Quantum Depth in the Random Oracle Model
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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) 𝖡𝖯𝖯^𝖰𝖭𝖢^𝖡𝖯𝖯≠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) 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. (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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