Efficient ancilla-free reversible and quantum circuits for the Hidden Weighted Bit function

TL;DR

This paper presents polynomial-size reversible and quantum ancilla-free circuits for the Hidden Weighted Bit function, challenging previous beliefs of exponential complexity and employing advanced theoretical and physical techniques.

Contribution

It develops the first polynomial-size reversible ancilla-free circuit and an efficient quantum circuit for the Hidden Weighted Bit function, refuting exponential hardness conjectures.

Findings

Reversible circuit size: O(n^{6.42})

Quantum circuit size: O(n^2)

Refutes exponential hardness conjecture

Abstract

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.