Quantum Walks: Schur Functions Meet Symmetry Protected Topological Phases

Area of Science:

• Quantum Physics
• Harmonic Analysis
• Topological Phases

Background:

• Symmetry protected topological phases are a key area in quantum matter.
• Quantum walks are quantum analogs of classical random walks with applications in quantum computing.
• Schur functions are central objects in harmonic analysis with potential links to other fields.

Purpose of the Study:

• To uncover and exploit a connection between Schur functions and symmetry protected topological phases.
• To develop a new method for calculating topological indices in quantum walks.
• To classify topological phases beyond translation-invariant systems.

Main Methods:

• Utilizing matrix Schur functions derived from quantum walks.
• Reducing topological index calculation to a linear algebra problem.
• Evaluating Schur functions at symmetry-protected points to obtain finite-dimensional unitaries.

Main Results:

• Topological indices for symmetry protected topological phases of quantum walks are encoded by matrix Schur functions.
• The Schur representation provides a complete set of symmetry indices for 1D quantum walks across various symmetry types.
• This method is valid even without translation invariance, enabling classification of non-translation invariant phases.

Conclusions:

• The link between Schur functions and topological phases in quantum walks offers a powerful new framework.
• The developed linear algebra approach simplifies the computation of topological indices.
• This work extends the classification of topological phases to non-translation invariant systems.