7,389 research outputs found

### Average case quantum lower bounds for computing the boolean mean

We study the average case approximation of the Boolean mean by quantum
algorithms. We prove general query lower bounds for classes of probability
measures on the set of inputs. We pay special attention to two probabilities,
where we show specific query and error lower bounds and the algorithms that
achieve them. We also study the worst expected error and the average expected
error of quantum algorithms and show the respective query lower bounds. Our
results extend the optimality of the algorithm of Brassard et al.Comment: 18 page

### Classical and Quantum Complexity of the Sturm-Liouville Eigenvalue Problem

We study the approximation of the smallest eigenvalue of a Sturm-Liouville
problem in the classical and quantum settings. We consider a univariate
Sturm-Liouville eigenvalue problem with a nonnegative function $q$ from the
class $C^2([0,1])$ and study the minimal number n(\e) of function evaluations
or queries that are necessary to compute an \e-approximation of the smallest
eigenvalue. We prove that n(\e)=\Theta(\e^{-1/2}) in the (deterministic)
worst case setting, and n(\e)=\Theta(\e^{-2/5}) in the randomized setting.
The quantum setting offers a polynomial speedup with {\it bit} queries and an
exponential speedup with {\it power} queries. Bit queries are similar to the
oracle calls used in Grover's algorithm appropriately extended to real valued
functions. Power queries are used for a number of problems including phase
estimation. They are obtained by considering the propagator of the discretized
system at a number of different time moments. They allow us to use powers of
the unitary matrix $\exp(\tfrac12 {\rm i}M)$, where $M$ is an $n\times n$
matrix obtained from the standard discretization of the Sturm-Liouville
differential operator. The quantum implementation of power queries by a number
of elementary quantum gates that is polylog in $n$ is an open issue.Comment: 33 page

### Qubit Complexity of Continuous Problems

The number of qubits used by a quantum algorithm will be a crucial
computational resource for the foreseeable future. We show how to obtain the
classical query complexity for continuous problems. We then establish a simple
formula for a lower bound on the qubit complexity in terms of the classical
query complexityComment: 6 pages, 2 figure

### Average Case Tractability of Non-homogeneous Tensor Product Problems

We study d-variate approximation problems in the average case setting with
respect to a zero-mean Gaussian measure. Our interest is focused on measures
having a structure of non-homogeneous linear tensor product, where covariance
kernel is a product of univariate kernels. We consider the normalized average
error of algorithms that use finitely many evaluations of arbitrary linear
functionals. The information complexity is defined as the minimal number n(h,d)
of such evaluations for error in the d-variate case to be at most h. The growth
of n(h,d) as a function of h^{-1} and d depends on the eigenvalues of the
covariance operator and determines whether a problem is tractable or not. Four
types of tractability are studied and for each of them we find the necessary
and sufficient conditions in terms of the eigenvalues of univariate kernels. We
illustrate our results by considering approximation problems related to the
product of Korobov kernels characterized by a weights g_k and smoothnesses r_k.
We assume that weights are non-increasing and smoothness parameters are
non-decreasing. Furthermore they may be related, for instance g_k=g(r_k) for
some non-increasing function g. In particular, we show that approximation
problem is strongly polynomially tractable, i.e., n(h,d)\le C h^{-p} for all d
and 0<h<1, where C and p are independent of h and d, iff liminf |ln g_k|/ln k
>1. For other types of tractability we also show necessary and sufficient
conditions in terms of the sequences g_k and r_k

### On the dc Magnetization, Spontaneous Vortex State and Specific Heat in the superconducting state of the weakly ferromagnetic superconductor RuSr$_{2}$GdCu$_{2}$O$_{8}$

Magnetic-field changes $<$ 0.2 Oe over the scan length in magnetometers that
necessitate sample movement are enough to create artifacts in the dc
magnetization measurements of the weakly ferromagnetic superconductor
RuSr$_{2}$GdCu$_{2}$O$_{8}$ (Ru1212) below the superconducting transition
temperature $T_{c} \approx$ 30 K. The observed features depend on the specific
magnetic-field profile in the sample chamber and this explains the variety of
reported behaviors for this compound below $T_{c}$. An experimental procedure
that combines improvement of the magnetic-field homogeneity with very small
scan lengths and leads to artifact-free measurements similar to those on a
stationary sample has been developed. This procedure was used to measure the
mass magnetization of Ru1212 as a function of the applied magnetic field H (-20
Oe $\le$ H $\le$ 20 Oe) at $T < T_{c}$ and discuss, in conjunction with
resistance and ac susceptibility measurements, the possibility of a spontaneous
vortex state (SVS) for this compound. Although the existence of a SVS can not
be excluded, an alternative interpretation of the results based on the granular
nature of the investigated sample is also possible. Specific-heat measurements
of Sr$_{2}$GdRuO$_{6}$ (Sr2116), the precursor for the preparation of Ru1212
and thus a possible impurity phase, show that it is unlikely that Sr2116 is
responsible for the specific-heat features observed for Ru1212 at $T_{c}$.Comment: 17 pages, 6 figure

### A rolling-horizon quadratic-programming approach to the signal control problem in large-scale congested urban road networks

The paper investigates the efficiency of a recently developed signal control methodology, which offers a computationally feasible technique for real-time network-wide signal control in large-scale urban traffic networks and is applicable also under congested traffic conditions. In this methodology, the traffic flow process is modeled by use of the store-and-forward modeling paradigm, and the problem of network-wide signal control (including all constraints) is formulated as a quadratic-programming problem that aims at minimizing and balancing the link queues so as to minimize the risk of queue spillback. For the application of the proposed methodology in real time, the corresponding optimization algorithm is embedded in a rolling-horizon (model-predictive) control scheme. The control strategyâs efficiency and real-time feasibility is demonstrated and compared with the Linear-Quadratic approach taken by the signal control strategy TUC (Traffic-responsive Urban Control) as well as with optimized fixed-control settings via their simulation-based application to the road network of the city centre of Chania, Greece, under a number of different demand scenarios. The comparative evaluation is based on various criteria and tools including the recently proposed fundamental diagram for urban network traffic

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