多项式空间中的酉复杂度
Unitary complexity in polynomial space
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中文总结 AI 辅助
本文证明若量子承诺存在，则酉综合问题无多项式时间解或BPP≠NEXP，并提出酉复杂度类unitaryP和unitaryPSPACE的新定义及其等价刻画。
中文摘要 AI 辅助
我们证明，如果量子承诺存在，那么要么酉综合问题没有多项式时间解，要么 $\mathsf{BPP} \neq \mathsf{NEXP}$。因此，无条件证明量子承诺的存在将需要回答复杂性理论中至少两个长期悬而未决的问题之一。我们将主要结果证明为一个更一般引理的推论，该引理表明 $\mathsf{unitaryPSPACE}$ 中的每个酉变换要么不能相对于任何经典预言机被高效综合，要么可以借助 $\mathsf{NEXP}$ 搜索问题的预言机被高效综合。我们的引理还有其他值得注意的推论，包括某些涉及 $\mathsf{unitaryPSPACE}$ 的预言机分离将意味着突破性的经典下界，如 $\mathsf{NC} \neq \mathsf{NP}$。在此过程中，我们为酉复杂度类 $\mathsf{unitaryP}$ 和 $\mathsf{unitaryPSPACE}$ 提出了新的定义。我们的修改解决了先前工作中提出的定义所面临的最大概念性问题，并引出了优雅的证明。我们同时研究了擦除垃圾的实现和允许垃圾的实现，因为我们无法排除这两种定义可能不同的可能性。尽管如此，我们表明这两种定义都可以被视为彼此的特例。我们还展示了我们的定义在许多其他方面的稳健性。例如，我们证明 $\mathsf{unitaryPSPACE}$ 有一个等价的刻画：其条目可以在多项式空间内以任意精度计算的酉变换集合。因此，我们推断 $\mathsf{unitaryPSPACE}$ 通常可以擦除垃圾，这一结果相对于酉预言机被证明是失败的。
英文摘要
We show that if quantum commitments exist, then either there is no polynomial-time solution to the unitary synthesis problem, or $\mathsf{BPP} \neq \mathsf{NEXP}$. Thus, showing unconditionally that quantum commitments exist would require answering at least one of two longstanding open questions in complexity theory. We prove our main result as a consequence of a more general lemma, which shows that every unitary in $\mathsf{unitaryPSPACE}$ either cannot be synthesized efficiently relative to any classical oracle, or can be synthesized efficiently with an oracle for $\mathsf{NEXP}$ search problems. Our lemma has other noteworthy consequences, including that certain oracle separations involving $\mathsf{unitaryPSPACE}$ would imply breakthrough classical lower bounds such as $\mathsf{NC} \neq \mathsf{NP}$. Along the way, we propose new definitions for the unitary complexity classes $\mathsf{unitaryP}$ and $\mathsf{unitaryPSPACE}$. Our changes address the biggest conceptual issues with definitions suggested in prior work, and lead to elegant proofs. We study both implementations that erase garbage and implementations that allow it, because we cannot rule out the possibility that the two definitions differ. Nevertheless, we show that both definitions can be viewed as special cases of each other. We also showcase many other ways in which our definitions are robust. For example, we show that $\mathsf{unitaryPSPACE}$ has an equivalent characterization as the set of unitary transformations whose entries can be computed to arbitrary precision in polynomial space. Consequently, we deduce that $\mathsf{unitaryPSPACE}$ can generically erase garbage, a result that provably fails relative to unitary oracles.
发表机构
- University of Texas at Austin（德克萨斯大学奥斯汀分校）
- Princeton University（普林斯顿大学）
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