22,847 research outputs found
Quantum singular value transformation and beyond: exponential improvements for quantum matrix arithmetics
Quantum computing is powerful because unitary operators describing the
time-evolution of a quantum system have exponential size in terms of the number
of qubits present in the system. We develop a new "Singular value
transformation" algorithm capable of harnessing this exponential advantage,
that can apply polynomial transformations to the singular values of a block of
a unitary, generalizing the optimal Hamiltonian simulation results of Low and
Chuang. The proposed quantum circuits have a very simple structure, often give
rise to optimal algorithms and have appealing constant factors, while usually
only use a constant number of ancilla qubits. We show that singular value
transformation leads to novel algorithms. We give an efficient solution to a
certain "non-commutative" measurement problem and propose a new method for
singular value estimation. We also show how to exponentially improve the
complexity of implementing fractional queries to unitaries with a gapped
spectrum. Finally, as a quantum machine learning application we show how to
efficiently implement principal component regression. "Singular value
transformation" is conceptually simple and efficient, and leads to a unified
framework of quantum algorithms incorporating a variety of quantum speed-ups.
We illustrate this by showing how it generalizes a number of prominent quantum
algorithms, including: optimal Hamiltonian simulation, implementing the
Moore-Penrose pseudoinverse with exponential precision, fixed-point amplitude
amplification, robust oblivious amplitude amplification, fast QMA
amplification, fast quantum OR lemma, certain quantum walk results and several
quantum machine learning algorithms. In order to exploit the strengths of the
presented method it is useful to know its limitations too, therefore we also
prove a lower bound on the efficiency of singular value transformation, which
often gives optimal bounds.Comment: 67 pages, 1 figur
Generalised Regret Optimal Controller Synthesis for Constrained Systems
This paper presents a synthesis method for the generalised dynamic regret
problem, comparing the performance of a strictly causal controller to the
optimal non-causal controller under a weighted disturbance. This framework
encompasses both the dynamic regret problem, considering the difference of the
incurred costs, as well as the competitive ratio, which considers their ratio,
and which have both been proposed as inherently adaptive alternatives to
classical control methods. Furthermore, we extend the synthesis to the case of
pointwise-in-time bounds on the disturbance and show that the optimal solution
is no worse than the bounded energy optimal solution and is lower bounded by a
constant factor, which is only dependent on the disturbance weight. The
proposed optimisation-based synthesis allows considering systems subject to
state and input constraints. Finally, we provide a numerical example which
compares the synthesised controller performance to - and
-controllers.Comment: Accepted at IFAC WC 202
User-Base Station Association in HetSNets: Complexity and Efficient Algorithms
This work considers the problem of user association to small-cell base
stations (SBSs) in a heterogeneous and small-cell network (HetSNet). Two
optimization problems are investigated, which are maximizing the set of
associated users to the SBSs (the unweighted problem) and maximizing the set of
weighted associated users to the SBSs (the weighted problem), under
signal-to-interference-plus-noise ratio (SINR) constraints. Both problems are
formulated as linear integer programs. The weighted problem is known to be
NP-hard and, in this paper, the unweighted problem is proved to be NP-hard as
well. Therefore, this paper develops two heuristic polynomial-time algorithms
to solve both problems. The computational complexity of the proposed algorithms
is evaluated and is shown to be far more efficient than the complexity of the
optimal brute-force (BF) algorithm. Moreover, the paper benchmarks the
performance of the proposed algorithms against the BF algorithm, the
branch-and-bound (B\&B) algorithm and standard algorithms, through numerical
simulations. The results demonstrate the close-to-optimal performance of the
proposed algorithms. They also show that the weighted problem can be solved to
provide solutions that are fair between users or to balance the load among
SBSs
Almost Optimal Stochastic Weighted Matching With Few Queries
We consider the {\em stochastic matching} problem. An edge-weighted general
(i.e., not necessarily bipartite) graph is given in the input, where
each edge in is {\em realized} independently with probability ; the
realization is initially unknown, however, we are able to {\em query} the edges
to determine whether they are realized. The goal is to query only a small
number of edges to find a {\em realized matching} that is sufficiently close to
the maximum matching among all realized edges. This problem has received a
considerable attention during the past decade due to its numerous real-world
applications in kidney-exchange, matchmaking services, online labor markets,
and advertisements.
Our main result is an {\em adaptive} algorithm that for any arbitrarily small
, finds a -approximation in expectation, by
querying only edges per vertex. We further show that our approach leads
to a -approximate {\em non-adaptive} algorithm that also
queries only edges per vertex. Prior to our work, no nontrivial
approximation was known for weighted graphs using a constant per-vertex budget.
The state-of-the-art adaptive (resp. non-adaptive) algorithm of Maehara and
Yamaguchi [SODA 2018] achieves a -approximation (resp.
-approximation) by querying up to edges per
vertex where denotes the maximum integer edge-weight. Our result is a
substantial improvement over this bound and has an appealing message: No matter
what the structure of the input graph is, one can get arbitrarily close to the
optimum solution by querying only a constant number of edges per vertex.
To obtain our results, we introduce novel properties of a generalization of
{\em augmenting paths} to weighted matchings that may be of independent
interest
Approximating shortest paths in large networks
In the classroom students are introduced to shortest route calculation using small
datasets (those that can be hand-drawn.) For demonstrating the application of an
algorithm a small dataset is typically sufficient. However, real-world applications
of shortest path calculations seem to be useful only when applied to large datasets.
This paper presents research on a computer based implementation of a modified
Dijkstra algorithm as applied to large datasets including tens of thousands of arcs.
In an attempt to improve the performance of calculating paths two heuristics are
also examined. The intuition behind the heuristics is to remove the arcs that will
likely not be traversed by the optimal path from the set of arcs that can possibly
be traversed by the optimal path. By reducing this number less labeling is required,
resulting in fewer CPU cycles being used to generate a route. This paper compares
the results of the optimal against those of the two heuristics
Near-Optimal UGC-hardness of Approximating Max k-CSP_R
In this paper, we prove an almost-optimal hardness for Max -CSP based
on Khot's Unique Games Conjecture (UGC). In Max -CSP, we are given a set
of predicates each of which depends on exactly variables. Each variable can
take any value from . The goal is to find an assignment to
variables that maximizes the number of satisfied predicates.
Assuming the Unique Games Conjecture, we show that it is NP-hard to
approximate Max -CSP to within factor for any . To the best of our knowledge, this result
improves on all the known hardness of approximation results when . In this case, the previous best hardness result was
NP-hardness of approximating within a factor by Chan. When , our result matches the best known UGC-hardness result of Khot, Kindler,
Mossel and O'Donnell.
In addition, by extending an algorithm for Max 2-CSP by Kindler, Kolla
and Trevisan, we provide an -approximation algorithm
for Max -CSP. This algorithm implies that our inapproximability result
is tight up to a factor of . In comparison,
when is a constant, the previously known gap was , which is
significantly larger than our gap of .
Finally, we show that we can replace the Unique Games Conjecture assumption
with Khot's -to-1 Conjecture and still get asymptotically the same hardness
of approximation
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