Quantum computing introduces many problems rooted in physics, asking to compute information from input quantum states. Determining the complexity of these problems has implications for both computer science and physics. However, as existing complexity theory primarily addresses problems with classical inputs and outputs, it lacks the framework to fully capture the complexity of quantum-input problems. This gap is relevant when studying the relationship between quantum cryptography and complexity theory, especially within Impagliazzo's five worlds framework, as characterizing the security of quantum cryptographic primitives requires complexity classes for problems involving quantum inputs. To bridge this gap, we examine the complexity theory of quantum promise problems, which determine if input quantum states have certain properties. We focus on complexity classes p/mBQP, p/mQ(C)MA, $\mathrm{p/mQSZK_{hv}}$, p/mQIP, and p/mPSPACE, where "p/mC" denotes classes with pure (p) or mixed (m) states corresponding to any classical class C. We establish structural results, including complete problems, search-to-decision reductions, and relationships between classes. Notably, our findings reveal differences from classical counterparts, such as p/mQIP $\neq$ p/mPSPACE and $\mathrm{mcoQSZK_{hv}} \neq \mathrm{mQSZK_{hv}}$. As an application, we apply this framework to cryptography, showing that breaking one-way state generators, pseudorandom states, and EFI is bounded by mQCMA or $\mathrm{mQSZK_{hv}}$. We also show that the average-case hardness of $\mathrm{pQCZK_{hv}}$ implies the existence of EFI. These results provide new insights into Impagliazzo's worlds, establishing a connection between quantum cryptography and quantum promise complexity theory. We also extend our findings to quantum property testing and unitary synthesis, highlighting further applications of this new framework.

翻译：量子计算引入了许多植根于物理学的问题，要求从输入的量子态中计算信息。确定这些问题的复杂性对计算机科学和物理学均有重要意义。然而，由于现有的复杂性理论主要处理经典输入和输出的问题，其缺乏能够完全刻画量子输入问题复杂性的理论框架。这一空白在研究量子密码学与复杂性理论的关系时尤为突出，特别是在Impagliazzo的五世界框架内，因为刻画量子密码原语的安全性需要针对涉及量子输入问题的复杂性类。为填补这一空白，我们研究了量子承诺问题的复杂性理论，这类问题判定输入的量子态是否具有特定性质。我们重点关注复杂性类 p/mBQP、p/mQ(C)MA、$\mathrm{p/mQSZK_{hv}}$、p/mQIP 和 p/mPSPACE，其中 "p/mC" 表示对应于任意经典类 C 的纯态（p）或混合态（m）类。我们建立了结构性的结果，包括完全问题、搜索到判定的归约以及类之间的关系。值得注意的是，我们的发现揭示了与经典对应物的差异，例如 p/mQIP $\neq$ p/mPSPACE 以及 $\mathrm{mcoQSZK_{hv}} \neq \mathrm{mQSZK_{hv}}$。作为应用，我们将此框架应用于密码学，证明了攻破单向态生成器、伪随机态和EFI受限于 mQCMA 或 $\mathrm{mQSZK_{hv}}$。我们还证明了 $\mathrm{pQCZK_{hv}}$ 的平均情况困难性意味着EFI的存在。这些结果为Impagliazzo的世界提供了新的见解，建立了量子密码学与量子承诺复杂性理论之间的联系。我们还将研究结果扩展到量子性质测试和酉合成领域，突显了这一新框架的进一步应用前景。