Quantum computing is a popular topic in computer science. In recent years, many new
studies came out in different areas such as machine learning, network and cryptography.
However, the topics in the database area seem long neglected. There is an open problem
in the database area: Can we make an improvement on existing data structures or data
mining algorithms by quantum techniques?

Assume that there is a dataset of cars, where each car has its price and horsepower.
Consider two different applications. One application is that a user gives acceptable ranges
of the price and horsepower, and then we could find all cars within the ranges. Another
application is that a user gives his or her preference (e.g., a cheap car with a high horsepower),
and then we could find a set of cars satisfying...[

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Quantum computing is a popular topic in computer science. In recent years, many new
studies came out in different areas such as machine learning, network and cryptography.
However, the topics in the database area seem long neglected. There is an open problem
in the database area: Can we make an improvement on existing data structures or data
mining algorithms by quantum techniques?

Assume that there is a dataset of cars, where each car has its price and horsepower.
Consider two different applications. One application is that a user gives acceptable ranges
of the price and horsepower, and then we could find all cars within the ranges. Another
application is that a user gives his or her preference (e.g., a cheap car with a high horsepower),
and then we could find a set of cars satisfying this preference. We study the
problem called the quantum range query problem for the first application and the problem
called the quantum preference query problem for the second application. In this thesis, we
introduce these two quantum problems in the database area, and discuss how to solve
them with quantum algorithms to obtain better efficiency than classical algorithms.

The first quantum problem is the quantum range query problem. Consider a dataset of key-record pairs. Given an interval as a query range, a B+ tree can report all the records
with keys within this interval, which is called a range query. A classical B+ tree answers
a range query in O(log

_{B} N + k) time, where B is the branching factor of the B+ tree, N is the total number of records, and the output size k is the number of records in the interval.
It is asymptotically optimal in a classical computer but not efficient enough in a
quantum computer, because it is expected that the execution time is linear to the output
size. Motivated by the different scenarios in real-world applications, we study the quantum
range query problem in three different cases. The first case is the static quantum range
query, where the dataset is immutable. The second case is the dynamic quantum range query,
where insertions and deletions are supported. The third case is the high-dimensional static
quantum range query, where the keys are high-dimensional points. Firstly, we propose
the static quantum B+ tree that answers a static quantum range query in O(log

_{B} N) time,
which is asymptotically optimal in quantum computers. Since the execution time does
not depend on output size (i.e., k) and k = O(N), it is exponentially faster than the classical
data structure. To achieve this significant improvement on range queries, we design a
hybrid quantum-classical algorithm to do the range search on the static quantum B+ tree.
Secondly, we extend it to a dynamic quantum B+ tree. The dynamic quantum B+ tree
performs an insertion or a deletion in O(log

_{B} N) time and answers a dynamic quantum
range query in O(log

^{2}_{B} N) time. Thirdly, based on the static quantum B+ tree, we propose
the static quantum range tree which answers a d-dimensional static quantum range query
in O(log

^{d}_{B} N) time, which cannot be achieved by any classical data structure.

The second problem is the quantum preference query problem. The preference query
problem, which is to find the most preferred tuples from a dataset, is widely discussed in
the database area. In this problem, a utility function is given by the user to evaluate to
what extent the user prefers the tuple. However, considering a dataset consisting of N tuples,
the existing algorithms either need O(N) time to answer a query or need O(N) time
for a cold start to answer a query. The reason is that in a classical computer, a linear time
is needed to evaluate the utilities by the utility function for N tuples. In this thesis, we discuss
the quantum preference query (QPQ) problem. In this problem, the dataset is given
in a quantum memory, and we use a quantum computer to return the answers. Taking
the advantage of quantum parallelism, the quantum algorithm can theoretically perform
better than their classical competitors. To better cover all the possible study directions, we discuss this problem in different kinds of input and output. In the QPQ problem, the input can be either a number k or a threshold θ. Given k, the problem is to return k tuples
with the highest utilities. Given θ, the problem is to return all the tuples with utilities
at least θ. Also, in the QPQ problem, the output can be classical (i.e., a list of tuples) or
in a quantum state (i.e., a superposition in quantum bits). Based on amplitude amplification
and post-selection, we propose four quantum algorithms to solve the problems in
the above four scenarios. We give an accurate analysis of the number of memory accesses
needed for each quantum algorithm, which shows that the proposed quantum algorithms
are at least quadratically faster than their classical competitors.

For the quantum range query problem, in the experiment, we did simulations to show that to answer a range query, the three quantum data structures are up to 1000✕ faster than their classical competitors. To the best of our knowledge, the quantum B+ tree is
the first tree-like quantum data structure that achieves a better complexity than classical
data structures. For the QPQ problem, the experimental results also show that the QPQ
quantum algorithms are up to 1000✕ faster than their classical competitors, which proved
that QPQ problem could be a future direction of the study of preference query problems.

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