2,704 research outputs found
Tensor Networks for Lattice Gauge Theories with continuous groups
We discuss how to formulate lattice gauge theories in the Tensor Network
language. In this way we obtain both a consistent truncation scheme of the
Kogut-Susskind lattice gauge theories and a Tensor Network variational ansatz
for gauge invariant states that can be used in actual numerical computation.
Our construction is also applied to the simplest realization of the quantum
link models/gauge magnets and provides a clear way to understand their
microscopic relation with Kogut-Susskind lattice gauge theories. We also
introduce a new set of gauge invariant operators that modify continuously
Rokshar-Kivelson wave functions and can be used to extend the phase diagram of
known models. As an example we characterize the transition between the
deconfined phase of the lattice gauge theory and the Rokshar-Kivelson
point of the U(1) gauge magnet in 2D in terms of entanglement entropy. The
topological entropy serves as an order parameter for the transition but not the
Schmidt gap.Comment: 27 pages, 25 figures, 2nd version the same as the published versio
A Bestiary of Sets and Relations
Building on established literature and recent developments in the
graph-theoretic characterisation of its CPM category, we provide a treatment of
pure state and mixed state quantum mechanics in the category fRel of finite
sets and relations. On the way, we highlight the wealth of exotic beasts that
hide amongst the extensive operational and structural similarities that the
theory shares with more traditional arenas of categorical quantum mechanics,
such as the category fdHilb. We conclude our journey by proving that fRel is
local, but not without some unexpected twists.Comment: In Proceedings QPL 2015, arXiv:1511.0118
Combinable Extensions of Abelian Groups
The design of decision procedures for combinations of theories sharing some arithmetic fragment is a challenging problem in verification. One possible solution is to apply a combination method Ă la Nelson-Oppen, like the one developed by Ghilardi for unions of non-disjoint theories. We show how to apply this non-disjoint combination method with the theory of abelian groups as shared theory. We consider the completeness and the effectiveness of this non-disjoint combination method. For the completeness, we show that the theory of abelian groups can be embedded into a theory admitting quantifier elimination. For achieving effectiveness, we rely on a superposition calculus modulo abelian groups that is shown complete for theories of practical interest in verification
The Quantum Frontier
The success of the abstract model of computation, in terms of bits, logical
operations, programming language constructs, and the like, makes it easy to
forget that computation is a physical process. Our cherished notions of
computation and information are grounded in classical mechanics, but the
physics underlying our world is quantum. In the early 80s researchers began to
ask how computation would change if we adopted a quantum mechanical, instead of
a classical mechanical, view of computation. Slowly, a new picture of
computation arose, one that gave rise to a variety of faster algorithms, novel
cryptographic mechanisms, and alternative methods of communication. Small
quantum information processing devices have been built, and efforts are
underway to build larger ones. Even apart from the existence of these devices,
the quantum view on information processing has provided significant insight
into the nature of computation and information, and a deeper understanding of
the physics of our universe and its connections with computation.
We start by describing aspects of quantum mechanics that are at the heart of
a quantum view of information processing. We give our own idiosyncratic view of
a number of these topics in the hopes of correcting common misconceptions and
highlighting aspects that are often overlooked. A number of the phenomena
described were initially viewed as oddities of quantum mechanics. It was
quantum information processing, first quantum cryptography and then, more
dramatically, quantum computing, that turned the tables and showed that these
oddities could be put to practical effect. It is these application we describe
next. We conclude with a section describing some of the many questions left for
future work, especially the mysteries surrounding where the power of quantum
information ultimately comes from.Comment: Invited book chapter for Computation for Humanity - Information
Technology to Advance Society to be published by CRC Press. Concepts
clarified and style made more uniform in version 2. Many thanks to the
referees for their suggestions for improvement
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