Power of quantum computing with restricted postselections.
Tomoyuki Morimae, Harumichi Nishimura · arXiv (Cornell University) · 2015
We consider restricted versions of ${\rm postBQP}$ where postselection probabilities can be efficiently calculated by classical or quantum computers. We show that such restricted ${\rm postBQP}$ classes are in ${\rm AWPP}(\subseteq{\rm PP}={\rm postBQP})$. This result suggests that postselecting an event with an exponentially small probability does not necessarily boost ${\rm BQP}$ to PP. The best upperbound of ${\rm BQP}$ is ${\rm AWPP}$, and therefore restricted ${\rm postBQP}$ classes are other examples of complexity classes slightly above ${\rm BQP}$. In [C. M. Lee and J. Barrett, arXiv:1412.8671], it was shown that the computational capacity of a general probabilistic theory which satisfies the tomographic locality is in ${\rm AWPP}$. Our result therefore implies that quantum physics with restricted postselections is an example of a super quantum theory which seems to be outside of their general probabilistic theory but its computational capacity is also in ${\rm AWPP}$. Although it is physically natural to expect that such restricted ${\rm postBQP}$ classes are not equivalent to BQP, we show that ${\rm UP}\cap{\rm coUP}$, which is unlikely to be in BQP, is contained in the restricted ${\rm postBQP}$ classes. Finally, we also consider another restricted class of ${\rm postBQP}$, where the postselected probability depends only on the size of inputs. We show that this restricted version is contained in ${\rm APP}$, where ${\rm AWPP}\subseteq{\rm APP}\subseteq{\rm PP}={\rm postBQP}$.