Randomized Linear Algebra Approaches to Estimate the Von Neumann Entropy of Density Matrices

Area of Science:

• Quantum Information Theory
• Numerical Analysis
• Quantum Computing

Background:

• The von Neumann entropy is a crucial quantum mechanical measure.
• Calculating it requires computing eigenvalues of density matrices, which is computationally expensive for large systems.

Purpose of the Study:

• To develop and analyze efficient randomized algorithms for approximating von Neumann entropy.
• To leverage advancements in Randomized Numerical Linear Algebra (RandNLA) for quantum mechanical computations.

Main Methods:

• Three novel randomized algorithms based on RandNLA techniques.
• Utilizing randomized trace estimators and random projections.
• Provable accuracy guarantees and theoretical analysis.

Main Results:

• Demonstrated considerable speedup in approximating von Neumann entropy.
• Experimental evaluations confirmed theoretical findings on accuracy and efficiency.
• Algorithms provide provable accuracy bounds.

Conclusions:

• The proposed randomized algorithms offer a practical and efficient approach to compute von Neumann entropy.
• These methods are valuable for large-scale quantum mechanical systems where exact computation is infeasible.
• Advancements in RandNLA can be effectively applied to quantum information theory problems.