30,039 research outputs found
Optimum Quantum Error Recovery using Semidefinite Programming
Quantum error correction (QEC) is an essential element of physical quantum
information processing systems. Most QEC efforts focus on extending classical
error correction schemes to the quantum regime. The input to a noisy system is
embedded in a coded subspace, and error recovery is performed via an operation
designed to perfectly correct for a set of errors, presumably a large subset of
the physical noise process. In this paper, we examine the choice of recovery
operation. Rather than seeking perfect correction on a subset of errors, we
seek a recovery operation to maximize the entanglement fidelity for a given
input state and noise model. In this way, the recovery operation is optimum for
the given encoding and noise process. This optimization is shown to be
calculable via a semidefinite program (SDP), a well-established form of convex
optimization with efficient algorithms for its solution. The error recovery
operation may also be interpreted as a combining operation following a quantum
spreading channel, thus providing a quantum analogy to the classical diversity
combining operation.Comment: 7 pages, 3 figure
Simple approach to approximate quantum error correction based on the transpose channel
We demonstrate that there exists a universal, near-optimal recovery map—the transpose channel—for approximate quantum error-correcting codes, where optimality is defined using the worst-case fidelity. Using the transpose channel, we provide an alternative interpretation of the standard quantum error correction (QEC) conditions and generalize them to a set of conditions for approximate QEC (AQEC) codes. This forms the basis of a simple algorithm for finding AQEC codes. Our analytical approach is a departure from earlier work relying on exhaustive numerical search for the optimal recovery map, with optimality defined based on entanglement fidelity. For the practically useful case of codes encoding a single qubit of information, our algorithm is particularly easy to implement
Channel-Optimized Quantum Error Correction
We develop a theory for finding quantum error correction (QEC) procedures
which are optimized for given noise channels. Our theory accounts for
uncertainties in the noise channel, against which our QEC procedures are
robust. We demonstrate via numerical examples that our optimized QEC procedures
always achieve a higher channel fidelity than the standard error correction
method, which is agnostic about the specifics of the channel. This demonstrates
the importance of channel characterization before QEC procedures are applied.
Our main novel finding is that in the setting of a known noise channel the
recovery ancillas are redundant for optimized quantum error correction. We show
this using a general rank minimization heuristic and supporting numerical
calculations. Therefore, one can further improve the fidelity by utilizing all
the available ancillas in the encoding block.Comment: 12 pages, 9 figure
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
Quantum Error Correction via Convex Optimization
We show that the problem of designing a quantum information error correcting
procedure can be cast as a bi-convex optimization problem, iterating between
encoding and recovery, each being a semidefinite program. For a given encoding
operator the problem is convex in the recovery operator. For a given method of
recovery, the problem is convex in the encoding scheme. This allows us to
derive new codes that are locally optimal. We present examples of such codes
that can handle errors which are too strong for codes derived by analogy to
classical error correction techniques.Comment: 16 page
Structured Near-Optimal Channel-Adapted Quantum Error Correction
We present a class of numerical algorithms which adapt a quantum error
correction scheme to a channel model. Given an encoding and a channel model, it
was previously shown that the quantum operation that maximizes the average
entanglement fidelity may be calculated by a semidefinite program (SDP), which
is a convex optimization. While optimal, this recovery operation is
computationally difficult for long codes. Furthermore, the optimal recovery
operation has no structure beyond the completely positive trace preserving
(CPTP) constraint. We derive methods to generate structured channel-adapted
error recovery operations. Specifically, each recovery operation begins with a
projective error syndrome measurement. The algorithms to compute the structured
recovery operations are more scalable than the SDP and yield recovery
operations with an intuitive physical form. Using Lagrange duality, we derive
performance bounds to certify near-optimality.Comment: 18 pages, 13 figures Update: typos corrected in Appendi
Concatenation of Error Avoiding with Error Correcting Quantum Codes for Correlated Noise Models
We study the performance of simple error correcting and error avoiding
quantum codes together with their concatenation for correlated noise models.
Specifically, we consider two error models: i) a bit-flip (phase-flip) noisy
Markovian memory channel (model I); ii) a memory channel defined as a memory
degree dependent linear combination of memoryless channels with Kraus
decompositions expressed solely in terms of tensor products of X-Pauli
(Z-Pauli) operators (model II). The performance of both the three-qubit bit
flip (phase flip) and the error avoiding codes suitable for the considered
error models is quantified in terms of the entanglement fidelity. We explicitly
show that while none of the two codes is effective in the extreme limit when
the other is, the three-qubit bit flip (phase flip) code still works for high
enough correlations in the errors, whereas the error avoiding code does not
work for small correlations. Finally, we consider the concatenation of such
codes for both error models and show that it is particularly advantageous for
model II in the regime of partial correlations.Comment: 16 pages, 3 figure
Exact Performance of Concatenated Quantum Codes
When a logical qubit is protected using a quantum error-correcting code, the
net effect of coding, decoherence (a physical channel acting on qubits in the
codeword) and recovery can be represented exactly by an effective channel
acting directly on the logical qubit. In this paper we describe a procedure for
deriving the map between physical and effective channels that results from a
given coding and recovery procedure. We show that the map for a concatenation
of codes is given by the composition of the maps for the constituent codes.
This perspective leads to an efficient means for calculating the exact
performance of quantum codes with arbitrary levels of concatenation. We present
explicit results for single-bit Pauli channels. For certain codes under the
symmetric depolarizing channel, we use the coding maps to compute exact
threshold error probabilities for achievability of perfect fidelity in the
infinite concatenation limit.Comment: An expanded presentation of the analytic methods and results from
quant-ph/0111003; 13 pages, 6 figure
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