Topological quantum hashing with the icosahedral group

Area of Science:

• Quantum Information Science
• Condensed Matter Physics
• Group Theory

Background:

• Single-qubit gates are fundamental operations in quantum computation.
• Fibonacci anyons and icosahedral groups offer novel frameworks for quantum information processing.
• Efficient algorithms are crucial for practical quantum compiling.

Purpose of the Study:

• To develop an efficient algorithm for hashing single-qubit gates into Fibonacci anyon braids.
• To represent these braids using products of icosahedral group elements.
• To analyze the approximation accuracy and computational cost.

Main Methods:

• Representing icosahedral group elements as braid segments of varying lengths.
• Introducing a series of pseudogroups for braid representation.
• Employing a renormalization group approach to form a Gaussian unitary ensemble of random-matrix representations.
• Analyzing the approximation error (epsilon) and time complexity (O(log(1/epsilon))).

Main Results:

• An efficient algorithm is established to map any single-qubit gate to a Fibonacci anyon braid.
• The algorithm achieves an average error of epsilon for approximating SU(2) matrices using braids of length O(log2(1/epsilon)).
• The computational cost is found to be O(log(1/epsilon)) in time.

Conclusions:

• The developed algorithm provides an efficient method for quantum gate approximation using anyon braids.
• This approach is applicable to generic quantum compiling tasks.
• The connection between anyon braids, group theory, and random-matrix theory offers new avenues for quantum computation research.