This note complements the paper "One-Way Ticket to Las Vegas and the Quantum Adversary" (arxiv:2301.02003). I develop the ideas behind the adversary bound - universal algorithm duality therein in a different form, using the same perspective as Barnum-Saks-Szegedy in which query algorithms are defined as sequences of feasible reduced density matrices rather than sequences of unitaries. This form may be faster to understand for a general quantum information audience: It avoids defining the "unidirectional relative $\gamma_{2}$-bound" and relating it to query algorithms explicitly. This proof is also more general because the lower bound (and universal query algorithm) apply to a class of optimal control problems rather than just query problems. That is in addition to the advantages to be discussed in Belovs-Yolcu, namely the more elementary algorithm and correctness proof that avoids phase estimation and spectral analysis, allows for limited treatment of noise, and removes another $\Theta(\log(1/\epsilon))$ factor from the runtime compared to the previous discrete-time algorithm.

翻译：本文是对论文《单程飞往拉斯维加斯与量子博弈界》（arXiv:2301.02003）的补充说明。我以另一种形式发展了博弈界——通用对偶算法的思想，采用与Barnum-Saks-Szegedy相同的视角，将查询算法定义为可行约化密度矩阵的序列而非酉算子序列。这种形式可能更易于被一般量子信息领域的读者理解：它避免了定义“单向相对γ2界”并将其与查询算法显式关联。该证明也更具普适性，因为下界（及通用查询算法）适用于一类最优控制问题，而不仅限于查询问题。此外，它还具有将在Belovs-Yolcu论文中讨论的优势，即更基础的算法与正确性证明——避免了相位估计与谱分析，允许对噪声进行有限处理，并相较于先前的离散时间算法额外消除了运行时间中的Θ(log(1/ε))因子。