Quantum circuits from non-unitary sparse binary matrices

Area of Science:

• Quantum Computing
• Theoretical Computer Science
• Matrix Theory

Background:

• Quantum computing relies on unitary matrices for reversible computations.
• Real-world applications often use non-unitary sparse matrices, creating a challenge for quantum implementation.
• Existing methods struggle to efficiently integrate non-unitary operations into quantum algorithms.

Purpose of the Study:

• To introduce a novel and efficient method for transforming non-unitary sparse binary matrices into unitary higher-dimensional permutation matrices.
• To demonstrate the practical applicability of this transformation for large-scale quantum problems.
• To explore the implications for quantum gate construction and modeling quantum systems.

Main Methods:

• Developed a technique to map non-unitary sparse binary matrices to higher-dimensional permutation matrices.
• Ensured the resulting permutation matrices are unitary, preserving quantum mechanical principles.
• Validated the method's efficiency in terms of space and time complexity.

Main Results:

• Successfully transformed a class of non-unitary sparse binary matrices into unitary matrices.
• Demonstrated the method's efficiency, making it suitable for large-scale applications.
• Showcased the transformation's utility in constructing quantum gates and modeling quantum finite state machines (QFSMs).

Conclusions:

• The proposed method provides a practical pathway for incorporating non-unitary transformations into quantum computing.
• This work bridges the gap between classical non-unitary matrix applications and quantum computational frameworks.
• Significant implications for automata theory and the broader field of quantum computation, particularly for systems utilizing sparse, non-unitary matrices.