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This note complements the paper "One-Way Ticket to Las Vegas and the QuantumAdversary" (arxiv:2301.02003). I develop the ideas behind the adversary bound -universal algorithm duality therein in a different form, using the sameperspective as Barnum-Saks-Szegedy in which query algorithms are defined assequences of feasible reduced density matrices rather than sequences ofunitaries. This form may be faster to understand for a general quantuminformation audience: It avoids defining the "unidirectional relative$\gamma_{2}$-bound" and relating it to query algorithms explicitly. This proofis 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, namelythe more elementary algorithm and correctness proof that avoids phaseestimation and spectral analysis, allows for limited treatment of noise, andremoves another $\Theta(\log(1/\epsilon))$ factor from the runtime compared tothe previous discrete-time algorithm.

The Adversary Bound Revisited: From Optimal Query Algorithms to Optimal Control