Reliable Quantum Memories with Unreliable Components

TL;DR

The paper investigates the feasibility of reliable quantum memories using unreliable components by formalizing stability, storage rate, and decoder complexity. It proves a strictly positive storage rate using quantum expander codes and connects storage problems to reliable quantum communication to derive upper bounds, including entropy-dissipation-based refinements. It also analyzes non-asymptotic limits under finite blocklength and time-constrained decoding, with numerical comparisons illustrating gaps between achievable and converse bounds. The work highlights both the potential of robust quantum memories and the practical challenges, pointing to future improvements in code constructions and decoding algorithms for tighter bounds and applicability to near-term devices.

Abstract

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.

Figures (3)

• Figure 1: Schematic of a quantum memory system.
• Figure 2: Geometric illustration of the optimization problem in \ref{['eqn:optimiz_depol']}. The values of the constants are set as follows: $\tau_r = 49\,\mu$s, $\tau_d = 95 \mu$s burnett2019decoherence, $1/\tau_c =$ 3.2 GHz, and $c_1 = 10$. The optimal point $(n^*, \tau^*) \approx (2.565 \times 10^6, 51.12~\text{ns})$ highlighted in the figure corresponds to the upper bound on storage capacity $\mathfrak{Q}^{(\epsilon, T)} \leq 0.9927320704447439$ for a target fidelity $\epsilon = 10^{-6}$.
• Figure 3: Comparison between the old and new upper bounds on classical storage capacity. The gap between the bounds is relatively significant in the range $\alpha \in [0, 0.2]$.

Theorems & Definitions (43)

• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Definition 5
• Remark 1
• Lemma 1
• proof
• Lemma 2
• proof