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}
}
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… 

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