Quantum Depth in the Random Oracle Model

@article{Arora2022QuantumDI,
  title={Quantum Depth in the Random Oracle Model},
  author={Atul Singh Arora and Andrea Coladangelo and Matthew Coudron and Alexandru Gheorghiu and Uttam Singh and Hendrik Waldner},
  journal={ArXiv},
  year={2022},
  volume={abs/2210.06454}
}
We give a comprehensive characterization of the computational power of shallow quantum circuits combined with classical computation. Specifically, for classes of search problems, we show that the following statements hold, relative to a random oracle: (a) $\mathsf{BPP}^{\mathsf{QNC}^{\mathsf{BPP}}} \neq \mathsf{BQP}$. This refutes Jozsa's conjecture [QIP 05] in the random oracle model. As a result, this gives the first instantiatable separation between the classes by replacing the oracle with a…