Permutation-invariant quantum codes

Abstract

A quantum code is a subspace of a Hilbert space of a physical system chosen to be correctable against a given class of errors, where information can be encoded. Ideally, the quantum code lies within the ground space of the physical system. When the physical model is the Heisenberg ferromagnet in the absence of an external magnetic field, the corresponding ground-space contains all permutation-invariant states. We use techniques from combinatorics and operator theory to construct families of permutation-invariant quantum codes. These codes have length proportional to $t^2$; one family of codes perfectly corrects arbitrary weight $t$ errors, while the other family of codes approximately correct $t$ spontaneous decay errors. The analysis of our codes' performance with respect to spontaneous decay errors utilizes elementary matrix analysis, where we revisit and extend the quantum error correction criterion of Knill and Laflamme, and Leung, Chuang, Nielsen and Yamamoto.

Figures are still available in this version. Introduction is revised, elementary proofs removed, and paper is close to its final form

References in corpus (10)

• Subsystem fault tolerance with the Bacon-Shor code
• Entanglement and permutational symmetry
• Permutationally invariant state reconstruction
• Quantum accuracy threshold for concatenated distance-3 codes
• Entanglement criteria for Dicke states
• Robust Quantum Error Correction via Convex Optimization
• Optimal quantum tomography of permutationally invariant qubits
• Nonadditive quantum error correcting codes adapted to the ampltitude damping channel
• Ordered Measurements of Permutationally-Symmetric Qubit Strings
• Truncated channel representations for coupled harmonic oscillators

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