Recovery With Incomplete Knowledge: Fundamental Bounds on Real-Time Quantum Memories

The recovery of fragile quantum states from decoherence is the basis of building a quantum memory, with applications ranging from quan-tum communications to quantum computing. Many recovery techniques, such as quantum error correction, rely on the apriori knowledge of the environment noise parameters to achieve their best performance. However, such param-eters are likely to drift in time in the context of implementing long-time quantum memories. This necessitates using a "spectator" system, which estimates the noise parameter in real-time, then feedforwards the outcome to the re-covery protocol as a classical side-information. The memory qubits and the spectator system hence comprise the building blocks for a real-time (i.e. drift-adapting) quantum memory. In this article, I consider spectator-based (in-complete knowledge) recovery protocols as a real-time parameter estimation problem (gen-erally with nuisance parameters present), fol-lowed by the application of the "best-guess" recovery map to the memory qubits, as in-formed by the estimation outcome. I present information-theoretic and metrological bounds on the performance of this protocol, quan-tified by the diamond distance between the "best-guess" recovery and optimal recovery outcomes, thereby identifying the cost of adap-tation in real-time quantum memories. Fi-nally, I provide fundamental bounds for multi -cycle recovery in the form of recurrence in-equalities. The latter suggests that incomplete knowledge of the noise could be an advantage, as errors from various cycles can cohere. These results are illustrated for the approximate [4,1] code of the amplitude-damping channel and re-lations to various fields are discussed.