Simple scheme for encoding and decoding a qubit in unknown state for various topological codes

Area of Science:

• Quantum Information Science
• Quantum Error Correction
• Topological Quantum Computing

Background:

• Stabilizer codes, such as CSS codes, are crucial for quantum error correction.
• Syndrome measurements are essential for detecting and correcting errors in quantum systems.
• Topological codes offer inherent robustness against local errors.

Purpose of the Study:

• To develop a generalizable scheme for encoding and decoding unknown quantum states using syndrome measurements in CSS codes.
• To analyze the performance of this scheme in both noiseless and noisy quantum environments.
• To evaluate the scheme's applicability to specific topological codes like Kitaev toric code and 3D Haah code.

Main Methods:

• Encoding and decoding of unknown states based on syndrome measurements.
• Application and illustration of the scheme to Kitaev toric code, defected-lattice code, topological subsystem code, and 3D Haah code.
• Analysis of a noisy scenario for subsystem codes, including bit-flip, phase-flip, state preparation, and syndrome measurement errors.

Main Results:

• The proposed protocol is local for codes where logical operator crossings involve nearest-neighbor pairs.
• In a noisy scenario for subsystem codes, the fidelity of the protected qubit scales as O(p) in the large code size limit, where p is the single-qubit error probability per time step.
• A noiseless scheme is presented for the Haah code, with the noisy case remaining an open problem.

Conclusions:

• The developed syndrome measurement scheme provides an effective method for encoding and decoding unknown states in various CSS codes.
• The scheme exhibits favorable performance in noisy conditions, with fidelity depending on error rates and code size.
• Further research is needed to address the noisy case for codes like the Haah code.