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Quantum memory systems are vital in quantum information processing for dependable storage and retrieval of quantum states. Inspired by classical reliability theories that synthesize reliable computing systems from unreliable components, we formalize the problem of reliable storage of quantum information using noisy components. We introduce the notion of stable quantum memories and define the storage rate as the ratio of the number of logical qubits to the total number of physical qubits, as well as the circuit complexity of the decoder, which includes both quantum gates and measurements. We demonstrate that a strictly positive storage rate can be achieved by constructing a quantum memory system with quantum expander codes. Moreover, by reducing the reliable storage problem to reliable quantum communication, we provide upper bounds on the achievable storage capacity. In the case of physical qubits corrupted by noise satisfying hypercontractivity conditions, we provide a tighter upper bound on storage capacity using an entropy dissipation argument. Furthermore, observing that the time complexity of the decoder scales non-trivially with the number of physical qubits, achieving asymptotic rates may not be possible due to the induced dependence of the noise on the number of physical qubits. In this constrained non-asymptotic setting, we derive upper bounds on storage capacity using finite blocklength communication bounds. Finally, we numerically analyze the gap between upper and lower bounds in both asymptotic and non-asymptotic cases, and provide suggestions to tighten the gap.
Nayak et al. (Sat,) studied this question.