3,296 research outputs found
The Quantum Monadology
The modern theory of functional programming languages uses monads for
encoding computational side-effects and side-contexts, beyond bare-bone program
logic. Even though quantum computing is intrinsically side-effectful (as in
quantum measurement) and context-dependent (as on mixed ancillary states),
little of this monadic paradigm has previously been brought to bear on quantum
programming languages.
Here we systematically analyze the (co)monads on categories of parameterized
module spectra which are induced by Grothendieck's "motivic yoga of operations"
-- for the present purpose specialized to HC-modules and further to set-indexed
complex vector spaces. Interpreting an indexed vector space as a collection of
alternative possible quantum state spaces parameterized by quantum measurement
results, as familiar from Proto-Quipper-semantics, we find that these
(co)monads provide a comprehensive natural language for functional quantum
programming with classical control and with "dynamic lifting" of quantum
measurement results back into classical contexts.
We close by indicating a domain-specific quantum programming language (QS)
expressing these monadic quantum effects in transparent do-notation, embeddable
into the recently constructed Linear Homotopy Type Theory (LHoTT) which
interprets into parameterized module spectra. Once embedded into LHoTT, this
should make for formally verifiable universal quantum programming with linear
quantum types, classical control, dynamic lifting, and notably also with
topological effects.Comment: 120 pages, various figure
Quantum information and statistical mechanics: an introduction to frontier
This is a short review on an interdisciplinary field of quantum information
science and statistical mechanics. We first give a pedagogical introduction to
the stabilizer formalism, which is an efficient way to describe an important
class of quantum states, the so-called stabilizer states, and quantum
operations on them. Furthermore, graph states, which are a class of stabilizer
states associated with graphs, and their applications for measurement-based
quantum computation are also mentioned. Based on the stabilizer formalism, we
review two interdisciplinary topics. One is the relation between quantum error
correction codes and spin glass models, which allows us to analyze the
performances of quantum error correction codes by using the knowledge about
phases in statistical models. The other is the relation between the stabilizer
formalism and partition functions of classical spin models, which provides new
quantum and classical algorithms to evaluate partition functions of classical
spin models.Comment: 15pages, 4 figures, to appear in Proceedings of 4th YSM-SPIP (Sendai,
14-16 December 2012
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