Clifford Hierarchy Stabilizer Codes: Transversal Non-Clifford Gates and Magic States

Area of Science:

• Quantum Information Science
• Condensed Matter Physics
• High Energy Physics

Background:

• A key challenge in fault-tolerant quantum computation is balancing code universality with dimensionality.
• The Bravyi-König bound highlights this tradeoff for n-dimensional topological stabilizer codes.

Purpose of the Study:

• To extend topological Pauli stabilizer codes to higher levels of the Clifford hierarchy in n-dimensional space.
• To construct transversal non-Clifford gates for enhanced quantum error correction.

Main Methods:

• Generalization of topological stabilizer codes to n-dimensional Clifford hierarchy stabilizer codes.
• Utilizing automorphism symmetries of twisted gauge theories (e.g., Z_{2}^{3}, Z_{2}^{4}) to realize transversal gates.
• Employing code switching with just-in-time decoders for magic state preparation.

Main Results:

• First demonstration of transversal non-Clifford logical gates (t, cnot) for Clifford stabilizer codes in 2D.
• Construction of a transversal logical sqrt[T] gate in a 3D non-Clifford stabilizer code.
• Achieved logical gates at the (n+1)th Clifford hierarchy level in n spatial dimensions, surpassing the Bravyi-König bound.

Conclusions:

• The developed Clifford hierarchy stabilizer codes offer a pathway to overcome the universality-dimensionality tradeoff in quantum computation.
• These findings pave the way for more robust and efficient fault-tolerant quantum computers.