Quantum Depth in the Random Oracle Model

We give a comprehensive characterisation 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) BPP QNC BPP ≠ BQP. This refutes Jozsa's conjecturein 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) BPP QNC ⊈ QNC BPP and QNC BPP ⊈ BPP 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. We also show that BPP QNC and QNC BPP are both strictly contained in BPP QNC BPP . (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 by Yamakawa and Zhandry.