19,420 research outputs found
On a complete set of generators for dot-depth two
AbstractA complete set of generators for Straubing's dot-depth-two monoids has been characterized as a set of quotients of the form A∗/∼(n,m), where n and m denote positive integers, A∗ denotes the free monoid generated by a finite alphabet A, and ∼(n,m) denote congruences related to a version of the Ehrenfeucht—Fraïssé game. This paper studies combinatorial properties of the ∼(n,m)'s and in particular the inclusion relations between them. Several decidability and inclusion consequences are discussed
Quantum error correction in crossbar architectures
A central challenge for the scaling of quantum computing systems is the need
to control all qubits in the system without a large overhead. A solution for
this problem in classical computing comes in the form of so called crossbar
architectures. Recently we made a proposal for a large scale quantum
processor~[Li et al. arXiv:1711.03807 (2017)] to be implemented in silicon
quantum dots. This system features a crossbar control architecture which limits
parallel single qubit control, but allows the scheme to overcome control
scaling issues that form a major hurdle to large scale quantum computing
systems. In this work, we develop a language that makes it possible to easily
map quantum circuits to crossbar systems, taking into account their
architecture and control limitations. Using this language we show how to map
well known quantum error correction codes such as the planar surface and color
codes in this limited control setting with only a small overhead in time. We
analyze the logical error behavior of this surface code mapping for estimated
experimental parameters of the crossbar system and conclude that logical error
suppression to a level useful for real quantum computation is feasible.Comment: 29 + 9 pages, 13 figures, 9 tables, 8 algorithms and 3 big boxes.
Comments are welcom
A Diagrammatic Temperley-Lieb Categorification
The monoidal category of Soergel bimodules categorifies the Hecke algebra of
a finite Weyl group. In the case of the symmetric group, morphisms in this
category can be drawn as graphs in the plane. We define a quotient category,
also given in terms of planar graphs, which categorifies the Temperley-Lieb
algebra. Certain ideals appearing in this quotient are related both to the
1-skeleton of the Coxeter complex and to the topology of 2D cobordisms. We
demonstrate how further subquotients of this category will categorify the cell
modules of the Temperley-Lieb algebra.Comment: long awaited update to published versio
A ZX-Calculus with Triangles for Toffoli-Hadamard, Clifford+T, and Beyond
We consider a ZX-calculus augmented with triangle nodes which is well-suited
to reason on the so-called Toffoli-Hadamard fragment of quantum mechanics. We
precisely show the form of the matrices it represents, and we provide an
axiomatisation which makes the language complete for the Toffoli-Hadamard
quantum mechanics. We extend the language with arbitrary angles and show that
any true equation involving linear diagrams which constant angles are multiple
of Pi are derivable. We show that a single axiom is then necessary and
sufficient to make the language equivalent to the ZX-calculus which is known to
be complete for Clifford+T quantum mechanics. As a by-product, it leads to a
new and simple complete axiomatisation for Clifford+T quantum mechanics.Comment: In Proceedings QPL 2018, arXiv:1901.09476. Contains Appendi
ZH: A Complete Graphical Calculus for Quantum Computations Involving Classical Non-linearity
We present a new graphical calculus that is sound and complete for a
universal family of quantum circuits, which can be seen as the natural
string-diagrammatic extension of the approximately (real-valued) universal
family of Hadamard+CCZ circuits. The diagrammatic language is generated by two
kinds of nodes: the so-called 'spider' associated with the computational basis,
as well as a new arity-N generalisation of the Hadamard gate, which satisfies a
variation of the spider fusion law. Unlike previous graphical calculi, this
admits compact encodings of non-linear classical functions. For example, the
AND gate can be depicted as a diagram of just 2 generators, compared to ~25 in
the ZX-calculus. Consequently, N-controlled gates, hypergraph states,
Hadamard+Toffoli circuits, and diagonal circuits at arbitrary levels of the
Clifford hierarchy also enjoy encodings with low constant overhead. This
suggests that this calculus will be significantly more convenient for reasoning
about the interplay between classical non-linear behaviour (e.g. in an oracle)
and purely quantum operations. After presenting the calculus, we will prove it
is sound and complete for universal quantum computation by demonstrating the
reduction of any diagram to an easily describable normal form.Comment: In Proceedings QPL 2018, arXiv:1901.0947
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