5,044 research outputs found
Classical simulation of commuting quantum computations implies collapse of the polynomial hierarchy
We consider quantum computations comprising only commuting gates, known as
IQP computations, and provide compelling evidence that the task of sampling
their output probability distributions is unlikely to be achievable by any
efficient classical means. More specifically we introduce the class post-IQP of
languages decided with bounded error by uniform families of IQP circuits with
post-selection, and prove first that post-IQP equals the classical class PP.
Using this result we show that if the output distributions of uniform IQP
circuit families could be classically efficiently sampled, even up to 41%
multiplicative error in the probabilities, then the infinite tower of classical
complexity classes known as the polynomial hierarchy, would collapse to its
third level. We mention some further results on the classical simulation
properties of IQP circuit families, in particular showing that if the output
distribution results from measurements on only O(log n) lines then it may in
fact be classically efficiently sampled.Comment: 13 page
A Full Characterization of Quantum Advice
We prove the following surprising result: given any quantum state rho on n
qubits, there exists a local Hamiltonian H on poly(n) qubits (e.g., a sum of
two-qubit interactions), such that any ground state of H can be used to
simulate rho on all quantum circuits of fixed polynomial size. In terms of
complexity classes, this implies that BQP/qpoly is contained in QMA/poly, which
supersedes the previous result of Aaronson that BQP/qpoly is contained in
PP/poly. Indeed, we can exactly characterize quantum advice, as equivalent in
power to untrusted quantum advice combined with trusted classical advice.
Proving our main result requires combining a large number of previous tools --
including a result of Alon et al. on learning of real-valued concept classes, a
result of Aaronson on the learnability of quantum states, and a result of
Aharonov and Regev on "QMA+ super-verifiers" -- and also creating some new
ones. The main new tool is a so-called majority-certificates lemma, which is
closely related to boosting in machine learning, and which seems likely to find
independent applications. In its simplest version, this lemma says the
following. Given any set S of Boolean functions on n variables, any function f
in S can be expressed as the pointwise majority of m=O(n) functions f1,...,fm
in S, such that each fi is the unique function in S compatible with O(log|S|)
input/output constraints.Comment: We fixed two significant issues: 1. The definition of YQP machines
needed to be changed to preserve our results. The revised definition is more
natural and has the same intuitive interpretation. 2. We needed properties of
Local Hamiltonian reductions going beyond those proved in previous works
(whose results we'd misstated). We now prove the needed properties. See p. 6
for more on both point
Operator renewal theory and mixing rates for dynamical systems with infinite measure
We develop a theory of operator renewal sequences in the context of infinite
ergodic theory. For large classes of dynamical systems preserving an infinite
measure, we determine the asymptotic behaviour of iterates of the
transfer operator. This was previously an intractable problem.
Examples of systems covered by our results include (i) parabolic rational
maps of the complex plane and (ii) (not necessarily Markovian) nonuniformly
expanding interval maps with indifferent fixed points.
In addition, we give a particularly simple proof of pointwise dual ergodicity
(asymptotic behaviour of ) for the class of systems under
consideration.
In certain situations, including Pomeau-Manneville intermittency maps, we
obtain higher order expansions for and rates of mixing. Also, we obtain
error estimates in the associated Dynkin-Lamperti arcsine laws.Comment: Preprint, August 2010. Revised August 2011. After publication, a
minor error was pointed out by Kautzsch et al, arXiv:1404.5857. The updated
version includes minor corrections in Sections 10 and 11, and corresponding
modifications of certain statements in Section 1. All main results are
unaffected. In particular, Sections 2-9 are unchanged from the published
versio
Geometries for universal quantum computation with matchgates
Matchgates are a group of two-qubit gates associated with free fermions. They
are classically simulatable if restricted to act between nearest neighbors on a
one-dimensional chain, but become universal for quantum computation with
longer-range interactions. We describe various alternative geometries with
nearest-neighbor interactions that result in universal quantum computation with
matchgates only, including subtle departures from the chain. Our results pave
the way for new quantum computer architectures that rely solely on the simple
interactions associated with matchgates.Comment: 6 pages, 4 figures. Updated version includes an appendix extending
one of the result
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
On interpolations from SUSY to non-SUSY strings and their properties
The interpolation from supersymmetric to non-supersymmetric heterotic theories is studied, via the Scherk-Schwarz compactification of supersymmetric theories to . A general modular-invariant Scherk-Schwarz deformation is deduced from the properties of the theories at the endpoints, which significantly extends previously known examples. This wider class of non-supersymmetric theories opens up new possibilities for model building. The full one-loop cosmological constant of such theories is studied as a function of compactification radius for a number of cases, and the following interpolating configurations are found: two supersymmetric theories related by a -duality transformation, with intermediate maximum or minimum at the string scale; a non-supersymmetric theory interpolating to a supersymmetric theory, with the theory possibly having an AdS minimum; a ``metastable'' non-supersymmetric theory interpolating via a theory to a supersymmetric theory
On Hausdorff dimension of the set of closed orbits for a cylindrical transformation
We deal with Besicovitch's problem of existence of discrete orbits for
transitive cylindrical transformations
where is an
irrational rotation on the circle \T and \varphi:\T\to\R is continuous,
i.e.\ we try to estimate how big can be the set
D(\alpha,\varphi):=\{x\in\T:|\varphi^{(n)}(x)|\to+\infty\text{as}|n|\to+\infty\}.
We show that for almost every there exists such that the
Hausdorff dimension of is at least . We also provide a
Diophantine condition on that guarantees the existence of
such that the dimension of is positive. Finally, for some
multidimensional rotations on \T^d, , we construct smooth
so that the Hausdorff dimension of is positive.Comment: 32 pages, 1 figur
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