98 research outputs found
Communication protocols and quantum error-correcting codes from the perspective of topological quantum field theory
Topological quantum field theories (TQFTs) provide a general,
minimal-assumption language for describing quantum-state preparation and
measurement. They therefore provide a general language in which to express
multi-agent communication protocols, e.g. local operations, classical
communication (LOCC) protocols. Here we construct LOCC protocols using TQFT,
and show that LOCC protocols induce quantum error-correcting codes (QECCs) on
the agent-environment boundary. Such QECCs can be regarded as implementing, or
inducing the emergence of, spacetimes on such boundaries. We investigate this
connection between inter-agent communication and spacetime using BF and
Chern-Simons theories, and then using topological M-theory.Comment: 52 page
Transformers Learn Shortcuts to Automata
Algorithmic reasoning requires capabilities which are most naturally
understood through recurrent models of computation, like the Turing machine.
However, Transformer models, while lacking recurrence, are able to perform such
reasoning using far fewer layers than the number of reasoning steps. This
raises the question: what solutions are learned by these shallow and
non-recurrent models? We find that a low-depth Transformer can represent the
computations of any finite-state automaton (thus, any bounded-memory
algorithm), by hierarchically reparameterizing its recurrent dynamics. Our
theoretical results characterize shortcut solutions, whereby a Transformer with
layers can exactly replicate the computation of an automaton on an input
sequence of length . We find that polynomial-sized -depth
solutions always exist; furthermore, -depth simulators are surprisingly
common, and can be understood using tools from Krohn-Rhodes theory and circuit
complexity. Empirically, we perform synthetic experiments by training
Transformers to simulate a wide variety of automata, and show that shortcut
solutions can be learned via standard training. We further investigate the
brittleness of these solutions and propose potential mitigations
Foundations of Software Science and Computation Structures
This open access book constitutes the proceedings of the 25th International Conference on Foundations of Software Science and Computational Structures, FOSSACS 2022, which was held during April 4-6, 2022, in Munich, Germany, as part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2022. The 23 regular papers presented in this volume were carefully reviewed and selected from 77 submissions. They deal with research on theories and methods to support the analysis, integration, synthesis, transformation, and verification of programs and software systems
Quantum Computing and Communications
This book explains the concepts and basic mathematics of quantum computing and communication. Chapters cover such topics as quantum algorithms, photonic implementations of discrete-time quantum walks, how to build a quantum computer, and quantum key distribution and teleportation, among others
This Week's Finds in Mathematical Physics (1-50)
These are the first 50 issues of This Week's Finds of Mathematical Physics,
from January 19, 1993 to March 12, 1995. These issues focus on quantum gravity,
topological quantum field theory, knot theory, and applications of
-categories to these subjects. However, there are also digressions into Lie
algebras, elliptic curves, linear logic and other subjects. They were typeset
in 2020 by Tim Hosgood. If you see typos or other problems please report them.
(I already know the cover page looks weird).Comment: 242 page
Disentangling supercohomology symmetry-protected topological phases in three spatial dimensions
We build exactly solvable lattice Hamiltonians for fermionic
symmetry-protected topological (SPT) phases in (3+1)D classified by group
supercohomology. A central benefit of our construction is that it produces an
explicit finite-depth quantum circuit (FDQC) that prepares the ground state
from an unentangled symmetric state. The FDQC allows us to clearly demonstrate
the characteristic properties of supercohomology phases - namely, symmetry
fractionalization on fermion parity flux loops - predicted by continuum
formulations. By composing the corresponding FDQCs, we also recover the
stacking relations of supercohomology phases. Furthermore, we derive
topologically ordered gapped boundaries for the supercohomology models by
extending the protecting symmetries, analogous to the construction of
topologically ordered boundaries for bosonic SPT phases. Our approach relies
heavily on dualities that relate certain bosonic 2-group SPT phases with
supercohomology SPT phases. We develop physical motivation for the dualities in
terms of explicit lattice prescriptions for gauging a 1-form symmetry and for
condensing emergent fermions. We also comment on generalizations to
supercohomology phases in higher dimensions and to fermionic SPT phases outside
of the supercohomology framework.Comment: 28+25 pages, 31 figure
Disentangling supercohomology symmetry-protected topological phases in three spatial dimensions
We build exactly solvable lattice Hamiltonians for fermionic symmetry-protected topological (SPT) phases in (3+1)D classified by group supercohomology. A central benefit of our construction is that it produces an explicit finite-depth quantum circuit (FDQC) that prepares the ground state from an unentangled symmetric state. The FDQC allows us to clearly demonstrate the characteristic properties of supercohomology phasesânamely, symmetry fractionalization on fermion parity flux loopsâpredicted by continuum formulations. By composing the corresponding FDQCs, we also recover the stacking relations of supercohomology phases. Furthermore, we derive topologically ordered gapped boundaries for the supercohomology models by extending the protecting symmetries, analogous to the construction of topologically ordered boundaries for bosonic SPT phases. Our approach relies heavily on dualities that relate certain bosonic 2-group SPT phases with supercohomology SPT phases. We develop physical motivation for the dualities in terms of explicit lattice prescriptions for gauging a 1-form symmetry and for condensing emergent fermions. We also comment on generalizations to supercohomology phases in higher dimensions and to fermionic SPT phases outside of the supercohomology framework
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