量子预言机分离类型之间的区分
A Separation Between Types of Quantum Oracle Separations
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- University of Texas Austin（德克萨斯大学奥斯汀分校）
- Stanford University（斯坦福大学）
- Columbia University（哥伦比亚大学）
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中文总结 AI 辅助
本文引入量子相对化的元复杂度概念，对预言机万神殿家族进行分类，证明PostBQP与PreciseBQP在特定预言机下分离，在另一些下相等，凸显量子复杂度类相对化的微妙性。
中文摘要 AI 辅助
近期研究表明，量子预言机具有微妙的行为，因为访问逆、共轭或受控查询可以指数级地改变某些任务的查询复杂度。受这些工作的启发，我们引入了量子相对化的元复杂度概念。我们提出疑问：对于任意两个量子复杂度类，在哪些“类型”的量子预言机下它们是相等或分离的？不同的预言机万神殿（或量子预言机类型，例如酉 vs 态、多项式维 vs 超多项式维、在逆运算下封闭与否）基于它们在分离复杂度类方面的能力形成一个偏序集。此外，如果存在一对复杂度类，在来自万神殿A的预言机下被分离，但在来自万神殿B的所有预言机下等价，则两个预言机万神殿A和B是分离的。我们通过给出本文定义的预言机万神殿家族内的完整分类，证明这一元复杂度可以是非平凡的，该分类确定了哪些模型可以分离复杂度类$\mathsf{PostBQP}$和$\mathsf{PreciseBQP}$，后者是$\mathsf{BQP}$的指数精确变体。在非相对化设置中，这两个类都等于$\mathsf{PP}$。在我们的分类法中，相对于实数或多项式维酉预言机以及当提供逆或共轭访问时，它们保持相等。相反，我们给出了相对于超多项式维的仅前向复数对角酉的分离，以及相对于单量子比特态制备预言机的分离。我们将此视为预言机元复杂度的测试案例，它强调了量子复杂度类相对化中固有的微妙性。
英文摘要
Recent works have demonstrated that quantum oracles have subtle behavior, as access to inverse, conjugate or controlled queries can exponentially change the query complexity of certain tasks. Inspired by these works, we introduce the notion of meta-complexity of quantum relativization. We ask: for any two quantum complexity classes, under which "types" of quantum oracles are they equal or separated? Different pantheons of oracles (or quantum oracle types, e.g. unitary vs state, poly- vs superpoly-dimensional, closed under inverse or not) form a partially ordered set based on their power in separating complexity classes. Moreover, two oracle pantheons A and B are separated if there exists a pair of complexity classes that are separated under an oracle from pantheon A but yet the complexity classes are equivalent under all oracles from pantheon B. We show that this meta-complexity can be nontrivial by giving a complete classification, within the family of oracle pantheons defined in this paper, of which models can separate the complexity classes $\mathsf{PostBQP}$ and $\mathsf{PreciseBQP}$, the exponentially precise variant of $\mathsf{BQP}$. Both classes equal $\mathsf{PP}$ in the unrelativized setting. Within our taxonomy, they remain equal relative to real or polynomial-dimensional unitary oracles and whenever inverse or conjugate access is supplied. In contrast, we give a separation relative to forward-only complex diagonal unitaries of superpolynomial dimension, as well as a separation relative to single-qubit state-preparation oracles. We view this as a test case for the meta-complexity of oracles which underscores the subtlety inherent to the relativization of quantum complexity classes.
