We give a natural problem over input quantum oracles $U$ which cannot be solved with exponentially many black-box queries to $U$ and $U^\dagger$, but which can be solved with constant many queries to $U$ and $U^*$, or $U$ and $U^{\mathrm{T}}$. We also demonstrate a quantum commitment scheme that is secure against adversaries that query only $U$ and $U^\dagger$, but is insecure if the adversary can query $U^*$. These results show that conjugate and transpose queries do give more power to quantum algorithms, lending credence to the idea put forth by Zhandry that cryptographic primitives should prove security against these forms of queries. Our key lemma is that any circuit using $q$ forward and inverse queries to a state preparation unitary for a state $\sigma$ can be simulated to $\varepsilon$ error with $n = \mathcal{O}(q^2/\varepsilon)$ copies of $\sigma$. Consequently, for decision tasks, algorithms using (forward and inverse) state preparation queries only ever perform quadratically better than sample access. These results follow from straightforward combinations of existing techniques; our contribution is to state their consequences in their strongest, most counter-intuitive form. In doing so, we identify a motif where generically strengthening a quantum resource can be possible if the output is allowed to be random, bypassing no-go theorems for deterministic algorithms. We call this the acorn trick.

翻译：我们提出一个关于输入量子预言机$U$的自然问题，该问题无法通过指数次对$U$和$U^\dagger$的黑盒查询求解，但可通过常数次对$U$和$U^*$或$U$和$U^{\mathrm{T}}$的查询解决。我们还构建了一个量子承诺方案，该方案对仅能查询$U$和$U^\dagger$的敌手是安全的，但当敌手能查询$U^*$时则不安全。这些结果表明，共轭查询和转置查询确实能增强量子算法的能力，从而支持了Zhandry提出的观点：密码原语应证明能抵抗此类查询形式的攻击。我们的核心引理是：任何使用$q$次正向和逆向查询到态制备酉算子（用于制备态$\sigma$）的电路，都可通过$n = \mathcal{O}(q^2/\varepsilon)$份$\sigma$的副本以$\varepsilon$误差进行模拟。因此，对于决策任务，使用（正向和逆向）态制备查询的算法最多只能比样本访问方式获得二次方的性能提升。这些结果源于对现有技术的直接组合；我们的贡献在于以最有力且最反直觉的形式阐述其推论。通过这一过程，我们识别出一种模式：若允许输出具有随机性，则通常可以增强量子资源的能力，从而绕过确定性算法的不可能性定理。我们将此称为"橡果技巧"。