Quantum Depth in the Random Oracle Model

• URL: http://arxiv.org/abs/2210.06454v1
• Date: Wed, 12 Oct 2022 17:54:02 GMT
• Title: Quantum Depth in the Random Oracle Model
• Authors: Atul Singh Arora and Andrea Coladangelo and Matthew Coudron and Alexandru Gheorghiu and Uttam Singh and Hendrik Waldner
• Abstract summary: We give a comprehensive characterization of the computational power of shallow quantum circuits combined with classical computation. For some problems, the ability to perform adaptive measurements in a single shallow quantum circuit is more useful than the ability to perform many shallow quantum circuits without adaptive measurements.
• Score: 57.663890114335736
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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].

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