6 research outputs found
Conditions for the approximate correction of algebras
We study the approximate correctability of general algebras of observables,
which represent hybrid quantum-classical information. This includes approximate
quantum error correcting codes and subsystems codes. We show that the main
result of arXiv:quant-ph/0605009 yields a natural generalization of the
Knill-Laflamme conditions in the form of a dimension independent estimate of
the optimal reconstruction error for a given encoding, measured using the
trace-norm distance to a noiseless channel.Comment: Related to a talk given at TQC 2009 in Waterlo
Pauli Manipulation Detection codes and Applications to Quantum Communication over Adversarial Channels
We introduce and explicitly construct a quantum code we coin a "Pauli
Manipulation Detection" code (or PMD), which detects every Pauli error with
high probability. We apply them to construct the first near-optimal codes for
two tasks in quantum communication over adversarial channels. Our main
application is an approximate quantum code over qubits which can efficiently
correct from a number of (worst-case) erasure errors approaching the quantum
Singleton bound. Our construction is based on the composition of a PMD code
with a stabilizer code which is list-decodable from erasures.
Our second application is a quantum authentication code for "qubit-wise"
channels, which does not require a secret key. Remarkably, this gives an
example of a task in quantum communication which is provably impossible
classically. Our construction is based on a combination of PMD codes,
stabilizer codes, and classical non-malleable codes (Dziembowski et al., 2009),
and achieves "minimal redundancy" (rate )
Quantum Error Correction in the Lowest Landau Level
We develop finite-dimensional versions of the quantum error-correcting codes
proposed by Albert, Covey, and Preskill (ACP) for continuous-variable quantum
computation on configuration spaces with nonabelian symmetry groups. Our codes
can be realized by a charged particle in a Landau level on a spherical geometry
-- in contrast to the planar Landau level realization of the qudit codes of
Gottesman, Kitaev, and Preskill (GKP) -- or more generally by spin coherent
states. Our quantum error-correction scheme is inherently approximate, and the
encoded states may be easier to prepare than those of GKP or ACP.Comment: 27 + 29 pages, comments welcome; v2: close to published versio