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Quantum Computational
Geometry
Marco Lanzagorta
Jeffrey K. Uhlmann
Center for Computational Science
US Naval Research Laboratory
Department for Computer Science
University of Missouri-Columbia
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Introduction
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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.
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Computational
Geometry
• Computational Geometry is concerned with
the computational complexity of geometric
problems that arise in a variety of
disciplines.
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Computer Graphics
Computer Vision
Virtual Reality
Multi-Object Simulation and Visualization
Multi-Target Tracking.
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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.
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Relevance to Naval
Systems
• These computational geometry problems arise in a
wide variety of defense systems of interest to the
US Navy.
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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
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Grover’s Algorithm
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Grover’s 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 O(N1/2)
• Instead of the O(N) required by brute force methods in
classical computing.
• It can be shown that Grover’s algorithm is optimal: no
other quantum algorithm can solve the search problem
in less than O(N1/2).
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Classical Data
Structures
• Any comparative analysis between CC and QC should
acknowledge 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.
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Exact-Match
Retrieval Queries
• Exact-match retrieval queries: Is a specific
element present in the database?
• 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.
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What Grover’s Algorithm
Isn’t Good For
• If there is no way to sort and/or structure the
dataset, then Grover’s algorithm for exact-match
retrieval is unbeatable.
• However, all the known scientific, industrial,
military and financial datasets of practical interest
are alphanumerical strings that somehow can be
sorted, structured and ordered.
• Grover’s algorithm is most appropriate for some
multidimensional spatial search problems found in
the realm of computational geometry.
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Quantum Multidimensional
Range Searches
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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.
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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.
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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.
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Quantum Range
Queries
• It can be shown that Grover’s 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.
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Spatial Search
Comparisons
Search Type
Classical
General
Preprocessing
Time
Space
Resources
O(N1-1/d )
O(N)
O(logd-1 N)
O(N logd-1 N)
O(N1/2)
O(log N)
O(N)
Classical
O(N log N)
Linear Space
Classical Non- O(N logd-1 N)
Linear Space
Quantum
General
Query
Time
O(N)
O(N)
d = space dimensions
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Practical
Considerations
• 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.
• The quantum O(N1/2) complexity may prove
to be problematic for large N if the QC runtime coefficients are not extremely small.
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Quantum Determination of
the Convex Hull
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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.
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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 Grover’s
algorithm.
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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 y=0 and
every point in the dataset. The line with the
smallest angle goes through the next point in
the convex hull.
• Overall complexity is O(N h).
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Quantum JarvisMarch 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 Grover’s algorithm.
• Overall complexity of O(N1/2 h).
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Comparison of Convex
Hull Algorithms
Method
Computation
Time
Space Resources
Classical
O(N log(h))
O(N)
Quantum
O(N1/2 h)
O(log N)
N = Total number of points.
h = Number of points comprising the hull
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Quantum Determination of
Object-Object Intersections
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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.
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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 Grover’s algorithm to determine which
objects intersect and retrieve them.
• This algorithm has O(N/m1/2 + m log m) time
complexity.
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Comparison of Intersection
Detection Algorithms
Method
Computation Time
Space Resources
Classical for General
Objects
O(N2)
O(N)
Quantum for General
Objects
Classical for CoordAlign Boxes
O(N/m1/2 + m log m)
O(log N)
O(N logd-2 (N) + m)
O(N)
Quantum for CoordAlign Boxes
O(N/m1/2 + m log m)
O(log N)
N = Total number of
objects
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m = Total number of
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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.
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Conclusions
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On the positive side…
• We have discussed efficient applications of
Grover’s 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.
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On the negative side…
• The success of these algorithms presumes:
– A smooth integration between classical and quantum
computational systems.
– The realization of an efficient (approximate) quantum
register copying circuit.
– Quantum software able to compile general purpose
Grover’s “black box” functions and oracles.
– The engineering and manufacturing of stable (quantum
noise resistant) quantum registers with logarithmic
space complexity.
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