Oracle separations for non-adaptive collapse-free quantum computing

Abstract

Aaronson et al. (2016) introduced the quantum complexity class $\mathrm{naCQP}$ (non-adaptive Collapse-free Quantum Polynomial time), also known as $\mathrm{PDQP}$ (Product Dynamical Quantum Polynomial time), aiming to define a class larger than $\mathrm{BQP}$ (Bounded-error Quantum Polynomial time), but not large enough to include $\mathrm{NP}$-complete problems. Aaronson et al. showed that $\mathrm{SZK}$ (Statistical Zero Knowledge) is contained in $\mathrm{naCQP}$ and that there is an oracle A for which ${\mathrm{NP}}^A⊈{\mathrm{naCQP}}^A$. We prove that: there is an oracle A for which $P^A={\mathrm{BQP}}^A={\mathrm{SZK}}^A={\mathrm{naCQP}}^A\ne {(\mathrm{UP}\cap \mathrm{coUP})}^A={\mathrm{EXP}}^A$, where $\mathrm{UP}$ (Unambiguous Polynomial time) is an important subclass of $\mathrm{NP}$; relative to an oracle A chosen uniformly at random, it holds ${(\mathrm{UP}\cap \mathrm{coUP})}^A⊈{\mathrm{naCQP}}^A$ with probability 1. Our results are a strengthening of the results by Bennett et al. (1997), Fortnow and Rogers (1999), Tamon and Yamakami (2001), and Aaronson et al. (2016).