• DOI:10.48550/arXiv.2209.14530
• Corpus ID: 252596125

Low-Stabilizer-Complexity Quantum States Are Not Pseudorandom

@article{Grewal2022LowStabilizerComplexityQS,
  title={Low-Stabilizer-Complexity Quantum States Are Not Pseudorandom},
  author={Sabee Grewal and Vishnu Iyer and William Kretschmer and Daniel Liang},
  journal={ArXiv},
  year={2022},
  volume={abs/2209.14530}
}

• Sabee Grewal, Vishnu Iyer, +1 author Daniel Liang
• Published 29 September 2022
• Computer Science
• ArXiv

We show that quantum states with “low stabilizer complexity” can be efficiently distinguished from Haar-random. Specifically, given an n -qubit pure state | ψ i , we give an efficient algorithm that distinguishes whether | ψ i is (i) Haar-random or (ii) a state with stabilizer fidelity at least 1 k (i.e., has fidelity at least 1 k with some stabilizer state), promised that one of these is the case. With black-box access to | ψ i , our algorithm uses O (cid:0) k 12 log(1 /δ ) (cid:1) copies of | ψ i and O (cid:0) nk 12 log(1 /δ ) (cid:1) time to succeed with probability at least 1 − δ , and, with access to a state preparation unitary for | ψ i (and its inverse), O (cid:0) k 3 log(1 /δ ) (cid:1) queries and O (cid:0) nk 3 log(1 /δ ) (cid:1) time suffice. As a corollary, we prove that ω (log( n )) T -gates are necessary for any Clifford+ T circuit to prepare computationally pseudorandom quantum states, a first-of-its-kind lower bound.