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. For a logical observable read through a physical meter M at measurement strength γ, we separate the induced logical decoherence into a conjugate cost — dephasing of the partner logical operators, which is bounded below by the information-gain rate, saturating it for quantum-limited readouts — and a self cost: corruption of the very operator being read. We derive the readability limit, a lower bound on the self cost set by the code structure: Γ_self ≥ γ max(2κ², ½ε²_KL) − O(γ²/g), where κ is the within-codespace off-diagonal action of M and ε_KL is the logical-confusion (off-diagonal) part of the Knill–Laflamme residual of the measurement back-action E = P⊥MP. The within-code term is an exact operator identity; the leak term is a recovery-independent converse from the code's Knill–Laflamme data, established as a rate at d = 2 for direct-return stabilizers and supported numerically beyond them, saturating at the cells where the within-code term dominates and loose by up to two orders of magnitude where it does not. The which-path (diagonal) part of the residual, by contrast, feeds only the conjugate cost. A readout corrupts what it reads at least to the extent that it rotates the logical within the code space or its back-action falls outside the code's correctable error set. We verify the bound cell-by-cell across cat and Gottesman–Kitaev–Preskill (GKP) codes and survey number-type codes, which within the Hermitian meter menu considered here are essentially unreadable-while-protected: their back-action is a Knill–Laflamme-uncorrectable error, so the confusion floor is maximal. The canonical phase measurement conventionally used to read number-phase codes is a POVM — the spectrum of the non-Hermitian Susskind–Glogower phase operator — and so lies outside the present framework; whether the bound extends to POVM meters is left open. Among the readable families, the 7.8× readout advantage of cat over GKP qubits we find at matched photon number decomposes into two contributions of distinct origin: a meter-scale gain — the cat's protected variable is an unbounded quadrature whose signal grows with photon number, against GKP's bounded modular one — and a ≈1.5× residual ratio of self-disturbance rates (at a single photon number) that a meter-normalized readability isolates. We map the resulting readability–protection trade-off across the GKP lattice family. The readability limit turns readout quality into a code-design objective and gives a recipe for self-reporting memories — engineer the logical readout so that its back-action is an error the code already corrects. Within the standard bosonic menu this is achieved exactly by symmetry (the cat parity readout is exactly quantum-non-demolition) and, for tuned modular meters, to the controlled leading order.

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.