Quantum Algorithms, Lower Bounds, and Time-Space Tradeoffs
Robert Špalek
Abstract:
Computers are physical objects and thus obey the laws of
physics. Although most computers nowadays are built of semiconductors
governed by quantum effects, their computation is purely classical -
at every instant the computer is in a classical state and the
computational steps are fully deterministic. Quantum computers, on the
other hand, are computers that exploit quantum effects to do
computation. Unlike classical computers, they can be at any moment in
a superposition of many classical states and perform several
computations is parallel.
In 1994, Peter Shor discovered a ground-breaking polynomial time
quantum algorithm for factoring integers, a task for which no
efficient classical algorithm is known. In 1996, Lov Grover discovered
another important quantum algorithm that searches an unordered
database quadratically faster than the best classical algorithm. Since
then, the field of quantum computing has significantly matured. Other
important discoveries have also been made in related fields, such as
quantum cryptography, information theory, error-correction codes, and
communication complexity.
In this thesis, we investigate the power of quantum computers. We
present a few new quantum algorithms, improve some quantum lower-bound
techniques, and prove the first known quantum time-space tradeoffs.
Part I: Algorithms
The quantum search algorithm by Grover is very versatile and can be
used as a subroutine in many classical algorithms. We apply it on the
following graph problems: finding a maximal matching and finding a
maximal flow in an integer network. In both cases we obtain a quantum
algorithm that is polynomially faster than the best known classical
algorithm. We address the question of verifying whether a product of
two n×n matrices is equal to a third one. This is easier than
computing the actual matrix product, as there is a classical algorithm
that runs in time O(n^2), whereas the best known matrix multiplication
algorithm runs in time O(n^2.376). Our quantum algorithm for matrix
verification is polynomially faster and it runs in time O(n^(5/3)).
Part II: Lower Bounds
A lower bound is a proof that some task cannot be computed faster than
some value. There are two basic techniques for proving quantum lower
bounds. The adversary method is based on the complexity of
distinguishing inputs that lead to different outcomes. It is easy to
use and often gives tight bounds, for example for the unordered search
problem. The polynomial method expresses the acceptance probability of
a quantum algorithm as a low-degree polynomial, and then bounds on the
degree of approximate polynomials imply lower bounds on quantum
computation. It is generally harder to apply, but gives stronger
bounds for some problems.
Many variants of the adversary method were known. We clean up the
forest and prove that there is just one method and that all known
variants are just different formulations. We prove a limitation on the
adversary bound in terms of the certificate complexity of the
function, which implies that most of the best known lower bounds that
are not known to be tight cannot be further improved by this
method. We give tight formulas for adversary bounds of composite
functions.
We prove tight direct product theorems for several functions. A DPT
states that the complexity of computing k independent instances of a
problem is k times bigger than the complexity of computing one
instance, even if we are willing to only succeed with exponentially
small probability. A statement of this type sounds plausible, however
it is very hard to prove in general.
Besides answering a fundamental question about computation, DPT\u2019s
have applications to time-space tradeoffs. A time-space tradeoff is a
relation between the running time and space complexity of an
algorithm. We use the polynomial method to obtain a DPT for the OR
function, and apply it to get tight quantum time-space tradeoffs for
sorting (in particular T^2S = \Theta(n3), whereas TS = \Theta(n^2)
classically) and Boolean matrix multiplication, and also several
communicationspace tradeoffs. These are the first such tradeoffs known
for quantum computation. We develop a new generalization of the
adversary method in terms of analysis of subspaces of density
matrices, and use it to get a DPT for all symmetric functions
(functions that only depend on the number of ones in the input). This
DPT implies a time-space tradeoff for the evaluation of solutions of
systems of linear inequalities, which is tight and polynomially better
than classical when the space is small.
Keywords:
Research for this dissertation was done at the Centrum voor Wiskunde en
Informatica (CWI) in Amsterdam.