670 research outputs found
Generalized Performance of Concatenated Quantum Codes -- A Dynamical Systems Approach
We apply a dynamical systems approach to concatenation of quantum error
correcting codes, extending and generalizing the results of Rahn et al. [1] to
both diagonal and nondiagonal channels. Our point of view is global: instead of
focusing on particular types of noise channels, we study the geometry of the
coding map as a discrete-time dynamical system on the entire space of noise
channels. In the case of diagonal channels, we show that any code with distance
at least three corrects (in the infinite concatenation limit) an open set of
errors. For Calderbank-Shor-Steane (CSS) codes, we give a more precise
characterization of that set. We show how to incorporate noise in the gates,
thus completing the framework. We derive some general bounds for noise
channels, which allows us to analyze several codes in detail.Comment: 12 pages two-column format, no figures, slightly revised versio
Single-bit Feedback and Quantum Dynamical Decoupling
Synthesizing an effective identity evolution in a target system subjected to
unwanted unitary or non-unitary dynamics is a fundamental task for both quantum
control and quantum information processing applications. Here, we investigate
how single-bit, discrete-time feedback capabilities may be exploited to enact
or to enhance quantum procedures for effectively suppressing unwanted dynamics
in a finite-dimensional open quantum system. An explicit characterization of
the joint unitary propagators correctable by a single-bit feedback strategy for
arbitrary evolution time is obtained. For a two-dimensional target system, we
show how by appropriately combining quantum feedback with dynamical decoupling
methods, concatenated feedback-decoupling schemes may be built, which can
operate under relaxed control assumptions and can outperform purely closed-loop
and open-loop protocols.Comment: 12 pages, 2 figure
Theory of Decoherence-Free Fault-Tolerant Universal Quantum Computation
Universal quantum computation on decoherence-free subspaces and subsystems
(DFSs) is examined with particular emphasis on using only physically relevant
interactions. A necessary and sufficient condition for the existence of
decoherence-free (noiseless) subsystems in the Markovian regime is derived here
for the first time. A stabilizer formalism for DFSs is then developed which
allows for the explicit understanding of these in their dual role as quantum
error correcting codes. Conditions for the existence of Hamiltonians whose
induced evolution always preserves a DFS are derived within this stabilizer
formalism. Two possible collective decoherence mechanisms arising from
permutation symmetries of the system-bath coupling are examined within this
framework. It is shown that in both cases universal quantum computation which
always preserves the DFS (*natural fault-tolerant computation*) can be
performed using only two-body interactions. This is in marked contrast to
standard error correcting codes, where all known constructions using one or
two-body interactions must leave the codespace during the on-time of the
fault-tolerant gates. A further consequence of our universality construction is
that a single exchange Hamiltonian can be used to perform universal quantum
computation on an encoded space whose asymptotic coding efficiency is unity.
The exchange Hamiltonian, which is naturally present in many quantum systems,
is thus *asymptotically universal*.Comment: 40 pages (body: 30, appendices: 3, figures: 5, references: 2). Fixed
problem with non-printing figures. New references added, minor typos
correcte
Quantum memories based on engineered dissipation
Storing quantum information for long times without disruptions is a major
requirement for most quantum information technologies. A very appealing
approach is to use self-correcting Hamiltonians, i.e. tailoring local
interactions among the qubits such that when the system is weakly coupled to a
cold bath the thermalization process takes a long time. Here we propose an
alternative but more powerful approach in which the coupling to a bath is
engineered, so that dissipation protects the encoded qubit against more general
kinds of errors. We show that the method can be implemented locally in four
dimensional lattice geometries by means of a toric code, and propose a simple
2D set-up for proof of principle experiments.Comment: 6 +8 pages, 4 figures, Includes minor corrections updated references
and aknowledgement
Internal Consistency of Fault-Tolerant Quantum Error Correction in Light of Rigorous Derivations of the Quantum Markovian Limit
We critically examine the internal consistency of a set of minimal
assumptions entering the theory of fault-tolerant quantum error correction for
Markovian noise. These assumptions are: fast gates, a constant supply of fresh
and cold ancillas, and a Markovian bath. We point out that these assumptions
may not be mutually consistent in light of rigorous formulations of the
Markovian approximation. Namely, Markovian dynamics requires either the
singular coupling limit (high temperature), or the weak coupling limit (weak
system-bath interaction). The former is incompatible with the assumption of a
constant and fresh supply of cold ancillas, while the latter is inconsistent
with fast gates. We discuss ways to resolve these inconsistencies. As part of
our discussion we derive, in the weak coupling limit, a new master equation for
a system subject to periodic driving.Comment: 19 pages. v2: Significantly expanded version. New title. Includes a
debate section in response to comments on the previous version, many of which
appeared here http://dabacon.org/pontiff/?p=959 and here
http://dabacon.org/pontiff/?p=1028. Contains a new derivation of the
Markovian master equation with periodic drivin
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