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Orthogonal polynomials and perfect state transfer
1Department of Mathematical Sciences, Bentley University, Waltham, MA, USA.
Summary
This review explores orthogonal polynomials and their applications in quantum information processing, including quantum walks and perfect state transfer detection. It highlights connections to classical processes and advanced concepts like exceptional orthogonal polynomials.
Area of Science:
- Quantum Information Processing
- Mathematical Physics
- Spectral Graph Theory
Background:
- Orthogonal polynomials are fundamental in various mathematical fields.
- Quantum information processing leverages quantum mechanics for computation and information tasks.
- Continuous-time quantum walks on graphs are a key area in quantum algorithms.
Purpose of the Study:
- To provide a self-contained introduction to orthogonal polynomials and continuous-time quantum walks.
- To discuss applications of orthogonal polynomials in quantum information processing.
- To explore connections between orthogonal polynomials, quantum walks, and classical processes.
Main Methods:
- Focus on Jacobi operators associated with orthogonal polynomials.
- Analyze the detection of perfect state transfer (PST) using these operators.
- Extend concepts to quantum walks with non-nearest-neighbor interactions using exceptional orthogonal polynomials (XOPs).
Main Results:
- Jacobi operators can be used to detect perfect state transfer.
- Orthogonal polynomials yield results analogous to Karlin-McGregor birth and death processes.
- Exceptional orthogonal polynomials enable extensions to more complex quantum walks.
Conclusions:
- Orthogonal polynomials offer powerful tools for quantum information processing.
- The review provides a foundation for understanding these applications.
- Open questions in the field are identified for future research.
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