11,105 research outputs found
Universal Quantum Hamiltonians
Quantum many-body systems exhibit an extremely diverse range of phases and
physical phenomena. Here, we prove that the entire physics of any other quantum
many-body system is replicated in certain simple, "universal" spin-lattice
models. We first characterise precisely what it means for one quantum many-body
system to replicate the entire physics of another. We then show that certain
very simple spin-lattice models are universal in this very strong sense.
Examples include the Heisenberg and XY models on a 2D square lattice (with
non-uniform coupling strengths). We go on to fully classify all two-qubit
interactions, determining which are universal and which can only simulate more
restricted classes of models. Our results put the practical field of analogue
Hamiltonian simulation on a rigorous footing and take a significant step
towards justifying why error correction may not be required for this
application of quantum information technology.Comment: 78 pages, 9 figures, 44 theorems etc. v2: Trivial fixes. v3: updated
and simplified proof of Thm. 9; 82 pages, 47 theorems etc. v3: Small fix in
proof of time-evolution lemma (this fix not in published version
Fault-tolerant quantum computation versus Gaussian noise
We study the robustness of a fault-tolerant quantum computer subject to
Gaussian non-Markovian quantum noise, and we show that scalable quantum
computation is possible if the noise power spectrum satisfies an appropriate
"threshold condition." Our condition is less sensitive to very-high-frequency
noise than previously derived threshold conditions for non-Markovian noise.Comment: 30 pages, 6 figure
Particle Computation: Complexity, Algorithms, and Logic
We investigate algorithmic control of a large swarm of mobile particles (such
as robots, sensors, or building material) that move in a 2D workspace using a
global input signal (such as gravity or a magnetic field). We show that a maze
of obstacles to the environment can be used to create complex systems. We
provide a wide range of results for a wide range of questions. These can be
subdivided into external algorithmic problems, in which particle configurations
serve as input for computations that are performed elsewhere, and internal
logic problems, in which the particle configurations themselves are used for
carrying out computations. For external algorithms, we give both negative and
positive results. If we are given a set of stationary obstacles, we prove that
it is NP-hard to decide whether a given initial configuration of unit-sized
particles can be transformed into a desired target configuration. Moreover, we
show that finding a control sequence of minimum length is PSPACE-complete. We
also work on the inverse problem, providing constructive algorithms to design
workspaces that efficiently implement arbitrary permutations between different
configurations. For internal logic, we investigate how arbitrary computations
can be implemented. We demonstrate how to encode dual-rail logic to build a
universal logic gate that concurrently evaluates and, nand, nor, and or
operations. Using many of these gates and appropriate interconnects, we can
evaluate any logical expression. However, we establish that simulating the full
range of complex interactions present in arbitrary digital circuits encounters
a fundamental difficulty: a fan-out gate cannot be generated. We resolve this
missing component with the help of 2x1 particles, which can create fan-out
gates that produce multiple copies of the inputs. Using these gates we provide
rules for replicating arbitrary digital circuits.Comment: 27 pages, 19 figures, full version that combines three previous
conference article
Making mathematics phenomenal : Based on an Inaugural Professorial Lecture delivered at the Institute of Education, University of London, on 14 March 2012
Mathematics is often portrayed as an 'abstract' cerebral subject, beyond the reach of many. In response, research with digital technology has led to innovative design in which mathematics can be experienced to some extent like everyday phenomena. I examine how careful design can 'phenomenalise' mathematics - that is to say create mathematical artefacts that can be directly experienced to support not only engagement but also focus on key ideas. I argue that mathematical knowledge gained through interaction with suitably designed tools can prioritise powerful reasons for doing mathematics, imbuing it with a sort of utility and offering learners hooks on which they can gradually develop fluency and connected understanding. Illustrative examples are taken from conventional topics such as number, algebra, geometry and statistics but also from novel situations where mathematical methods are juxtaposed with social values. The suggestion that prioritising utility supports a more natural way of learning mathematics emerges directly from constructionist pedagogy and inferentialist philosophy
Annoyancetech Vigilante Torts and Policy
The twenty-first century has ushered in demand by some Americans for annoyancetech devicesânovel electronic gadgets that secretly fend off, punish, or comment upon perceived antisocial and annoying behaviors of others. Manufacturers, marketers, and users of certain annoyancetech devices, however, face potential tort liability for personal and property damages suffered by the targets of this ârevenge by gadget.â Federal, state, and local policymakers should start the process of coming to pragmatic terms with the troubling rise in the popularity of annoyancetech devices. This is an area of social policy that cries out for thoughtful and creative legislative solutions
Pricing in Social Networks with Negative Externalities
We study the problems of pricing an indivisible product to consumers who are
embedded in a given social network. The goal is to maximize the revenue of the
seller. We assume impatient consumers who buy the product as soon as the seller
posts a price not greater than their values of the product. The product's value
for a consumer is determined by two factors: a fixed consumer-specified
intrinsic value and a variable externality that is exerted from the consumer's
neighbors in a linear way. We study the scenario of negative externalities,
which captures many interesting situations, but is much less understood in
comparison with its positive externality counterpart. We assume complete
information about the network, consumers' intrinsic values, and the negative
externalities. The maximum revenue is in general achieved by iterative pricing,
which offers impatient consumers a sequence of prices over time.
We prove that it is NP-hard to find an optimal iterative pricing, even for
unweighted tree networks with uniform intrinsic values. Complementary to the
hardness result, we design a 2-approximation algorithm for finding iterative
pricing in general weighted networks with (possibly) nonuniform intrinsic
values. We show that, as an approximation to optimal iterative pricing, single
pricing can work rather well for many interesting cases, but theoretically it
can behave arbitrarily bad
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