From reversible computation to quantum computation by Lagrange interpolation

TL;DR

This paper explores a mathematical framework that connects classical reversible circuits to quantum circuits using Lagrange interpolation, providing a pathway from permutation groups to the unitary group U(n).

Contribution

It introduces a novel interpolation method that transitions from classical permutation matrices to quantum unitary operations, bridging classical and quantum computation.

Findings

Lagrange interpolation transforms permutation matrices into elements of the unitary group.

The method provides a mathematical pathway from classical reversible circuits to quantum circuits.

It offers a new perspective on the relationship between classical and quantum computation.

Abstract

Classical reversible circuits, acting on $w$~bits, are represented by permutation matrices of size $2^w \times 2^w$. Those matrices form the group P($2^w$), isomorphic to the symmetric group {\bf S}$_{2^w}$. The permutation group P($n$), isomorphic to {\bf S}$_n$, contains cycles with length~$p$, ranging from~1 to $L(n)$, where $L(n)$ is the so-called Landau function. By Lagrange interpolation between the $p$~matrices of the cycle, we step from a finite cyclic group of order~$p$ to a 1-dimensional Lie group, subgroup of the unitary group U($n$). As U($2^w$) is the group of all possible quantum circuits, acting on $w$~qubits, such interpolation is a natural way to step from classical computation to quantum computation.