• DOI:10.48550/arXiv.2210.06454
• Corpus ID: 252846391

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}
}

• A. S. Arora, A. Coladangelo, +3 authors Hendrik Waldner
• Published 12 October 2022
• Mathematics, Computer Science
• ArXiv

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 cryptographic hash function, yielding a resolution to one of Aaronson's ten semi-grand challenges in quantum computing. (b) $\mathsf{BPP}^{\mathsf{QNC}} \nsubseteq \mathsf{QNC}^{\mathsf{BPP}}$ and $\mathsf{QNC}^{\mathsf{BPP}} \nsubseteq \mathsf{BPP}^{\mathsf{QNC}}$. This shows that there is a subtle interplay between classical computation and shallow quantum computation. In fact, for the second separation, we establish that, for some problems, the ability to perform adaptive measurements in a single shallow quantum circuit, is more useful than the ability to perform polynomially many shallow quantum circuits without adaptive measurements. (c) There exists a 2-message proof of quantum depth protocol. Such a protocol allows a classical verifier to efficiently certify that a prover must be performing a computation of some minimum quantum depth. Our proof of quantum depth can be instantiated using the recent proof of quantumness construction by Yamakawa and Zhandry [STOC 22].