We show that any Boolean function with approximate rank r can be computed by bounded-error quantum protocols without prior entanglement of complexity \(O( \sqrt{r} \log r)\). In addition, we show that any Boolean function with approximate rank r and discrepancy \(\delta \) can be computed by deterministic protocols of complexity O(r), and private coin bounded-error randomized protocols of complexity \(O((\frac{1}{\delta })^2 + \log r)\). Our deterministic upper bound in terms of approximate rank is tight up to constant factors, and the dependence on discrepancy in our randomized upper bound is tight up to taking square-roots. Our results can be used to obtain lower bounds on approximate rank. We also obtain a strengthening of Newman’s theorem with respect to approximate rank.

我们证明，任何具有近似秩r的布尔函数都可以通过有界误差量子协议来计算，而无需事先纠缠复杂性\(O( \sqrt{r} \log r)\)。此外，我们表明任何具有近似秩r和差异\(\delta \)的布尔函数都可以通过复杂度为O ( r ) 的确定性协议和复杂度为\(O((\ frac{1}{\delta })^2 + \log r)\)。我们在近似等级方面的确定性上限与常数因子紧密相关，并且对随机上限差异的依赖性与平方根紧密相关。我们的结果可用于获得近似排名的下限。我们还得到了关于近似等级的纽曼定理的强化。