the
scene, we create a sphere (or cube) that
completely surrounds it.
• Then, the problem is reduced to calculate the
ray-bb intersections for each ray.
• Using the equations of a line and a sphere
(cube), it is easy to determine if a ray
intersects a bb.

68
Quantum Ray Casting (2)
• We create a quantum register that holds an
enumeration of the N bounding boxes
• We create a function f such that
• f(r, j i) 1 if the ray r intersects j i
• f(r, j i) 0 if the ray r does not
intersects j i

69
Quantum Ray Casting (3)
• We use a two-step quantum search algorithm, very
similar in nature to Grovers algorithm
• First we find the subset of k objects that
intersect the ray.
• Then, from these k objects we find the one that
lies closest to the screen.
• Algorithm gives the right answer with probability
close to 1 in O(N1/2 k1/2 ) computational steps.
• In most practical applications, k is close to be
a constant and k ltlt N.

70
Quantum Ray Casting (4)
• Quantum Ray Casting algorithm per ray (pixel)
• For each pixel in the screen
• Initialize the quantum register.
• Calculate ray from pixel position and center of
projection.
• Apply the function f(r,x) to the quantum
register.
• Perform the quantum search algorithm to find p,
the closest intersected polygon.
• Shade pixel according to p.

71
Complexity Analysis (1)
• Memory initialization
• Recall that each memory initialization may take
up to O(Log(N)), depending on the computers
architecture.
• This has to be done once per ray.
• Quantum search
• The quantum search algorithm for finding one
element out of N takes O(Sqrt(N)).
• This has to be done once per ray.

72
Complexity Analysis (2)
• Therefore, the quantum algorithm has the
following query time and space complexities
• Time O(NP Sqrt(N) ) (NP total number of rays)
• Space O(log(N))
• Naïve Classical Solution
• Time O(NP N)
• Space O(N)
• As N grows into the billions, the quantum
solution scales better in time and space
requirements

73
Using Data Structures in Classical Rendering
• So far, we have only used the most naïve
procedure for finding intersections and distances
in classical rendering algorithms.
• A most likely situation is the use of data
structures, such as kd-trees, to speedup the
calculations.
• In this case, the time complexity may be much
better than O(N) and even comparable to or better
than O(Sqrt(N)).

74
Quantum vs. Classical Rendering
• However, as we discussed in the previous section,
the only classical algorithms that achieve
query-time complexity superior to that of
Grovers algorithm do so at the expense of
non-linear space.
• This makes the O(Sqrt(N)) query complexity and
O(log(N)) space complexity of quantum rendering
highly attractive.
• Could be used to generate very detailed textures,
terrains, and virtual environments beyond what
could be stored in a classical computer.

75
Beyond Grovers algorithm based quantum rendering
• While Grovers O(Sqrt(N)) is proved to be an
optimal result, there may be hidden symmetries
in the problem that could allow better query
times.
• For example, Shors algorithm for factorizing
large numbers in O(log(N)) is based on the
realization that factorization can be seen as a
periodicity problem.
• A similar association may exist for rendering
algorithms.
• Quantum Data Structures may also be developed to
speed up the rendering processing.
• Further research in quantum rendering algorithms
is necessary!

76
Summary
• Even with a brute force approach, quantum
computers offer some benefits to computer
graphics.
• Superior algorithmic speed and storage
• The large computational space available on a
quantum computer could allow the creation of
photorealistic virtual environments (like the one
showcased in The Matrix), far more complex than
what could be achieved with classical computers.

77
Conclusions
• Quantum Computational Geometry, is it as good as
it sounds?

78
On the positive side
• We have discussed efficient applications of
Grovers algorithm to computational geometry
problems.
• We have described quantum algorithms which
outperform the best classical computing
algorithms currently known.
• The algorithms describe combine classical and
quantum computing techniques, and resources from
both types of hardware.

79
On the negative side
• The success of these algorithms presumes
• A smooth integration between classical and
quantum computational systems.
• The physical realization of an efficient
(approximate) quantum register copying circuit.
• Quantum software able to compile general purpose
Grovers black box functions and oracles.
• The engineering and manufacturing of stable
(quantum noise resistant) quantum registers with
logarithmic space complexity.