未知等距信道的最优复共轭
Optimal complex conjugation of unknown isometry channels
AI总结：
该研究针对未知等距信道，确定了实现其复共轭的最优确定性协议，推导了最优保真度闭式表达式，给出了电路构造及多副本扩展方案，还得到了未知秩-$r$量子信道复共轭的最优查询复杂度协议。
AI中文摘要：
获取未知量子信道的复共轭是量子预言机问题中的一项有用资源，这催生了一个问题：如何仅通过有限次调用原始信道来模拟这种访问。我们利用未知等距信道 $V: \bigoplus^d \to \bigoplus^D$ 的 $n$ 次使用，确定了近似实现复共轭等距 $\bar{V}$ 的最优确定性协议。我们推导了最优保真度的闭式表达式，并证明并行协议在包括自适应和不定因果序策略在内的一般量子超信道中仍是最优的。该公式意味着，为了实现不相容性 $\boldsymbol{\tau}$，查询复杂度为 $n = \boldsymbol{\theta}(d[(D-d)/\boldsymbol{\tau} + 1])$。我们还提出了一种基于量子舒尔变换和对偶克莱布施-戈登变换的电路构造，其电路复杂度为 $O(\text{poly}(D,1/\boldsymbol{\tau}))$。该任务被扩展到多副本情况 $V^{\boxtimes n} \to \bar{V}^{\boxtimes k}$。对于固定的 $d<D$ 和 $k$，我们证明多副本情况的最优保真度为 $1 - kd(D-d)/n + o(n^{-1})$，且该值可由基于并行估计的协议渐近达到。最后，将等距协议与随机斯汀斯普林膨胀相结合，得到了未知秩-$r$ 量子信道复共轭的协议，若克劳斯秩 $r$ 为常数，则其查询复杂度在常数因子内是最优的。
英文摘要：
Access to the complex conjugate of an unknown quantum channel is a useful resource in quantum oracle problems, motivating the question of how such access can be simulated using only a limited number of calls to the original channel. We determine the optimal deterministic protocol for approximately implementing the complex conjugate isometry $\overline{V}$ from $n$ uses of an unknown isometry channel $V: \mathbb{C}^d\to\mathbb{C}^D$. We derive a closed-form expression for the optimal fidelity and prove that a parallel protocol is optimal even among general quantum superchannels, including adaptive and indefinite-causal-order strategies. The formula implies a query complexity $n=Θ(d[(D-d)/ε+1])$ for achieving infidelity $ε$. We also present a circuit construction based on the quantum Schur transform and the dual Clebsch--Gordan transform, with circuit complexity $O(\mathrm{poly}(D,1/ε))$. This task is extended to the multi-copy case $V^{\otimes n}\mapsto \overline{V}^{\otimes k}$. For fixed $d<D$ and $k$, we show that the optimal fidelity for the multi-copy case is $1-kd(D-d)/n+o(n^{-1})$, and that this value is asymptotically attained by a parallel estimation-based protocol. Finally, combining the isometry protocol with random Stinespring dilations yields a protocol for complex conjugation of unknown rank-$r$ quantum channels whose query complexity is optimal up to a constant factor if the Kraus rank $r$ is constant.
