117,865 research outputs found
On the Hidden Subgroup Problem as a Pivot in Quantum Complexity Theory
Quantum computing has opened the way to new algorithms that can efficiently solve problems that have always been deemed intractable. However, since quantum algorithms are hard to design, the necessity to find a generalization of these problems arises. Such necessity is satisfied by the hidden subgroup problem (HSP), an abstract problem of group theory which successfully generalizes a large number of intractable problems. The HSP plays a significant role in quantum complexity theory, as efficient algorithms that solve it can be employed to efficiently solve other valuable problems, such as integer factorization, discrete logarithms, graph isomorphism and many others.
In this thesis we give a computational definition of the HSP. We then prove the reducibility of some of the aforementioned problems to the HSP. Lastly, we introduce some essential notions of quantum computing and we present two quantum algorithms that efficiently solve the HSP on Abelian groups
Robust Control and Hot Spots in Dynamic Spatially Interconnected Systems
This paper develops linear quadratic robust control theory for a class of spatially invariant distributed control systems that appear in areas of economics such as New Economic Geography, management of ecological systems, optimal harvesting of spatially mobile species, and the like. Since this class of problems has an infinite dimensional state and control space it would appear analytically intractable. We show that by Fourier transforming the problem, the solution decomposes into a countable number of finite state space robust control problems each of which can be solved by standard methods. We use this convenient property to characterize “hot spots” which are points in the transformed space that correspond to “breakdown” points in conventional finite dimensional robust control, or where instabilities appear or where the value function loses concavity. We apply our methods to a spatial extension of a well known optimal fishing model.Distributed Parameter Systems, Robust Control, Spatial Invariance, Hot Spot, Agglomeration
Enumerating fundamental normal surfaces: Algorithms, experiments and invariants
Computational knot theory and 3-manifold topology have seen significant
breakthroughs in recent years, despite the fact that many key algorithms have
complexity bounds that are exponential or greater. In this setting,
experimentation is essential for understanding the limits of practicality, as
well as for gauging the relative merits of competing algorithms.
In this paper we focus on normal surface theory, a key tool that appears
throughout low-dimensional topology. Stepping beyond the well-studied problem
of computing vertex normal surfaces (essentially extreme rays of a polyhedral
cone), we turn our attention to the more complex task of computing fundamental
normal surfaces (essentially an integral basis for such a cone). We develop,
implement and experimentally compare a primal and a dual algorithm, both of
which combine domain-specific techniques with classical Hilbert basis
algorithms. Our experiments indicate that we can solve extremely large problems
that were once though intractable. As a practical application of our
techniques, we fill gaps from the KnotInfo database by computing 398
previously-unknown crosscap numbers of knots.Comment: 17 pages, 5 figures; v2: Stronger experimental focus, restrict
attention to primal & dual algorithms only, larger and more detailed
experiments, more new crosscap number
Engineered 2D Ising interactions on a trapped-ion quantum simulator with hundreds of spins
The presence of long-range quantum spin correlations underlies a variety of
physical phenomena in condensed matter systems, potentially including
high-temperature superconductivity. However, many properties of exotic strongly
correlated spin systems (e.g., spin liquids) have proved difficult to study, in
part because calculations involving N-body entanglement become intractable for
as few as N~30 particles. Feynman divined that a quantum simulator - a
special-purpose "analog" processor built using quantum particles (qubits) -
would be inherently adept at such problems. In the context of quantum
magnetism, a number of experiments have demonstrated the feasibility of this
approach. However, simulations of quantum magnetism allowing controlled,
tunable interactions between spins localized on 2D and 3D lattices of more than
a few 10's of qubits have yet to be demonstrated, owing in part to the
technical challenge of realizing large-scale qubit arrays. Here we demonstrate
a variable-range Ising-type spin-spin interaction J_ij on a naturally occurring
2D triangular crystal lattice of hundreds of spin-1/2 particles (9Be+ ions
stored in a Penning trap), a computationally relevant scale more than an order
of magnitude larger than existing experiments. We show that a spin-dependent
optical dipole force can produce an antiferromagnetic interaction J_ij ~
1/d_ij^a, where a is tunable over 0<a<3; d_ij is the distance between spin
pairs. These power-laws correspond physically to infinite-range (a=0),
Coulomb-like (a=1), monopole-dipole (a=2) and dipole-dipole (a=3) couplings.
Experimentally, we demonstrate excellent agreement with theory for 0.05<a<1.4.
This demonstration coupled with the high spin-count, excellent quantum control
and low technical complexity of the Penning trap brings within reach simulation
of interesting and otherwise computationally intractable problems in quantum
magnetism.Comment: 10 pages, 10 figures; article plus Supplementary Material
Cliques, colouring and satisfiability : from structure to algorithms
We examine the implications of various structural restrictions on the computational
complexity of three central problems of theoretical computer science
(colourability, independent set and satisfiability), and their relatives. All problems
we study are generally NP-hard and they remain NP-hard under various restrictions.
Finding the greatest possible restrictions under which a problem is computationally
difficult is important for a number of reasons. Firstly, this can make it easier to
establish the NP-hardness of new problems by allowing easier transformations. Secondly,
this can help clarify the boundary between tractable and intractable instances
of the problem.
Typically an NP-hard graph problem admits an infinite sequence of narrowing
families of graphs for which the problem remains NP-hard. We obtain a number
of such results; each of these implies necessary conditions for polynomial-time
solvability of the respective problem in restricted graph classes. We also identify
a number of classes for which these conditions are sufficient and describe explicit
algorithms that solve the problem in polynomial time in those classes. For the
satisfiability problem we use the language of graph theory to discover the very first
boundary property, i.e. a property that separates tractable and intractable instances
of the problem. Whether this property is unique remains a big open problem
- …