Pseudorandom Quantum States (PRS) were introduced by Ji, Liu and Song as quantum analogous to Pseudorandom Generators. They are an ensemble of states efficiently computable but computationally indistinguishable from Haar random states. Subsequent works have shown that some cryptographic primitives can be constructed from PRSs. Moreover, recent classical and quantum oracle separations of PRS from One-Way Functions strengthen the interest in a purely quantum alternative building block for quantum cryptography, potentially weaker than OWFs. However, our lack of knowledge of extending or shrinking the number of qubits of the PRS output still makes it difficult to reproduce some of the classical proof techniques and results. Short-PRSs, that is PRSs with logarithmic size output, have been introduced in the literature along with cryptographic applications, but we still do not know how they relate to PRSs. Here we answer half of the question, by showing that it is not possible to shrink the output of a PRS from polynomial to logarithmic qubit length while still preserving the pseudorandomness property, in a relativized way. More precisely, we show that relative to Kretschmer's quantum oracle (TQC 2021) short-PRSs cannot exist (while PRSs exist, as shown by Kretschmer's work).

翻译：伪随机量子态（PRS）由Ji、Liu和Song提出，作为伪随机生成器的量子模拟。这类态构成一个可有效计算但计算上与Haar随机态不可区分的态系综。后续研究表明，某些密码学原语可由PRS构造。此外，近期经典与量子预言机分离PRS与单向函数的研究，强化了将纯量子替代构建模块（可能弱于OWF）应用于量子密码学的兴趣。然而，我们关于扩展或压缩PRS输出量子比特数的知识匮乏，仍使得复现某些经典证明技术与结果存在困难。短输出PRS（即输出具有对数规模长度的PRS）已随密码学应用一同被引入文献，但我们尚不清楚其与PRS的关系。本文通过相对化方式证明无法在保持伪随机性的前提下将PRS输出从多项式长度压缩至对数量子比特长度，从而回答了这一问题的半数。具体而言，我们证明相对于Kretschmer的量子预言机（TQC 2021），短输出PRS不能存在（而PRS可存在，如Kretschmer工作所示）。