Promise problems challenge traditional oracle comparisons in computing theory
Promises should be taken seriously: On relativization with promise problems
Computational Complexity
Summary
Computers sometimes solve problems with help from special 'oracles' that give answers to tricky questions. When the problem has a 'promise'—a condition limiting which inputs are valid—these oracles behave differently, making it harder to compare computer models. The researchers studied two ways to handle oracles for promise problems and found key differences from regular problems, showing some assumptions do not apply when promises are involved. They also improved our understanding of quantum and classical computing hierarchies by carefully handling these promise-based oracles.
relativizationoraclepromise problemcomplexity classquantum computingclassical computingquantum-classical polynomial hierarchypostselectioncounting hierarchy
Authors
David Miloschewsky, Supartha Podder, Dorian Rudolph
Abstract
Relativization is concerned with comparing computational models with black-box access to an oracle. For promise problems, black-box access is not canonical due to inputs outside of the promise being unconstrained. We study two semantics for such access. Under robust queries, a machine must correctly answer regardless of the completion of the problem,, while loose access requires that the internal choices of a machine do not change based on off-promise queries. Our first result separates the language and promise settings. Namely, we construct an oracle $O$ such that $\mathsf{P}^O = \mathsf{BQP}^O = \mathsf{AWPP}^O$, but $\mathsf{PromiseBQP}^O\not\subseteq\mathsf{PromiseP}^O_{\mathsf{/poly}}$. In particular, $\mathsf{BPP}^O = \mathsf{BQP}^O$, but $\mathsf{PromiseBQP}^O \neq \mathsf{PromiseBPP}^O$, showing that results for languages need not transfer to promises. Next, we use loose queries to strengthen the upper bound on the Quantum-Classical Polynomial Hierarchy from $\mathsf{P}^{\mathsf{PP}^{\mathsf{PP}}}$ to $\mathsf{QCPH} \subseteq \mathsf{BP\cdot PP} \subseteq \mathsf{PromiseBPP}^{\mathsf{PP}}$. The same proof also shows $\mathsf{PP}^\mathsf{PromiseBQP} = \mathsf{PP}$. Additionally, we show that $\mathsf{PromiseBQP}$, even when given quantum advice, is self-low under robust queries. Finally, we exhibit an obstruction to transferring language-level counting results to promise classes. Although $\mathsf{AWPP}$ and $\mathsf{APP}$ are low for $\mathsf{PP}$, a corresponding promise analogue would collapse the counting hierarchy as $\mathsf{GapP} \subseteq \mathsf{FP}^{\mathsf{PromiseAWPP}}$. This motivates the introduction of $\mathsf{PromisePostBQP^*}$, which restricts $\mathsf{PostBQP}$ to input-indepencent postselection. By showing that it is low for \PP, we obtain $\mathsf{PP}^{\mathsf{PromiseYQP^*}} = \mathsf{PP}$.