Abstract
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) BPPQNCBPP ≠ BQP. This refutes Jozsa’s conjecture 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) BPPQNC ⊈ QNCBPP and QNCBPP ⊈ BPPQNC. 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 BPPQNC and BPPQNC are both strictly contained in BPPQNCBPP. (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. Show more
Publication status
publishedExternal links
Book title
STOC 2023: Proceedings of the 55th Annual ACM Symposium on Theory of ComputingPages / Article No.
Publisher
Association for Computing MachineryEvent
Subject
Hybrid classical-quantum models of computation; proof of quantum depth; random oracle modelRelated publications and datasets
Is new version of: https://doi.org/10.3929/ethz-b-000584642
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