We study a natural complexity measure of Boolean functions known as the (exact) rational degree. For total functions $f$, it is conjectured that $\mathrm{rdeg}(f)$ is polynomially related to $\mathrm{deg}(f)$, where $\mathrm{deg}(f)$ is the Fourier degree. Towards this conjecture, we show that symmetric functions have rational degree at least $\mathrm{deg}(f)/2$ and monotone functions have rational degree at least $\sqrt{\mathrm{deg}(f)}$. We observe that both of these lower bounds are tight. In addition, we show that all read-once depth-$d$ Boolean formulae have rational degree at least $\Omega(\mathrm{deg}(f)^{1/d})$. Furthermore, we show that almost every Boolean function on $n$ variables has rational degree at least $n/2 - O(\sqrt{n})$. In contrast to total functions, we exhibit partial functions that witness unbounded separations between rational and approximate degree, in both directions. As a consequence, we show that for quantum computers, post-selection and bounded-error are incomparable resources in the black-box model.

翻译：我们研究了一种称为（精确）有理次数的布尔函数自然复杂度度量。对于全函数 $f$，猜想 $\mathrm{rdeg}(f)$ 与 $\mathrm{deg}(f)$（傅里叶次数）存在多项式关联。针对该猜想，我们证明对称函数的有理次数至少为 $\mathrm{deg}(f)/2$，单调函数的有理次数至少为 $\sqrt{\mathrm{deg}(f)}$，并指出这两个下界均为紧界。此外，我们证明所有只读深度为 $d$ 的布尔公式的有理次数至少为 $\Omega(\mathrm{deg}(f)^{1/d})$。进一步地，我们证明几乎所有 $n$ 元布尔函数的有理次数至少为 $n/2 - O(\sqrt{n})$。与全函数形成对比的是，我们构造了部分函数，其在有理次数与近似次数之间存在无界分离（双向均可）。作为推论，我们证明在量子计算机的黑盒模型中，后选择与有界错误是不可比较的计算资源。