Code-Visible Liouvillian Modes and Coherent Logical Errors in Quantum Codes

Abstract

Coherent Hamiltonian drifts and systematic errors can pass through quantum error correction differently than stochastic Pauli noise: they generate phase components of the logical channel that are lost under full Pauli twirling. We consider the Lindbladian dynamics of \(n\) physical qubits propagated through an effective logical channel of a fixed encoding, recovery, and decoding scheme. The work is methodological in character: standard ideas of observability and minimal realization from linear systems theory are applied to the channel \[ ω \mapsto Φ_t(ω) = \cD e^{t\cL}\cE(ω), \] where \(\cE\) encodes the logical state and \(\cD\) is the chosen recovery map with final decoding. The minimal code-visible space \(\cK_{\rm code}\) is defined as the Krylov closure of the decodable logical observables with respect to \(\cL^\dagger\); its observable factorization yields a minimal realization of the logical channel. In this form, the Knill--Laflamme conditions serve as a short-time consistency check, Pauli noise on stabilizer codes reduces to a chain on syndrome-logical classes, and the choice of recovery map acts as a spectral filter on the visible modes. For quantum memory, the same language yields an estimate of the storage time in terms of the dominant visible eigenvalues of the logical channel of the correction cycle. Particular attention is paid to the spectral signatures of coherent noise and to how these signatures depend on the chosen decoder.