The Hidden Weighted Bit function plays an important role in the study of classical models of computation. A common belief is that this function is exponentially hard for the implementation by reversible ancilla-free circuits, even though introducing a small number of ancillae allows a very efficient implementation. In this paper, we refute the exponential hardness conjecture by developing a polynomial-size reversible ancilla-free circuit computing the Hidden Weighted Bit function. Our circuit has size $O(n^{6.42})$, where $n$ is the number of input bits. We also show that the Hidden Weighted Bit function can be computed by a quantum ancilla-free circuit of size $O(n^2)$. The technical tools employed come from a combination of Theoretical Computer Science (Barrington's theorem) and Physics (simulation of fermionic Hamiltonians) techniques.

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用于隐藏加权位函数的高效无辅助可逆和量子电路

隐藏加权位函数在经典计算模型的研究中起着重要作用。一个普遍的看法是，尽管引入少量的辅助电路可以非常有效地实现，但通过可逆无辅助电路实现该功能的难度呈指数级增长。在本文中，我们通过开发计算隐藏加权位函数的多项式大小可逆无辅助电路来驳斥指数硬度猜想。我们的电路大小为 $O(n^{6.42})$，其中 $n$ 是输入位数。我们还表明，可以通过大小为 $O(n^2)$ 的无量子无辅助电路来计算隐藏加权位函数。所采用的技术工具来自理论计算机科学（巴林顿定理）和物理学（费米子哈密顿量模拟）技术的结合。