23,388 research outputs found
Is there a physically universal cellular automaton or Hamiltonian?
It is known that both quantum and classical cellular automata (CA) exist that
are computationally universal in the sense that they can simulate, after
appropriate initialization, any quantum or classical computation, respectively.
Here we introduce a different notion of universality: a CA is called physically
universal if every transformation on any finite region can be (approximately)
implemented by the autonomous time evolution of the system after the complement
of the region has been initialized in an appropriate way. We pose the question
of whether physically universal CAs exist. Such CAs would provide a model of
the world where the boundary between a physical system and its controller can
be consistently shifted, in analogy to the Heisenberg cut for the quantum
measurement problem. We propose to study the thermodynamic cost of computation
and control within such a model because implementing a cyclic process on a
microsystem may require a non-cyclic process for its controller, whereas
implementing a cyclic process on system and controller may require the
implementation of a non-cyclic process on a "meta"-controller, and so on.
Physically universal CAs avoid this infinite hierarchy of controllers and the
cost of implementing cycles on a subsystem can be described by mixing
properties of the CA dynamics. We define a physical prior on the CA
configurations by applying the dynamics to an initial state where half of the
CA is in the maximum entropy state and half of it is in the all-zero state
(thus reflecting the fact that life requires non-equilibrium states like the
boundary between a hold and a cold reservoir). As opposed to Solomonoff's
prior, our prior does not only account for the Kolmogorov complexity but also
for the cost of isolating the system during the state preparation if the
preparation process is not robust.Comment: 27 pages, 1 figur
Anticoncentration theorems for schemes showing a quantum speedup
One of the main milestones in quantum information science is to realise
quantum devices that exhibit an exponential computational advantage over
classical ones without being universal quantum computers, a state of affairs
dubbed quantum speedup, or sometimes "quantum computational supremacy". The
known schemes heavily rely on mathematical assumptions that are plausible but
unproven, prominently results on anticoncentration of random prescriptions. In
this work, we aim at closing the gap by proving two anticoncentration theorems
and accompanying hardness results, one for circuit-based schemes, the other for
quantum quench-type schemes for quantum simulations. Compared to the few other
known such results, these results give rise to a number of comparably simple,
physically meaningful and resource-economical schemes showing a quantum speedup
in one and two spatial dimensions. At the heart of the analysis are tools of
unitary designs and random circuits that allow us to conclude that universal
random circuits anticoncentrate as well as an embedding of known circuit-based
schemes in a 2D translation-invariant architecture.Comment: 12+2 pages, added applications sectio
Testing Linear-Invariant Non-Linear Properties
We consider the task of testing properties of Boolean functions that are
invariant under linear transformations of the Boolean cube. Previous work in
property testing, including the linearity test and the test for Reed-Muller
codes, has mostly focused on such tasks for linear properties. The one
exception is a test due to Green for "triangle freeness": a function
f:\cube^{n}\to\cube satisfies this property if do not all
equal 1, for any pair x,y\in\cube^{n}.
Here we extend this test to a more systematic study of testing for
linear-invariant non-linear properties. We consider properties that are
described by a single forbidden pattern (and its linear transformations), i.e.,
a property is given by points v_{1},...,v_{k}\in\cube^{k} and
f:\cube^{n}\to\cube satisfies the property that if for all linear maps
L:\cube^{k}\to\cube^{n} it is the case that do
not all equal 1. We show that this property is testable if the underlying
matroid specified by is a graphic matroid. This extends
Green's result to an infinite class of new properties.
Our techniques extend those of Green and in particular we establish a link
between the notion of "1-complexity linear systems" of Green and Tao, and
graphic matroids, to derive the results.Comment: This is the full version; conference version appeared in the
proceedings of STACS 200
Convex computation of the region of attraction of polynomial control systems
We address the long-standing problem of computing the region of attraction
(ROA) of a target set (e.g., a neighborhood of an equilibrium point) of a
controlled nonlinear system with polynomial dynamics and semialgebraic state
and input constraints. We show that the ROA can be computed by solving an
infinite-dimensional convex linear programming (LP) problem over the space of
measures. In turn, this problem can be solved approximately via a classical
converging hierarchy of convex finite-dimensional linear matrix inequalities
(LMIs). Our approach is genuinely primal in the sense that convexity of the
problem of computing the ROA is an outcome of optimizing directly over system
trajectories. The dual infinite-dimensional LP on nonnegative continuous
functions (approximated by polynomial sum-of-squares) allows us to generate a
hierarchy of semialgebraic outer approximations of the ROA at the price of
solving a sequence of LMI problems with asymptotically vanishing conservatism.
This sharply contrasts with the existing literature which follows an
exclusively dual Lyapunov approach yielding either nonconvex bilinear matrix
inequalities or conservative LMI conditions. The approach is simple and readily
applicable as the outer approximations are the outcome of a single semidefinite
program with no additional data required besides the problem description
Designing Software Architectures As a Composition of Specializations of Knowledge Domains
This paper summarizes our experimental research and software development activities in designing robust, adaptable and reusable software architectures. Several years ago, based on our previous experiences in object-oriented software development, we made the following assumption: âA software architecture should be a composition of specializations of knowledge domainsâ. To verify this assumption we carried out three pilot projects. In addition to the application of some popular domain analysis techniques such as use cases, we identified the invariant compositional structures of the software architectures and the related knowledge domains. Knowledge domains define the boundaries of the adaptability and reusability capabilities of software systems. Next, knowledge domains were mapped to object-oriented concepts. We experienced that some aspects of knowledge could not be directly modeled in terms of object-oriented concepts. In this paper we describe our approach, the pilot projects, the experienced problems and the adopted solutions for realizing the software architectures. We conclude the paper with the lessons that we learned from this experience
Generalized liquid crystals: giant fluctuations and the vestigial chiral order of , and matter
The physics of nematic liquid crystals has been subject of intensive research
since the late 19th century. However, because of the limitations of chemistry
the focus has been centered around uni- and biaxial nematics associated with
constituents bearing a or symmetry respectively. In
view of general symmetries, however, these are singularly special since nematic
order can in principle involve any point group symmetry. Given the progress in
tailoring nano particles with particular shapes and interactions, this vast
family of "generalized nematics" might become accessible in the laboratory.
Little is known since the order parameter theories associated with the highly
symmetric point groups are remarkably complicated, involving tensor order
parameters of high rank. Here we show that the generic features of the
statistical physics of such systems can be studied in a highly flexible and
efficient fashion using a mathematical tool borrowed from high energy physics:
discrete non-Abelian gauge theory. Explicitly, we construct a family of lattice
gauge models encapsulating nematic ordering of general three dimensional point
group symmetries. We find that the most symmetrical "generalized nematics" are
subjected to thermal fluctuations of unprecedented severity. As a result, novel
forms of fluctuation phenomena become possible. In particular, we demonstrate
that a vestigial phase carrying no more than chiral order becomes ubiquitous
departing from high point group symmetry chiral building blocks, such as ,
and symmetric matter.Comment: 14 pages, 5 figures; published versio
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