2,183 research outputs found
Lower Bounds on Quantum Query Complexity
Shor's and Grover's famous quantum algorithms for factoring and searching
show that quantum computers can solve certain computational problems
significantly faster than any classical computer. We discuss here what quantum
computers_cannot_ do, and specifically how to prove limits on their
computational power. We cover the main known techniques for proving lower
bounds, and exemplify and compare the methods.Comment: survey, 23 page
The geometry of quantum learning
Concept learning provides a natural framework in which to place the problems
solved by the quantum algorithms of Bernstein-Vazirani and Grover. By combining
the tools used in these algorithms--quantum fast transforms and amplitude
amplification--with a novel (in this context) tool--a solution method for
geometrical optimization problems--we derive a general technique for quantum
concept learning. We name this technique "Amplified Impatient Learning" and
apply it to construct quantum algorithms solving two new problems: BATTLESHIP
and MAJORITY, more efficiently than is possible classically.Comment: 20 pages, plain TeX with amssym.tex, related work at
http://www.math.uga.edu/~hunziker/ and http://math.ucsd.edu/~dmeyer
Quantum algorithms for testing properties of distributions
Suppose one has access to oracles generating samples from two unknown
probability distributions P and Q on some N-element set. How many samples does
one need to test whether the two distributions are close or far from each other
in the L_1-norm ? This and related questions have been extensively studied
during the last years in the field of property testing. In the present paper we
study quantum algorithms for testing properties of distributions. It is shown
that the L_1-distance between P and Q can be estimated with a constant
precision using approximately N^{1/2} queries in the quantum settings, whereas
classical computers need \Omega(N) queries. We also describe quantum algorithms
for testing Uniformity and Orthogonality with query complexity O(N^{1/3}). The
classical query complexity of these problems is known to be \Omega(N^{1/2}).Comment: 20 page
A subexponential-time quantum algorithm for the dihedral hidden subgroup problem
We present a quantum algorithm for the dihedral hidden subgroup problem with
time and query complexity . In this problem an oracle
computes a function on the dihedral group which is invariant under a
hidden reflection in . By contrast the classical query complexity of DHSP
is . The algorithm also applies to the hidden shift problem for an
arbitrary finitely generated abelian group.
The algorithm begins with the quantum character transform on the group, just
as for other hidden subgroup problems. Then it tensors irreducible
representations of and extracts summands to obtain target
representations. Finally, state tomography on the target representations
reveals the hidden subgroup.Comment: 11 pages. Revised in response to referee reports. Early sections are
more accessible; expanded section on other hidden subgroup problem
Symmetry-assisted adversaries for quantum state generation
We introduce a new quantum adversary method to prove lower bounds on the
query complexity of the quantum state generation problem. This problem
encompasses both, the computation of partial or total functions and the
preparation of target quantum states. There has been hope for quite some time
that quantum state generation might be a route to tackle the {\sc Graph
Isomorphism} problem. We show that for the related problem of {\sc Index
Erasure} our method leads to a lower bound of which matches
an upper bound obtained via reduction to quantum search on elements. This
closes an open problem first raised by Shi [FOCS'02].
Our approach is based on two ideas: (i) on the one hand we generalize the
known additive and multiplicative adversary methods to the case of quantum
state generation, (ii) on the other hand we show how the symmetries of the
underlying problem can be leveraged for the design of optimal adversary
matrices and dramatically simplify the computation of adversary bounds. Taken
together, these two ideas give the new result for {\sc Index Erasure} by using
the representation theory of the symmetric group. Also, the method can lead to
lower bounds even for small success probability, contrary to the standard
adversary method. Furthermore, we answer an open question due to \v{S}palek
[CCC'08] by showing that the multiplicative version of the adversary method is
stronger than the additive one for any problem. Finally, we prove that the
multiplicative bound satisfies a strong direct product theorem, extending a
result by \v{S}palek to quantum state generation problems.Comment: 35 pages, 5 figure
- …