Efficient Ancilla-Free Reversible and Quantum Circuits for the Hidden Weighted Bit Function.

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 $O(n^{6.42})$ O (n 6. 42) , where $n$ 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 $O(n^2)$ 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. [ABSTRACT FROM AUTHOR]