Quantum Algorithms for Tree Isomorphism and State Symmetrization

TL;DR

This paper presents quantum algorithms that efficiently solve tree isomorphism and state symmetrization problems, potentially paving the way for quantum solutions to more complex graph isomorphism challenges.

Contribution

It introduces polynomial-time quantum algorithms for tree isomorphism and state symmetrization, extending quantum computational techniques to these problems.

Findings

Polynomial-time quantum algorithm for rooted tree isomorphism.

Quantum state symmetrization over arbitrary permutation groups in polynomial time.

Potential for these techniques to inform quantum solutions for complex graph isomorphism problems.

Abstract

The graph isomorphism problem is theoretically interesting and also has many practical applications. The best known classical algorithms for graph isomorphism all run in time super-polynomial in the size of the graph in the worst case. An interesting open problem is whether quantum computers can solve the graph isomorphism problem in polynomial time. In this paper, an algorithm is shown which can decide if two rooted trees are isomorphic in polynomial time. Although this problem is easy to solve efficiently on a classical computer, the techniques developed may be useful as a basis for quantum algorithms for deciding isomorphism of more interesting types of graphs. The related problem of quantum state symmetrization is also studied. A polynomial time algorithm for the problem of symmetrizing a set of orthonormal states over an arbitrary permutation group is shown.