AI 中文总结
该研究针对酉合成问题，证明了1次查询对2次查询的显式分离、复相位酉矩阵的1次查询近似上界，还引入新密码游戏并得到量子程序近似难解性的新结果。
AI 中文摘要
酉合成问题（Aaronson-Kuperberg，CCC 2007）询问，是否每个n量子比特酉矩阵U都可通过依赖于U的某个经典预言机f = f_U实现的高效量子电路来计算。最近，Lombardi-Ma-Wright（STOC 2024）证明，哈尔随机酉矩阵无法被向任意经典预言机进行1次查询（或poly(n)次并行查询）的算法高效合成。在本研究中，我们证明了关于酉合成变体的难解性（以及易解性）的若干结果，包括：（1）1次查询与2次查询酉合成：我们证明了合成随机置换酉矩阵P|x⟩=|π(x)⟩以及随机交替基相位酉矩阵F₂·H⊗ⁿ·F₁的1次查询下界，这为具有高效（甚至2次查询）合成算法的“显式”酉矩阵族提供了1次查询下界；（2）复相位酉矩阵的上界：我们还考虑复相位酉矩阵|x⟩↦αₓ|x⟩，这类酉矩阵有一个清晰的2次查询合成算法，但无明显的1次查询算法，在这种情况下，我们证明了一个上界：存在1次查询算法（相对于二进制相位预言机）可在钻石距离下对这些酉矩阵进行常数近似。为证明我们的下界，我们引入并分析了两种新的密码游戏：预言机态搜索游戏和预言机Choi态游戏，与现有工作相比，我们的框架数学上更简单、可证明的内容更灵活，且能更准确地捕捉非“完全随机”酉矩阵的合成难解性。最后，我们还使用搜索游戏证明了量子程序（相对于量子建议合成酉矩阵）的相位酉矩阵近似难解性的新结果，在1次查询酉合成与量子程序之间提供了更清晰的分离。
英文摘要
The unitary synthesis problem (Aaronson-Kuperberg, CCC 2007) asks whether every $n$-qubit unitary $U$ is computable by efficient quantum circuits relative to some classical oracle $f = f_U$ depending on $U$. Recently, Lombardi-Ma-Wright (STOC 2024) proved that Haar-random unitaries cannot be efficiently synthesized by algorithms that make 1 query (or poly$(n)$ parallel queries) to an arbitrary classical oracle. In this work, we prove several results about the hardness (and easiness!) of variants of unitary synthesis. Our results include: (1) 1-query vs. 2-query unitary synthesis: we prove 1-query lower bounds for synthesizing random permutation unitaries $P\lvert x\rangle = \lvert π(x)\rangle$, as well as random alternating-basis phase unitaries $F_2 \cdot H^{\otimes n} \cdot F_1$. This gives 1-query lower bounds for "explicit" families of unitaries that have efficient (even 2-query) synthesis algorithms. (2) Upper bound for complex phase unitaries: we also consider complex phase unitaries $\lvert x\rangle\mapsto α_x \lvert x\rangle$, which have a clean 2-query synthesis algorithm with no obvious 1-query algorithm. In this case, we prove an upper bound: there are 1-query algorithms (relative to binary phase oracles) that constant-approximate these unitaries in diamond distance. In order to prove our lower bounds, we introduce and analyze two new cryptographic games: the oracle state search game and the oracle Choi state game. Compared to prior work, our framework is mathematically simple, more flexible in what it can prove, and more accurately captures the hardness of synthesizing unitaries that are not "fully random". Finally, we also use the search game to prove a new hardness-of-approximation result for quantum programs (synthesizing unitaries relative to quantum advice) for phase unitaries, giving a sharper separation between 1-query unitary synthesis and quantum programs.
