The Acrobatics of BQP
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One can fix the randomness used by a randomized algorithm, but there is no analogous notion of fixing the quantumness used by a quantum algorithm. Underscoring this fundamental difference, we show that, in the black-box setting, the behavior of quantum polynomial-time (𝖡𝖰𝖯) can be remarkably decoupled from that of classical complexity classes like 𝖭𝖯. Specifically: -There exists an oracle relative to which 𝖭𝖯^𝖡𝖰𝖯⊄𝖡𝖰𝖯^𝖯𝖧, resolving a 2005 problem of Fortnow. As a corollary, there exists an oracle relative to which 𝖯=𝖭𝖯 but 𝖡𝖰𝖯≠𝖰𝖢𝖬𝖠. -Conversely, there exists an oracle relative to which 𝖡𝖰𝖯^𝖭𝖯⊄𝖯𝖧^𝖡𝖰𝖯. -Relative to a random oracle, 𝖯𝖯=𝖯𝗈𝗌𝗍𝖡𝖰𝖯 is not contained in the "𝖰𝖬𝖠 hierarchy" 𝖰𝖬𝖠^𝖰𝖬𝖠^𝖰𝖬𝖠^⋯. -Relative to a random oracle, Σ_k+1^𝖯⊄𝖡𝖰𝖯^Σ_k^𝖯 for every k. -There exists an oracle relative to which 𝖡𝖰𝖯=𝖯^# 𝖯 and yet 𝖯𝖧 is infinite. -There exists an oracle relative to which 𝖯=𝖭𝖯≠𝖡𝖰𝖯=𝖯^# 𝖯. To achieve these results, we build on the 2018 achievement by Raz and Tal of an oracle relative to which 𝖡𝖰𝖯⊄𝖯𝖧, and associated results about the Forrelation problem. We also introduce new tools that might be of independent interest. These include a "quantum-aware" version of the random restriction method, a concentration theorem for the block sensitivity of 𝖠𝖢^0 circuits, and a (provable) analogue of the Aaronson-Ambainis Conjecture for sparse oracles.
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