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 to implement using 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(n6.42)O, where nn 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(n2). The technical tools employed come from a combination of Theoretical Computer Science (Barrington's theorem) and Physics (simulation of fermionic Hamiltonians) techniques.