89 research outputs found
Approximate quantum error correction for generalized amplitude damping errors
We present analytic estimates of the performances of various approximate
quantum error correction schemes for the generalized amplitude damping (GAD)
qubit channel. Specifically, we consider both stabilizer and nonadditive
quantum codes. The performance of such error-correcting schemes is quantified
by means of the entanglement fidelity as a function of the damping probability
and the non-zero environmental temperature. The recovery scheme employed
throughout our work applies, in principle, to arbitrary quantum codes and is
the analogue of the perfect Knill-Laflamme recovery scheme adapted to the
approximate quantum error correction framework for the GAD error model. We also
analytically recover and/or clarify some previously known numerical results in
the limiting case of vanishing temperature of the environment, the well-known
traditional amplitude damping channel. In addition, our study suggests that
degenerate stabilizer codes and self-complementary nonadditive codes are
especially suitable for the error correction of the GAD noise model. Finally,
comparing the properly normalized entanglement fidelities of the best
performant stabilizer and nonadditive codes characterized by the same length,
we show that nonadditive codes outperform stabilizer codes not only in terms of
encoded dimension but also in terms of entanglement fidelity.Comment: 44 pages, 8 figures, improved v
Degenerate Quantum Codes for Pauli Channels
A striking feature of quantum error correcting codes is that they can
sometimes be used to correct more errors than they can uniquely identify. Such
degenerate codes have long been known, but have remained poorly understood. We
provide a heuristic for designing degenerate quantum codes for high noise
rates, which is applied to generate codes that can be used to communicate over
almost any Pauli channel at rates that are impossible for a nondegenerate code.
The gap between nondegenerate and degenerate code performance is quite large,
in contrast to the tiny magnitude of the only previous demonstration of this
effect. We also identify a channel for which none of our codes outperform the
best nondegenerate code and show that it is nevertheless quite unlike any
channel for which nondegenerate codes are known to be optimal.Comment: Introduction changed to give more motivation and background. Figure 1
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Entanglement-Assisted Quantum Error-Correcting Codes with Imperfect Ebits
The scheme of entanglement-assisted quantum error-correcting (EAQEC) codes
assumes that the ebits of the receiver are error-free. In practical situations,
errors on these ebits are unavoidable, which diminishes the error-correcting
ability of these codes. We consider two different versions of this problem. We
first show that any (nondegenerate) standard stabilizer code can be transformed
into an EAQEC code that can correct errors on the qubits of both sender and
receiver. These EAQEC codes are equivalent to standard stabilizer codes, and
hence the decoding techniques of standard stabilizer codes can be applied.
Several EAQEC codes of this type are found to be optimal. In a second scheme,
the receiver uses a standard stabilizer code to protect the ebits, which we
call a "combination code." The performances of different quantum codes are
compared in terms of the channel fidelity over the depolarizing channel. We
give a formula for the channel fidelity over the depolarizing channel (or any
Pauli error channel), and show that it can be efficiently approximated by a
Monte Carlo calculation. Finally, we discuss the tradeoff between performing
extra entanglement distillation and applying an EAQEC code with imperfect
ebits.Comment: 15 pages, 12 figure
Universal fault-tolerant gates on concatenated stabilizer codes
It is an oft-cited fact that no quantum code can support a set of
fault-tolerant logical gates that is both universal and transversal. This no-go
theorem is generally responsible for the interest in alternative universality
constructions including magic state distillation. Widely overlooked, however,
is the possibility of non-transversal, yet still fault-tolerant, gates that
work directly on small quantum codes. Here we demonstrate precisely the
existence of such gates. In particular, we show how the limits of
non-transversality can be overcome by performing rounds of intermediate
error-correction to create logical gates on stabilizer codes that use no
ancillas other than those required for syndrome measurement. Moreover, the
logical gates we construct, the most prominent examples being Toffoli and
controlled-controlled-Z, often complete universal gate sets on their codes. We
detail such universal constructions for the smallest quantum codes, the 5-qubit
and 7-qubit codes, and then proceed to generalize the approach. One remarkable
result of this generalization is that any nondegenerate stabilizer code with a
complete set of fault-tolerant single-qubit Clifford gates has a universal set
of fault-tolerant gates. Another is the interaction of logical qubits across
different stabilizer codes, which, for instance, implies a broadly applicable
method of code switching.Comment: 18 pages + 5 pages appendix, 12 figure
Direct Characterization of Quantum Dynamics: General Theory
The characterization of the dynamics of quantum systems is a task of both
fundamental and practical importance. A general class of methods which have
been developed in quantum information theory to accomplish this task is known
as quantum process tomography (QPT). In an earlier paper [M. Mohseni and D. A.
Lidar, Phys. Rev. Lett. 97, 170501 (2006)] we presented a new algorithm for
Direct Characterization of Quantum Dynamics (DCQD) of two-level quantum
systems. Here we provide a generalization by developing a theory for direct and
complete characterization of the dynamics of arbitrary quantum systems. In
contrast to other QPT schemes, DCQD relies on quantum error-detection
techniques and does not require any quantum state tomography. We demonstrate
that for the full characterization of the dynamics of n d-level quantum systems
(with d a power of a prime), the minimal number of required experimental
configurations is reduced quadratically from d^{4n} in separable QPT schemes to
d^{2n} in DCQD.Comment: 17 pages, 6 figures, minor modifications are mad
Quantum Stabilizer Codes and Classical Linear Codes
We show that within any quantum stabilizer code there lurks a classical
binary linear code with similar error-correcting capabilities, thereby
demonstrating new connections between quantum codes and classical codes. Using
this result -- which applies to degenerate as well as nondegenerate codes --
previously established necessary conditions for classical linear codes can be
easily translated into necessary conditions for quantum stabilizer codes.
Examples of specific consequences are: for a quantum channel subject to a
delta-fraction of errors, the best asymptotic capacity attainable by any
stabilizer code cannot exceed H(1/2 + sqrt(2*delta*(1-2*delta))); and, for the
depolarizing channel with fidelity parameter delta, the best asymptotic
capacity attainable by any stabilizer code cannot exceed 1-H(delta).Comment: 17 pages, ReVTeX, with two figure
Multi-Error-Correcting Amplitude Damping Codes
We construct new families of multi-error-correcting quantum codes for the
amplitude damping channel. Our key observation is that, with proper encoding,
two uses of the amplitude damping channel simulate a quantum erasure channel.
This allows us to use concatenated codes with quantum erasure-correcting codes
as outer codes for correcting multiple amplitude damping errors. Our new codes
are degenerate stabilizer codes and have parameters which are better than the
amplitude damping codes obtained by any previously known construction.Comment: 5 pages. Submitted to ISIT 201
Codeword Stabilized Quantum Codes for Asymmetric Channels
We discuss a method to adapt the codeword stabilized (CWS) quantum code
framework to the problem of finding asymmetric quantum codes. We focus on the
corresponding Pauli error models for amplitude damping noise and phase damping
noise. In particular, we look at codes for Pauli error models that correct one
or two amplitude damping errors. Applying local Clifford operations on graph
states, we are able to exhaustively search for all possible codes up to length
. With a similar method, we also look at codes for the Pauli error model
that detect a single amplitude error and detect multiple phase damping errors.
Many new codes with good parameters are found, including nonadditive codes and
degenerate codes.Comment: 5 page
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