Towards a unified framework for approximate quantum error correction

Recent work on approximate quantum error correction (QEC) has opened up the possibility of constructing subspace codes that protect information with high fidelity in scenarios where perfect error correction is impossible. Motivated by this, we investigate the problem of approximate subsystem codes. Subsystem codes extend the standard formalism of subspace QEC to codes in which only a subsystem within a subspace of states is used to store information in a noise-resilient fashion. Here, we demonstrate easily checkable sufficient conditions for the existence of approximate subsystem codes. Furthermore, for certain classes of subsystem codes and noise processes, we prove the efficacy of the transpose channel as a simple-to-construct recovery map that works nearly as well as the optimal recovery channel. This work generalizes our earlier approach [H.K. Ng and P. Mandayam, Phys. Rev. A 81 062342 (2010)] of using the transpose channel for approximate correction of subspace codes to the case of subsystem codes, and brings us closer to a unifying framework for approximate QEC.

[1]  I. L. Chuang,et al.  Approximate quantum error correction can lead to better codes , 1997 .

[2]  R F Werner,et al.  Iterative optimization of quantum error correcting codes. , 2005, Physical review letters.

[3]  Cédric Bény,et al.  General conditions for approximate quantum error correction and near-optimal recovery channels. , 2009, Physical review letters.

[4]  Ekert,et al.  Quantum Error Correction for Communication. , 1996 .

[5]  David Poulin,et al.  Operator quantum error correction , 2006, Quantum Inf. Comput..

[6]  Shinji Hara,et al.  Suboptimal quantum-error-correcting procedure based on semidefinite programming , 2005 .

[7]  D. Poulin.  Stabilizer formalism for operator quantum error correction. , 2005, Physical review letters.

[8]  M. Nielsen,et al.  Algebraic and information-theoretic conditions for operator quantum error correction , 2005, quant-ph/0506069.

[9]  A. Winter,et al.  Communications in Mathematical Physics Structure of States Which Satisfy Strong Subadditivity of Quantum Entropy with Equality , 2022 .

[10]  Daniel A Lidar,et al.  Robust quantum error correction via convex optimization. , 2008, Physical review letters.

[11]  Moe Z. Win,et al.  Channel-Adapted Quantum Error Correction for the Amplitude Damping Channel , 2007, IEEE Transactions on Information Theory.

[12]  Jon Tyson,et al.  Two-sided bounds on minimum-error quantum measurement, on the reversibility of quantum dynamics, and on maximum overlap using directional iterates , 2009, 0907.3386.

[13]  Charles H. Bennett,et al.  Mixed-state entanglement and quantum error correction. , 1996, Physical review. A, Atomic, molecular, and optical physics.

[14]  D. Bacon.  Operator quantum error-correcting subsystems for self-correcting quantum memories , 2005, quant-ph/0506023.

[15]  E. Knill,et al.  Reversing quantum dynamics with near-optimal quantum and classical fidelity , 2000, quant-ph/0004088.

[16]  E. Knill,et al.  Theory of quantum error-correcting codes , 1997 .

[17]  Prabha Mandayam,et al.  Simple approach to approximate quantum error correction based on the transpose channel , 2009, 0909.0931.

[18]  Panos Aliferis,et al.  Subsystem fault tolerance with the Bacon-Shor code. , 2007, Physical review letters.

[19]  D. Petz.  Monotonicity of quantum relative entropy revisited , 2002, quant-ph/0209053.

[20]  David Poulin,et al.  Characterizing the structure of preserved information in quantum processes. , 2007, Physical review letters.

[21]  D. Poulin,et al.  Information-preserving structures: A general framework for quantum zero-error information , 2010, 1006.1358.

[22]  David Poulin,et al.  Unified and generalized approach to quantum error correction. , 2004, Physical review letters.

[23]  Daniel A. Lidar,et al.  Theory of initialization-free decoherence-free subspaces and subsystems (14 pages) , 2005, quant-ph/0505051.