Shannon proved that almost all Boolean functions require a circuit of size $\Theta(2^n/n)$. We prove a quantum analog of this classical result. Unlike in the classical case the number of quantum circuits of any fixed size that we allow is uncountably infinite. Our main tool is a classical result in real algebraic geometry bounding the number of realizable sign conditions of any finite set of real polynomials in many variables.

翻译：香农证明几乎所有布尔函数都需要规模为$\Theta(2^n/n)$的电路。我们证明了这一经典结果的量子类比。与经典情形不同，我们允许的任意固定规模的量子电路数量是不可数无穷的。我们的主要工具是实代数几何中的一个经典结果，该结果对任意有限组多元实多项式的可实现符号条件数量给出了上界。