7 research outputs found
A Quantum Time-Space Lower Bound for the Counting Hierarchy
We obtain the first nontrivial time-space lower bound for quantum algorithms
solving problems related to satisfiability. Our bound applies to MajSAT and
MajMajSAT, which are complete problems for the first and second levels of the
counting hierarchy, respectively. We prove that for every real d and every
positive real epsilon there exists a real c>1 such that either: MajMajSAT does
not have a quantum algorithm with bounded two-sided error that runs in time
n^c, or MajSAT does not have a quantum algorithm with bounded two-sided error
that runs in time n^d and space n^{1-\epsilon}. In particular, MajMajSAT cannot
be solved by a quantum algorithm with bounded two-sided error running in time
n^{1+o(1)} and space n^{1-\epsilon} for any epsilon>0. The key technical
novelty is a time- and space-efficient simulation of quantum computations with
intermediate measurements by probabilistic machines with unbounded error. We
also develop a model that is particularly suitable for the study of general
quantum computations with simultaneous time and space bounds. However, our
arguments hold for any reasonable uniform model of quantum computation.Comment: 25 page
Alternation-Trading Proofs, Linear Programming, and Lower Bounds
A fertile area of recent research has demonstrated concrete polynomial time
lower bounds for solving natural hard problems on restricted computational
models. Among these problems are Satisfiability, Vertex Cover, Hamilton Path,
Mod6-SAT, Majority-of-Majority-SAT, and Tautologies, to name a few. The proofs
of these lower bounds follow a certain proof-by-contradiction strategy that we
call alternation-trading. An important open problem is to determine how
powerful such proofs can possibly be.
We propose a methodology for studying these proofs that makes them amenable
to both formal analysis and automated theorem proving. We prove that the search
for better lower bounds can often be turned into a problem of solving a large
series of linear programming instances. Implementing a small-scale theorem
prover based on this result, we extract new human-readable time lower bounds
for several problems. This framework can also be used to prove concrete
limitations on the current techniques.Comment: To appear in STACS 2010, 12 page
Unbounded-error quantum computation with small space bounds
We prove the following facts about the language recognition power of quantum
Turing machines (QTMs) in the unbounded error setting: QTMs are strictly more
powerful than probabilistic Turing machines for any common space bound
satisfying . For "one-way" Turing machines, where the
input tape head is not allowed to move left, the above result holds for
. We also give a characterization for the class of languages
recognized with unbounded error by real-time quantum finite automata (QFAs)
with restricted measurements. It turns out that these automata are equal in
power to their probabilistic counterparts, and this fact does not change when
the QFA model is augmented to allow general measurements and mixed states.
Unlike the case with classical finite automata, when the QFA tape head is
allowed to remain stationary in some steps, more languages become recognizable.
We define and use a QTM model that generalizes the other variants introduced
earlier in the study of quantum space complexity.Comment: A preliminary version of this paper appeared in the Proceedings of
the Fourth International Computer Science Symposium in Russia, pages
356--367, 200
Proving the power of postselection
It is a widely believed, though unproven, conjecture that the capability of
postselection increases the language recognition power of both probabilistic
and quantum polynomial-time computers. It is also unknown whether
polynomial-time quantum machines with postselection are more powerful than
their probabilistic counterparts with the same resource restrictions. We
approach these problems by imposing additional constraints on the resources to
be used by the computer, and are able to prove for the first time that
postselection does augment the computational power of both classical and
quantum computers, and that quantum does outperform probabilistic in this
context, under simultaneous time and space bounds in a certain range. We also
look at postselected versions of space-bounded classes, as well as those
corresponding to error-free and one-sided error recognition, and provide
classical characterizations. It is shown that would equal
if the randomized machines had the postselection capability.Comment: 26 pages. This is a heavily improved version of arXiv:1102.066
On Counting Propositional Logic and Wagner's Hierarchy
We introduce an extension of classical propositional logic with counting quantifiers. These forms of quantification make it possible to express that a formula is true in a certain portion of the set of all its interpretations. Beyond providing a sound and complete proof system for this logic, we show that validity problems for counting propositional logic can be used to capture counting complexity classes. More precisely, we show that the complexity of the decision problems for validity of prenex formulas of this logic perfectly match the appropriate levels of Wagner's counting hierarchy