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Quantum Computational Geometry

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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.
  • To learn more GMU QIT Spring 2005 Course!
  • 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
  • This leads to
  • 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
    of the Hadamard operator.

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.
  • Navigation and Guidance 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
    in the form of query-answer
  • 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
Advantages of the Quantum Solution
  • 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.
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