Xinyu Wu
After presentations of Raz and Tal's oracle separation of BQP and PH result, several people (e.g. Ryan O'Donnell, James Lee, Avishay Tal) suggested that the proof may be simplified by stochastic calculus. In this short note, we describe such a simplification.
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In 2009, using the $\textsf{Fourier Checking}$ problem, Aaronson claimed to construct the relativized worlds such that $\textsf{BQP} \not\subset \mathsf{BPP_{path}}$ and $\textsf{BQP} \not\subset \textsf{SZK}$. However, there are subtle errors in the original proof. In this paper, we point out the issues, and rescue these two separations by using more sophisticated constructions. Meanwhile, we take the opportunity to study the complexity classes $\mathsf{BPP_{path}}$ and $\textsf{SZK}$. We give general ways to construct functions which are hard for $\textsf{SZK}$ and $\mathsf{BPP_{path}}$ (in …
The Acrobatics of BQP
本研究揭示了黑盒设置下量子多项式时间（BQP）与经典复杂度类之间的解耦现象，通过构建特定预言机解决了多个复杂度类包含问题，并引入了量子感知随机限制等新工具。
The relationship between BQP and PH has been an open problem since the earliest days of quantum computing. We present evidence that quantum computers can solve problems outside the entire polynomial hierarchy, by relating this question to topics in circuit complexity, pseudorandomness, and Fourier analysis. First, we show that there exists an oracle relation problem (i.e., a problem with many valid outputs) that is solvable in BQP, but not in PH. This also yields a non-oracle relation problem that is solvable in quantum logarithmic time, but not in AC0. Second, we show that an oracle decision …
We study the relationship between problems solvable by quantum algorithms in polynomial time and those for which zero-knowledge proofs exist. In prior work, Aaronson [arxiv:quant-ph/0111102] showed an oracle separation between BQP and SZK, i.e. an oracle $A$ such that $\mathrm{SZK}^A \not\subseteq \mathrm{BQP}^A$. In this paper we give a simple extension of Aaronson's result to non-interactive zero-knowledge proofs with perfect security. This class, NIPZK, is the most restrictive zero-knowledge class. We show that even for this class we can construct an $A$ with $\mathrm{NIPZK}^A \not\subseteq…
A Relativizing MIP for BQP
本文探讨BQP⊆IP的相对化问题，通过构造基于经典 oracle 的PCP证明系统，证明BQP^O⊆MIP，为量子计算的交互证明提供了新思路。
Classical vs Quantum Advice and Proofs under Classically-Accessible Oracle
本文构造经典可访问预言机，首次实现 BQP/qpoly 与 BQP/poly、QMA 与 QCMA 的相对分离，并给出分布式量子可访问预言机下 QMA 与 QCMA 分离的新证明。
It is a longstanding open problem to devise an oracle relative to which BQP does not lie in the Polynomial-Time Hierarchy (PH). We advance a natural conjecture about the capacity of the Nisan-Wigderson pseudorandom generator [NW94] to fool AC_0, with MAJORITY as its hard function. Our conjecture is essentially that the loss due to the hybrid argument (which is a component of the standard proof from [NW94]) can be avoided in this setting. This is a question that has been asked previously in the pseudorandomness literature [BSW03]. We then make three main contributions: (1) We show that our conj…
$\mathsf{BQP} \subseteq \mathsf{IP}$ Does Not Relativize
该研究构造预言机证明BQP包含于IP不相对化，并首次分离IP与MIP，基于Forrelation问题及凸函数近似Avg-Max电路的新分析，表明经典验证BQP需非相对化技术。
Separating Quantum and Classical Advice with Good Codes
本文通过构造优良码，实现了量子证明验证（QMA）与经典证明验证（QCMA）之间的无条件经典 oracle 分离，并扩展到其他 oracle 分离问题。
An important theoretical problem in the study of quantum computation, that is also practically relevant in the context of near-term quantum devices, is to understand the computational power of hybrid models, that combine poly-time classical computation with short-depth quantum computation. Here, we consider two such models: CQ_d which captures the scenario of a polynomial-time classical algorithm that queries a d-depth quantum computer many times; and QC_d which is more analogous to measurement-based quantum computation and captures the scenario of a d-depth quantum computer with the ability t…
An optimal oracle separation of classical and quantum hybrid schemes
本文证明存在预言机可将量子深度 d 与 d+1 分离，实现经典与量子混合方案的最优预言机分离，优于此前 d 与 2d+1 的分离结果。
A Separation Between Types of Quantum Oracle Separations
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。
We prove two sets of results concerning computational complexity classes. The first concerns a variation of the random oracle hypothesis posed by Bennett and Gill after they showed that relative to a randomly chosen oracle, P not equal NP with probability 1. This hypothesis was quickly disproven in several ways, most famously in 1992 with the result that IP equals PSPACE, in spite of the classes being shown unequal with probability 1. Here we propose a variation of what it means to be ``large'' using the Ellentuck topology. In this new context, we demonstrate that the set of oracles separating…
Oracle Separations in the Fourier Hierarchy
该论文证明对于每个常数k≥2，存在预言机使傅里叶层级第k层严格弱于第k+1层，基于Forrelation问题构造分离，并利用层数限制查询自适应性的结构性质建立下界。
Quantum Pseudorandomness and Classical Complexity
本研究构造了一个量子预言机，在其中BQP=QMA但伪随机量子态和酉变换仍存在，并证明伪随机态需要复杂性假设，揭示了量子与经典输入算法的区别及其对密码学和复杂性理论的影响。
Random Permutations in Computational Complexity
针对随机置换与随机预言机的复杂性分离问题，提出资源受限测度框架定义个体随机置换，证明其首个分离结果并加强量子分离，还揭示两者归约关系及相关未解决问题。
Oracle Separations for RPH
本文通过定义实数图灵机并发展预言机分离技术，证明了实数多项式谱系RPH与PSPACE、量子计算等类之间的分离，暗示ER严格包含于PSPACE且实数计算能力受限。
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