We introduce a model for a stacked quantum memory made with multi-qubit cells, inspired by multi-level flash cells in classical solid-state drive, and we design quantum error correction codes for this model by generalizing rank-metric codes to the quantum setting. Rank-metric codes are used to correct faulty links in classical communication networks. We propose a quantum generalization of Gabidulin codes, which is one of the most popular family of rank-metric codes, and we design a protocol to correct faults in Clifford circuits applied to a stacked quantum memory based on these codes. We envision potential applications to the optimization of stabilizer states and magic states factories, and to variational quantum algorithms. Further work is needed to make this protocol practical. It requires a hardware platform capable of hosting multi-qubit cells with low crosstalk between cells, a fault-tolerant syndrome extraction circuit for rank-metric codes and an associated efficient decoder.

翻译：我们提出了一种采用多量子比特单元构建的堆叠量子存储器模型，其灵感来源于经典固态硬盘中的多级闪存单元，并通过将秩度量码推广至量子领域，为该模型设计了量子纠错码。秩度量码在经典通信网络中用于校正故障链路。我们提出了Gabidulin码（最常用的秩度量码族之一）的量子推广方案，并基于这些码设计了一种协议，用于校正作用于堆叠量子存储器的Clifford电路中的故障。我们展望了该方法在稳定子态与魔术态工厂优化以及变分量子算法中的潜在应用。要使该协议具备实用性仍需进一步研究，这需要能够容纳多量子比特单元且单元间串扰较低的硬件平台、适用于秩度量码的容错校验子提取电路以及相应的高效解码器。