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

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This study introduces a syndrome measurement scheme for encoding and decoding unknown states in CSS codes, applicable to various quantum error correction codes. The method demonstrates high fidelity for protected qubits in noisy scenarios.

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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.