How gently can a protected quantum memory be read? Rate limits on logical readout from the structure of the code

Abstract

Quantum error-correcting codes protect logical information by hiding it from the environment; yet the same information must remain accessible to the experimenter. We quantify this tension for weak continuous readout of bosonic codes, to leading order in the measurement rate and in the single-mode Markovian limit. Reading a logical observable through a physical meter M at strength γ induces a logical decoherence that separates into two channels of distinct origin: a conjugate cost, dephasing of the partner logical operators, bounded below by the information-gain rate and saturating it for quantum-limited readouts; and a self cost, corruption of the very operator being read. Our central result is that the self cost is set by the structure of the code. Within the code space this is an exact operator identity, Γ_self ≥ 2γκ², nonperturbative in the within-codespace off-diagonal action κ of M and free of further hypotheses. A second and weaker contribution follows from the Knill–Laflamme residual of the back-action E = P⊥MP: leakage the code cannot correct forces any recovery, at any speed, to confuse the logical states, giving Γ_self ≥ ½γε²_KL. Together, Γ_self ≥ γ max(2κ², ½ε²_KL) − O(γ²/g) — the readability limit. The second term is a recovery-independent converse established as a rate at d = 2 for direct-return stabilizers and supported numerically beyond them; it saturates where the within-code term dominates and is loose by up to a factor of ~300 where it does not. The which-path part of the residual, by contrast, feeds only the conjugate cost. We verify both terms cell by cell across cat and Gottesman–Kitaev–Preskill (GKP) codes and map the readability–protection trade-off across the GKP lattice family; number-type codes, within the Hermitian meter menu considered here, are essentially unreadable-while-protected. The canonical phase measurement conventionally used to read number-phase codes is a POVM and lies outside this framework; whether the bound extends to POVM meters is left open. The consequence is that readout quality becomes a code-design objective: engineer the logical readout so that its back-action is an error the code already corrects — achieved exactly by symmetry for the cat parity readout, and to the controlled leading order for tuned modular meters.

Where it stops

Leading order in the measurement rate, single-mode Markovian limit, and the framework cannot express canonical phase measurement — which is how number-phase codes are conventionally read.