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 qubitsinto 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 Algorithms1.
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 RayCasting (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
ComplexityAnalysis (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.