869 research outputs found
Factorizations of Elements in Noncommutative Rings: A Survey
We survey results on factorizations of non zero-divisors into atoms
(irreducible elements) in noncommutative rings. The point of view in this
survey is motivated by the commutative theory of non-unique factorizations.
Topics covered include unique factorization up to order and similarity, 2-firs,
and modular LCM domains, as well as UFRs and UFDs in the sense of Chatters and
Jordan and generalizations thereof. We recall arithmetical invariants for the
study of non-unique factorizations, and give transfer results for arithmetical
invariants in matrix rings, rings of triangular matrices, and classical maximal
orders as well as classical hereditary orders in central simple algebras over
global fields.Comment: 50 pages, comments welcom
Fast Quantum Modular Exponentiation
We present a detailed analysis of the impact on modular exponentiation of
architectural features and possible concurrent gate execution. Various
arithmetic algorithms are evaluated for execution time, potential concurrency,
and space tradeoffs. We find that, to exponentiate an n-bit number, for storage
space 100n (twenty times the minimum 5n), we can execute modular exponentiation
two hundred to seven hundred times faster than optimized versions of the basic
algorithms, depending on architecture, for n=128. Addition on a neighbor-only
architecture is limited to O(n) time when non-neighbor architectures can reach
O(log n), demonstrating that physical characteristics of a computing device
have an important impact on both real-world running time and asymptotic
behavior. Our results will help guide experimental implementations of quantum
algorithms and devices.Comment: to appear in PRA 71(5); RevTeX, 12 pages, 12 figures; v2 revision is
substantial, with new algorithmic variants, much shorter and clearer text,
and revised equation formattin
Resource Requirements for Fault-Tolerant Quantum Simulation: The Transverse Ising Model Ground State
We estimate the resource requirements, the total number of physical qubits
and computational time, required to compute the ground state energy of a 1-D
quantum Transverse Ising Model (TIM) of N spin-1/2 particles, as a function of
the system size and the numerical precision. This estimate is based on
analyzing the impact of fault-tolerant quantum error correction in the context
of the Quantum Logic Array (QLA) architecture. Our results show that due to the
exponential scaling of the computational time with the desired precision of the
energy, significant amount of error correciton is required to implement the TIM
problem. Comparison of our results to the resource requirements for a
fault-tolerant implementation of Shor's quantum factoring algorithm reveals
that the required logical qubit reliability is similar for both the TIM problem
and the factoring problem.Comment: 19 pages, 8 figure
Implementing Shor's algorithm on Josephson Charge Qubits
We investigate the physical implementation of Shor's factorization algorithm
on a Josephson charge qubit register. While we pursue a universal method to
factor a composite integer of any size, the scheme is demonstrated for the
number 21. We consider both the physical and algorithmic requirements for an
optimal implementation when only a small number of qubits is available. These
aspects of quantum computation are usually the topics of separate research
communities; we present a unifying discussion of both of these fundamental
features bridging Shor's algorithm to its physical realization using Josephson
junction qubits. In order to meet the stringent requirements set by a short
decoherence time, we accelerate the algorithm by decomposing the quantum
circuit into tailored two- and three-qubit gates and we find their physical
realizations through numerical optimization.Comment: 12 pages, submitted to Phys. Rev.
The lesson of causal discovery algorithms for quantum correlations: Causal explanations of Bell-inequality violations require fine-tuning
An active area of research in the fields of machine learning and statistics
is the development of causal discovery algorithms, the purpose of which is to
infer the causal relations that hold among a set of variables from the
correlations that these exhibit. We apply some of these algorithms to the
correlations that arise for entangled quantum systems. We show that they cannot
distinguish correlations that satisfy Bell inequalities from correlations that
violate Bell inequalities, and consequently that they cannot do justice to the
challenges of explaining certain quantum correlations causally. Nonetheless, by
adapting the conceptual tools of causal inference, we can show that any attempt
to provide a causal explanation of nonsignalling correlations that violate a
Bell inequality must contradict a core principle of these algorithms, namely,
that an observed statistical independence between variables should not be
explained by fine-tuning of the causal parameters. In particular, we
demonstrate the need for such fine-tuning for most of the causal mechanisms
that have been proposed to underlie Bell correlations, including superluminal
causal influences, superdeterminism (that is, a denial of freedom of choice of
settings), and retrocausal influences which do not introduce causal cycles.Comment: 29 pages, 28 figs. New in v2: a section presenting in detail our
characterization of Bell's theorem as a contradiction arising from (i) the
framework of causal models, (ii) the principle of no fine-tuning, and (iii)
certain operational features of quantum theory; a section explaining why a
denial of hidden variables affords even fewer opportunities for causal
explanations of quantum correlation
Star Architecture as Socio-Material Assemblage
Taking inspiration from new materialism and assemblage, the chapter deals with star architects and iconic buildings as socio-material network effects that do not pre-exist action, but are enacted in practice, in the materiality of design crafting and city building. Star architects are here conceptualized as part of broader assemblages of actors and practices ‘making star architecture’ a reality, and the buildings they design are considered not just as unique and iconic objects, but dis-articulated as complex crafts mobilizing skills, technologies, materials, and forms of knowledge not necessarily ascribable to architecture. Overcoming narrow criticism focusing on the symbolic order of icons as unique creations and alienated repetitions of capitalist development, the chapter’s main aim is to widen the scope of critique by bridging culture and economy, symbolism and practicality, making star architecture available to a broad, fragmented arena of (potential) critics, unevenly equipped with critical tools and differentiated experiences
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