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 $σ$ can be simulated to $\varepsilon$ error with $n = \mathcal{O}(q^2/\varepsilon)$ copies of $σ$. Consequently, for decision tasks, algorithms using (forward and inverse) state preparation queries only ever perform quadratically better than sample access. We also identify a motif, which we call the "acorn trick", 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 demonstrate this idea for several settings, including controlization and purification.

翻译：我们针对输入量子黑箱 $U$ 给出一个自然问题：该问题无法通过对 $U$ 和 $U^\dagger$ 进行指数次黑箱查询求解，但可通过常数次对 $U$ 与 $U^*$ 或 $U$ 与 $U^{\mathrm{T}}$ 的查询解决。同时，我们展示了一个量子承诺方案：该方案能抵御仅查询 $U$ 和 $U^\dagger$ 的对手，但在对手可查询 $U^*$ 时不再安全。这些结果表明共轭与转置查询确实增强了量子算法的能力，为 Zhandry 提出的"密码学原语应证明针对这类查询的安全性"观点提供了有力支持。我们的核心引理是：任何使用 $q$ 次正向和逆向查询状态制备酉算子（对应量子态 $\sigma$）的量子电路，均可通过 $n = \mathcal{O}(q^2/\varepsilon)$ 份 $\sigma$ 副本模拟至 $\varepsilon$ 误差。因此对于决策类问题，使用（正向与逆向）状态制备查询的算法性能仅平方倍优于采样访问。我们还发现一种称为"橡子技巧"的范式：当允许随机输出时，通常可突破确定性算法的不可行性定理，从而强化量子资源。我们通过控制化与纯化等多个场景验证了这一思想。