Topological Quantum Codes from Lattices Partition on the n-Dimensional Flat Tori

Area of Science:

• Mathematics
• Quantum Information Science
• Condensed Matter Physics

Background:

• Lattices and their partitions are fundamental in various scientific domains.
• Toric codes and color codes are crucial for quantum error correction.
• Flat tori provide a versatile mathematical framework for theoretical studies.

Purpose of the Study:

• To explore the relationship between lattice partitions and tessellations in flat tori.
• To construct new classes of quantum error-correcting codes using lattice techniques.
• To investigate the application of these methods to hexagonal lattices in R2.

Main Methods:

• Inducing tessellations in flat tori (Tβ'=Rn/Λ') via sublattice partitioning (Λ'=mΛ).
• Utilizing group theory (G isomorphic to Λ/Λ') to characterize tessellations.
• Applying lattice partitioning techniques to construct toric codes and color codes.

Main Results:

• Demonstrated that n-dimensional sublattice partitioning induces G=Zmn tessellations.
• Derived toric codes with parameters [[2m^2,2,m]], [[3m^3,3,m]], and [[6m^4,6,m^2]] from Z2, Z3, and Z4 lattices.
• Presented new toric codes [[3m^2,2,m]] and color codes [[18m^2,4,4m]] using hexagonal lattices in R2.

Conclusions:

• Lattice partitioning is a powerful technique for generating tessellations and constructing quantum codes.
• The study provides novel methods for creating specific quantum error-correcting codes.
• Findings contribute to the development of advanced quantum computing architectures.