The Unitary Synthesis Problem (Aaronson-Kuperberg 2007) asks whether any $n$-qubit unitary $U$ can be implemented by an efficient quantum algorithm $A$ augmented with an oracle that computes an arbitrary Boolean function $f$. In other words, can the task of implementing any unitary be efficiently reduced to the task of implementing any Boolean function? In this work, we prove a one-query lower bound for unitary synthesis. We show that there exist unitaries $U$ such that no quantum polynomial-time oracle algorithm $A^f$ can implement $U$, even approximately, if it only makes one (quantum) query to $f$. Our approach also has implications for quantum cryptography: we prove (relative to a random oracle) the existence of quantum cryptographic primitives that remain secure against all one-query adversaries $A^{f}$. Since such one-query algorithms can decide any language, solve any classical search problem, and even prepare any quantum state, our result suggests that implementing random unitaries and breaking quantum cryptography may be harder than all of these tasks. To prove this result, we formulate unitary synthesis as an efficient challenger-adversary game, which enables proving lower bounds by analyzing the maximum success probability of an adversary $A^f$. Our main technical insight is to identify a natural spectral relaxation of the one-query optimization problem, which we bound using tools from random matrix theory. We view our framework as a potential avenue to rule out polynomial-query unitary synthesis, and we state conjectures in this direction.

翻译：酉合成问题（Aaronson-Kuperberg 2007）探讨的是：是否任意$n$比特酉算子$U$都能通过一个配备可计算任意布尔函数$f$的预言机的有效量子算法$A$来实现？换言之，实现任意酉算子的任务能否高效归约为实现任意布尔函数的任务？本文证明了酉合成的一查询下界。我们指出存在某些酉算子$U$，使得任何仅对$f$进行一次（量子）查询的量子多项式时间预言机算法$A^f$都无法实现$U$（甚至无法近似实现）。该方法对量子密码学也有重要意义：我们证明了（相对于随机预言机）存在可抵抗所有一查询敌手$A^{f}$攻击的量子密码学原语。由于此类一查询算法能够判定任意语言、解决任意经典搜索问题，甚至制备任意量子态，我们的结果表明实现随机酉算子与破解量子密码学可能比上述所有任务更难。为证明该结论，我们将酉合成表述为一种高效的挑战者-敌手博弈模型，通过分析敌手$A^f$的最大成功概率来推导下界。主要技术突破在于识别出一查询优化问题的自然谱松弛形式，并利用随机矩阵理论工具对其进行定界。我们视此框架为排除多项式查询酉合成的潜在途径，并据此提出相关猜想。