The general adversary dual is a powerful tool in quantum computing because it gives a query-optimal bounded-error quantum algorithm for deciding any Boolean function. Unfortunately, the algorithm uses linear qubits in the worst case, and only works if the constraints of the general adversary dual are exactly satisfied. The challenge of improving the algorithm is that it is brittle to arbitrarily small errors since it relies on a reflection over a span of vectors. We overcome this challenge and build a robust dual adversary algorithm that can handle approximately satisfied constraints. As one application of our robust algorithm, we prove that for any Boolean function with polynomially many 1-valued inputs (or in fact a slightly weaker condition) there is a query-optimal algorithm that uses logarithmic qubits. As another application, we prove that numerically derived, approximate solutions to the general adversary dual give a bounded-error quantum algorithm under certain conditions. Further, we show that these conditions empirically hold with reasonable iterations for Boolean functions with small domains. We also develop several tools that may be of independent interest, including a robust approximate spectral gap lemma, a method to compress a general adversary dual solution using the Johnson-Lindenstrauss lemma, and open-source code to find solutions to the general adversary dual.

翻译：通用对抗对偶是量子计算中的有力工具，因为它能给出判定任意布尔函数的最优查询复杂度的有界误差量子算法。但该算法在最坏情况下需使用线性数量的量子比特，且仅当通用对抗对偶的约束条件被精确满足时有效。改进该算法的难点在于，由于它依赖于向量空间上的反射操作，因此对任意微小的误差都非常脆弱。我们克服了这一挑战，构建了一种能处理近似满足约束条件的鲁棒双对抗算法。作为该鲁棒算法的一个应用，我们证明对于任意具有多项式数量取值为1的输入（或更弱条件）的布尔函数，存在使用对数数量量子比特的最优查询算法。作为另一项应用，我们证明通过数值方法推导的通用对抗对偶近似解，能在特定条件下给出有界误差量子算法。进一步地，我们通过实验表明，对于小规模域的布尔函数，这些条件在合理迭代次数下经验性地成立。我们还开发了几种可能具有独立价值的工具，包括鲁棒近似谱隙引理、利用Johnson-Lindenstrauss引理压缩通用对抗对偶解的方法，以及寻找通用对抗对偶解的开源代码。