With the rapid advances in quantum computer architectures and the emerging prospect of large-scale quantum memory, it is becoming essential to classically verify that remote devices genuinely allocate the promised quantum memory with specified number of qubits and coherence time. In this paper, we introduce a new concept, proofs of quantum memory (PoQM). A PoQM is an interactive protocol between a classical probabilistic polynomial-time (PPT) verifier and a quantum polynomial-time (QPT) prover over a classical channel where the verifier can verify that the prover has possessed a quantum memory with a certain number of qubits during a specified period of time. PoQM generalize the notion of proofs of quantumness (PoQ) [Brakerski, Christiano, Mahadev, Vazirani, and Vidick, JACM 2021]. Our main contributions are a formal definition of PoQM and its constructions based on hardness of LWE. Specifically, we give two constructions of PoQM. The first is of a four-round and has negligible soundness error under subexponential-hardness of LWE. The second is of a polynomial-round and has inverse-polynomial soundness error under polynomial-hardness of LWE. As a lowerbound of PoQM, we also show that PoQM imply one-way puzzles. Moreover, a certain restricted version of PoQM implies quantum computation classical communication (QCCC) key exchange.

翻译：随着量子计算机架构的快速发展和大规模量子存储这一新兴前景的出现，经典地验证远程设备是否真实分配了所承诺的、具有指定量子比特数和相干时间的量子存储，正变得至关重要。本文引入了一个新概念——量子存储证明（PoQM）。PoQM是一种在经典概率多项式时间（PPT）验证者和量子多项式时间（QPT）证明者之间通过经典信道进行的交互协议，验证者可以借此验证证明者在指定时间段内确实拥有具有特定量子比特数的量子存储。PoQM推广了量子性证明（PoQ）的概念 [Brakerski, Christiano, Mahadev, Vazirani, and Vidick, JACM 2021]。我们的主要贡献是给出了PoQM的形式化定义，并基于LWE问题的困难性构造了PoQM协议。具体而言，我们给出了两种PoQM构造。第一种是四轮协议，在LWE具有亚指数级困难性的假设下，其可靠性误差可忽略。第二种是多项式轮协议，在LWE具有多项式级困难性的假设下，其可靠性误差为逆多项式。作为PoQM的下界，我们还证明了PoQM蕴含单向谜题。此外，某个受限版本的PoQM蕴含量子计算经典通信（QCCC）密钥交换。