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Approximate Quantum Error Correcting Codes from Conformal Field Theory.

Shengqi Sang1,2, Timothy H Hsieh1, Yijian Zou1

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Conformal field theory (CFT) codes offer quantum error correction. A finite decoding threshold exists if the scaling dimension exceeds 1/2, protecting logarithmic qubits.

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Area of Science:

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

Background:

  • Conformal field theory (CFT) low-energy subspaces act as quantum error correcting codes.
  • These codes have significant implications for holography and quantum gravity research.
  • Understanding error correction in CFT codes is crucial for their practical application.

Purpose of the Study:

  • Analyze the error correctability of generic (1+1)D CFT codes under dephasing channels.
  • Determine the conditions for a finite decoding threshold in the thermodynamic limit.
  • Investigate the number of protected logical qubits in such systems.

Main Methods:

  • Studied generic (1+1)D conformal field theory (CFT) codes.
  • Applied extensive local dephasing channels.
  • Analyzed error correctability in the thermodynamic limit.
  • Investigated the fusion algebra of the channel's jump operator.

Main Results:

  • A finite decoding threshold exists if and only if the minimal nonzero scaling dimension in the fusion algebra is greater than 1/2.
  • The number of protected logical qubits scales as k≥Ω(loglogn), where n is the number of physical qubits.
  • The one-dimensional quantum critical Ising model demonstrates a finite threshold for specific dephasing noise.

Conclusions:

  • CFT codes provide a robust framework for quantum error correction.
  • The derived conditions and qubit scaling offer insights into the capacity of CFT codes.
  • Results have direct applications in quantum critical systems and covariant code recovery fidelity.