907 research outputs found
A Rational Deconstruction of Landin's SECD Machine with the J Operator
Landin's SECD machine was the first abstract machine for applicative
expressions, i.e., functional programs. Landin's J operator was the first
control operator for functional languages, and was specified by an extension of
the SECD machine. We present a family of evaluation functions corresponding to
this extension of the SECD machine, using a series of elementary
transformations (transformation into continu-ation-passing style (CPS) and
defunctionalization, chiefly) and their left inverses (transformation into
direct style and refunctionalization). To this end, we modernize the SECD
machine into a bisimilar one that operates in lockstep with the original one
but that (1) does not use a data stack and (2) uses the caller-save rather than
the callee-save convention for environments. We also identify that the dump
component of the SECD machine is managed in a callee-save way. The caller-save
counterpart of the modernized SECD machine precisely corresponds to Thielecke's
double-barrelled continuations and to Felleisen's encoding of J in terms of
call/cc. We then variously characterize the J operator in terms of CPS and in
terms of delimited-control operators in the CPS hierarchy. As a byproduct, we
also present several reduction semantics for applicative expressions with the J
operator, based on Curien's original calculus of explicit substitutions. These
reduction semantics mechanically correspond to the modernized versions of the
SECD machine and to the best of our knowledge, they provide the first syntactic
theories of applicative expressions with the J operator
Fast optimization of parametrized quantum optical circuits
Parametrized quantum optical circuits are a class of quantum circuits in
which the carriers of quantum information are photons and the gates are optical
transformations. Classically optimizing these circuits is challenging due to
the infinite dimensionality of the photon number vector space that is
associated to each optical mode. Truncating the space dimension is unavoidable,
and it can lead to incorrect results if the gates populate photon number states
beyond the cutoff. To tackle this issue, we present an algorithm that is orders
of magnitude faster than the current state of the art, to recursively compute
the exact matrix elements of Gaussian operators and their gradient with respect
to a parametrization. These operators, when augmented with a non-Gaussian
transformation such as the Kerr gate, achieve universal quantum computation.
Our approach brings two advantages: first, by computing the matrix elements of
Gaussian operators directly, we don't need to construct them by combining
several other operators; second, we can use any variant of the gradient descent
algorithm by plugging our gradients into an automatic differentiation framework
such as TensorFlow or PyTorch. Our results will find applications in quantum
optical hardware research, quantum machine learning, optical data processing,
device discovery and device design.Comment: 21 pages, 10 figure
Continuous symmetry reduction and return maps for high-dimensional flows
We present two continuous symmetry reduction methods for reducing
high-dimensional dissipative flows to local return maps. In the Hilbert
polynomial basis approach, the equivariant dynamics is rewritten in terms of
invariant coordinates. In the method of moving frames (or method of slices) the
state space is sliced locally in such a way that each group orbit of
symmetry-equivalent points is represented by a single point. In either
approach, numerical computations can be performed in the original state-space
representation, and the solutions are then projected onto the symmetry-reduced
state space. The two methods are illustrated by reduction of the complex Lorenz
system, a 5-dimensional dissipative flow with rotational symmetry. While the
Hilbert polynomial basis approach appears unfeasible for high-dimensional
flows, symmetry reduction by the method of moving frames offers hope.Comment: 32 pages, 7 figure
Beyond Geometry: Comparing the Temporal Structure of Computation in Neural Circuits with Dynamical Similarity Analysis
How can we tell whether two neural networks are utilizing the same internal
processes for a particular computation? This question is pertinent for multiple
subfields of both neuroscience and machine learning, including neuroAI,
mechanistic interpretability, and brain-machine interfaces. Standard approaches
for comparing neural networks focus on the spatial geometry of latent states.
Yet in recurrent networks, computations are implemented at the level of neural
dynamics, which do not have a simple one-to-one mapping with geometry. To
bridge this gap, we introduce a novel similarity metric that compares two
systems at the level of their dynamics. Our method incorporates two components:
Using recent advances in data-driven dynamical systems theory, we learn a
high-dimensional linear system that accurately captures core features of the
original nonlinear dynamics. Next, we compare these linear approximations via a
novel extension of Procrustes Analysis that accounts for how vector fields
change under orthogonal transformation. Via four case studies, we demonstrate
that our method effectively identifies and distinguishes dynamic structure in
recurrent neural networks (RNNs), whereas geometric methods fall short. We
additionally show that our method can distinguish learning rules in an
unsupervised manner. Our method therefore opens the door to novel data-driven
analyses of the temporal structure of neural computation, and to more rigorous
testing of RNNs as models of the brain.Comment: 21 pages, 10 figure
Quantum information protocols in complex entangled networks
Quantum entangled networks represent essential tools for Quantum Communication, i.e. the exchange of Quantum Information between parties. This work consists in the theoretical study of continuous variables (CV) entangled networks - which can be deterministically generated via multimode squeezed light - with complex topology. In particular we investigate CV complex quantum networks properties for quantum communication protocols. We focused on the role played by the topology in the implementation and the optimization of given characteristics of our entangled resource that are useful for a specific quantum communication task, i.e. the creation of an entanglement link between two arbitrary nodes of the resource we are provided with. We implemented an analytical procedure for the generation of entangled complex networks, their optimization and their manipulation via global linear optics operations. We also developed a numerical procedure, based on an evolutionary algorithm, for manipulating entanglement connections via local linear optics operations. Finally, we analyzed the re-shaping of our entangled resource via homodyne measurements
Tachyon Condensation on the Elliptic Curve
We use the framework of matrix factorizations to study topological B-type
D-branes on the cubic curve. Specifically, we elucidate how the brane RR
charges are encoded in the matrix factors, by analyzing their structure in
terms of sections of vector bundles in conjunction with equivariant R-symmetry.
One particular advantage of matrix factorizations is that explicit moduli
dependence is built in, thus giving us full control over the open-string moduli
space. It allows one to study phenomena like discontinuous jumps of the
cohomology over the moduli space, as well as formation of bound states at
threshold. One interesting aspect is that certain gauge symmetries inherent to
the matrix formulation lead to a non-trivial global structure of the moduli
space. We also investigate topological tachyon condensation, which enables us
to construct, in a systematic fashion, higher-dimensional matrix factorizations
out of smaller ones; this amounts to obtaining branes with higher RR charges as
composites of ones with minimal charges. As an application, we explicitly
construct all rank-two matrix factorizations.Comment: 69p, 6 figs, harvmac; v2: minor change
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