36 research outputs found
Unbounded-Error Classical and Quantum Communication Complexity
Since the seminal work of Paturi and Simon \cite[FOCS'84 & JCSS'86]{PS86},
the unbounded-error classical communication complexity of a Boolean function
has been studied based on the arrangement of points and hyperplanes. Recently,
\cite[ICALP'07]{INRY07} found that the unbounded-error {\em quantum}
communication complexity in the {\em one-way communication} model can also be
investigated using the arrangement, and showed that it is exactly (without a
difference of even one qubit) half of the classical one-way communication
complexity. In this paper, we extend the arrangement argument to the {\em
two-way} and {\em simultaneous message passing} (SMP) models. As a result, we
show similarly tight bounds of the unbounded-error two-way/one-way/SMP
quantum/classical communication complexities for {\em any} partial/total
Boolean function, implying that all of them are equivalent up to a
multiplicative constant of four. Moreover, the arrangement argument is also
used to show that the gap between {\em weakly} unbounded-error quantum and
classical communication complexities is at most a factor of three.Comment: 11 pages. To appear at Proc. ISAAC 200
Unbounded-error One-way Classical and Quantum Communication Complexity
This paper studies the gap between quantum one-way communication complexity
and its classical counterpart , under the {\em unbounded-error}
setting, i.e., it is enough that the success probability is strictly greater
than 1/2. It is proved that for {\em any} (total or partial) Boolean function
, , i.e., the former is always exactly one half
as large as the latter. The result has an application to obtaining (again an
exact) bound for the existence of -QRAC which is the -qubit random
access coding that can recover any one of original bits with success
probability . We can prove that -QRAC exists if and only if
. Previously, only the construction of QRAC using one qubit,
the existence of -RAC, and the non-existence of
-QRAC were known.Comment: 9 pages. To appear in Proc. ICALP 200
Better short-seed quantum-proof extractors
We construct a strong extractor against quantum storage that works for every
min-entropy , has logarithmic seed length, and outputs bits,
provided that the quantum adversary has at most qubits of memory, for
any \beta < \half. The construction works by first condensing the source
(with minimal entropy-loss) and then applying an extractor that works well
against quantum adversaries when the source is close to uniform.
We also obtain an improved construction of a strong quantum-proof extractor
in the high min-entropy regime. Specifically, we construct an extractor that
uses a logarithmic seed length and extracts bits from any source
over \B^n, provided that the min-entropy of the source conditioned on the
quantum adversary's state is at least , for any \beta < \half.Comment: 14 page
Exponential Separation of Quantum and Classical Online Space Complexity
Although quantum algorithms realizing an exponential time speed-up over the
best known classical algorithms exist, no quantum algorithm is known performing
computation using less space resources than classical algorithms. In this
paper, we study, for the first time explicitly, space-bounded quantum
algorithms for computational problems where the input is given not as a whole,
but bit by bit. We show that there exist such problems that a quantum computer
can solve using exponentially less work space than a classical computer. More
precisely, we introduce a very natural and simple model of a space-bounded
quantum online machine and prove an exponential separation of classical and
quantum online space complexity, in the bounded-error setting and for a total
language. The language we consider is inspired by a communication problem (the
set intersection function) that Buhrman, Cleve and Wigderson used to show an
almost quadratic separation of quantum and classical bounded-error
communication complexity. We prove that, in the framework of online space
complexity, the separation becomes exponential.Comment: 13 pages. v3: minor change
Short seed extractors against quantum storage
Some, but not all, extractors resist adversaries with limited quantum
storage. In this paper we show that Trevisan's extractor has this property,
thereby showing an extractor against quantum storage with logarithmic seed
length
Simultaneous Communication Protocols with Quantum and Classical Messages
We study the simultaneous message passing (SMP) model of communication
complexity, for the case where one party is quantum and the other is classical.
We show that in an SMP protocol that computes some function with the first
party sending q qubits and the second sending c classical bits, the quantum
message can be replaced by a randomized message of O(qc) classical bits, as
well as by a deterministic message of O(q c log q) classical bits. Our proofs
rely heavily on earlier results due to Scott Aaronson.
In particular, our results imply that quantum-classical protocols need to
send Omega(sqrt{n/log n}) bits/qubits to compute Equality on n-bit strings, and
hence are not significantly better than classical-classical protocols (and are
much worse than quantum-quantum protocols such as quantum fingerprinting). This
essentially answers a recent question of Wim van Dam. Our results also imply,
more generally, that there are no superpolynomial separations between
quantum-classical and classical-classical SMP protocols for functional
problems. This contrasts with the situation for relational problems, where
exponential gaps between quantum-classical and classical-classical SMP
protocols are known. We show that this surprising situation cannot arise in
purely classical models: there, an exponential separation for a relational
problem can be converted into an exponential separation for a functional
problem.Comment: 11 pages LaTeX. 2nd version: author added and some changes to the
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