Channel-Adapted Quantum Error Correction
arXiv:0706.3400
Abstract
Quantum error correction (QEC) is an essential concept for any quantum information processing device. Typically, QEC is designed with minimal assumptions about the noise process; this generic assumption exacts a high cost in efficiency and performance. In physical systems, errors are not likely to be arbitrary; rather we will have reasonable models for the structure of quantum decoherence. We may choose quantum error correcting codes and recovery operations that specifically target the most likely errors. We present a convex optimization method to determine the optimal (in terms of average entanglement fidelity) recovery operation for a given channel, encoding, and information source. This is solvable via a semidefinite program (SDP). We present computational algorithms to generate near-optimal recovery operations structured to begin with a projective syndrome measurement. These structured operations are more computationally scalable than the SDP required for computing the optimal; we can thus numerically analyze longer codes. Using Lagrange duality, we bound the performance of the structured recovery operations and show that they are nearly optimal in many relevant cases. We present two classes of channel-adapted quantum error correcting codes specifically designed for the amplitude damping channel. These have significantly higher rates with shorter block lengths than corresponding generic quantum error correcting codes. Both classes are stabilizer codes, and have good fidelity performance with stabilizer recovery operations. The encoding, syndrome measurement, and syndrome recovery operations can all be implemented with Clifford group operations.
Ph.D. dissertation, 163 pages, 38 figures
References in corpus (18)
- Mixed State Entanglement and Quantum Error Correction
- Good Quantum Error-Correcting Codes Exist
- A Class of Quantum Error-Correcting Codes Saturating the Quantum Hamming Bound
- A Theory of Fault-Tolerant Quantum Computation
- Quantum Error Correction and Orthogonal Geometry
- Distinguishing separable and entangled states
- Reversing quantum dynamics with near-optimal quantum and classical fidelity
- Approximate quantum error correction can lead to better codes
- Designing Optimal Quantum Detectors Via Semidefinite Programming
- Procedures for Converting among Lindblad, Kraus and Matrix Representations of Quantum Dynamical Semigroups
- Detecting multipartite entanglement
- Optimal Non-Universally Covariant Cloning
- Finding optimal strategies for minimum-error quantum-state discrimination
- An Introduction to Quantum Error Correction
- Mixed quantum state detection with inconclusive results
- Separable Multipartite Mixed States - Operational Asymptotically Necessary and Sufficient Conditions
- Optimal quantum detectors for unambiguous detection of mixed states
- Improved quantum algorithms for the ordered search problem via semidefinite programming
Cited by in corpus (10)
- Performance and structure of single-mode bosonic codes
- Quantum Channel Construction with Circuit Quantum Electrodynamics
- Approximate quantum error correction for generalized amplitude damping errors
- Channel-Optimized Quantum Error Correction
- Achieving fault tolerance against amplitude-damping noise
- Error rates of Belavkin weighted quantum measurements and a converse to Holevo's asymptotic optimality theorem
- Nonadditive quantum error correcting codes adapted to the ampltitude damping channel
- Robustness-optimized quantum error correction
- Quantum Error Correction Code in the Hamiltonian Formulation
- Channel-Adapted Quantum Error Correction for the Amplitude Damping Channel