15 research outputs found
High Performance Single-Error-Correcting Quantum Codes for Amplitude Damping
Original manuscript July 29, 2009We construct families of high performance quantum amplitude damping codes. All of our codes are nonadditive and most modestly outperform the best possible additive codes in terms of encoded dimension. One family is built from nonlinear error-correcting codes for classical asymmetric channels, with which we systematically construct quantum amplitude damping codes with parameters better than any prior construction known for any block length n ≥ 8 except n=2r-1. We generalize this construction to employ classical codes over GF(3) with which we numerically obtain better performing codes up to length 14. Because the resulting codes are of the codeword stabilized (CWS) type, conceptually simple (though potentially computationally expensive) encoding and decoding circuits are available
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
Reliable channel-adapted error correction: Bacon-Shor code recovery from amplitude damping
We construct two simple error correction schemes adapted to amplitude damping
noise for Bacon-Shor codes and investigate their prospects for fault-tolerant
implementation. Both consist solely of Clifford gates and require far fewer
qubits, relative to the standard method, to achieve correction to a desired
order in the damping rate. The first, employing one-bit teleportation and
single-qubit measurements, needs only one fourth as many physical qubits, while
the second, using just stabilizer measurements and Pauli corrections, needs
only half. We show that existing fault-tolerance methods can be employed for
the latter, while the former can be made to avoid potential catastrophic errors
and can easily cope with damping faults in ancilla qubits.Comment: 8 pages, 1 figur
Concatenated Codes for Amplitude Damping
We discuss a method to construct quantum codes correcting amplitude damping
errors via code concatenation. The inner codes are chosen as asymmetric
Calderbank-Shor-Steane (CSS) codes. By concatenating with outer codes
correcting symmetric errors, many new codes with good parameters are found,
which are better than the amplitude damping codes obtained by any previously
known construction.Comment: 5 page
Linear programming bounds for quantum amplitude damping codes
Given that approximate quantum error-correcting (AQEC) codes have a
potentially better performance than perfect quantum error correction codes, it
is pertinent to quantify their performance. While quantum weight enumerators
establish some of the best upper bounds on the minimum distance of quantum
error-correcting codes, these bounds do not directly apply to AQEC codes.
Herein, we introduce quantum weight enumerators for amplitude damping (AD)
errors and work within the framework of approximate quantum error correction.
In particular, we introduce an auxiliary exact weight enumerator that is
intrinsic to a code space and moreover, we establish a linear relationship
between the quantum weight enumerators for AD errors and this auxiliary exact
weight enumerator. This allows us to establish a linear program that is
infeasible only when AQEC AD codes with corresponding parameters do not exist.
To illustrate our linear program, we numerically rule out the existence of
three-qubit AD codes that are capable of correcting an arbitrary AD error.Comment: 5 page
Reliable channel-adapted error correction: Bacon-Shor code recovery from amplitude damping
We construct two simple error correction schemes adapted to amplitude damping noise for Bacon-Shor codes and investigate their prospects for fault-tolerant implementation. Both consist solely of Clifford gates and require far fewer qubits, relative to the standard method, to achieve correction to a desired order in the damping rate. The first, employing on
A simple comparative analysis of exact and approximate quantum error correction
We present a comparative analysis of exact and approximate quantum error
correction by means of simple unabridged analytical computations. For the sake
of clarity, using primitive quantum codes, we study the exact and approximate
error correction of the two simplest unital (Pauli errors) and nonunital
(non-Pauli errors) noise models, respectively. The similarities and differences
between the two scenarios are stressed. In addition, the performances of
quantum codes quantified by means of the entanglement fidelity for different
recovery schemes are taken into consideration in the approximate case. Finally,
the role of self-complementarity in approximate quantum error correction is
briefly addressed.Comment: 29 pages, 1 figure, improved v2; accepted for publication in Open
Systems and Information Dynamics (2014
New Constructions of Codes for Asymmetric Channels via Concatenation
We present new constructions of codes for asymmetric channels for both binary and nonbinary alphabets, based on methods of generalized code concatenation. For the binary asymmetric channel, our methods construct nonlinear single-error-correcting codes from ternary outer codes. We show that some of the Varshamov-Tenengol'ts-Constantin-Rao codes, a class of binary nonlinear codes for this channel, have a nice structure when viewed as ternary codes. In many cases, our ternary construction yields even better codes. For the nonbinary asymmetric channel, our methods construct linear codes for many lengths and distances which are superior to the linear codes of the same length capable of correcting the same number of symmetric errors