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Approximate Quantum Error Correcting Codes from Conformal Field Theory
Shengqi Sang1,2, Timothy H Hsieh1, Yijian Zou1
1<a href="https://ror.org/013m0ej23">Perimeter Institute for Theoretical Physics</a>, Waterloo, Ontario N2L 2Y5, Canada.
Abstract:
The low-energy subspace of a conformal field theory (CFT) can serve as a quantum error correcting code, with important consequences in holography and quantum gravity. We consider generic (1+1)D CFT codes under extensive local dephasing channels and analyze their error correctability in the thermodynamic limit. We show that (i) there is a finite decoding threshold if and only if the minimal nonzero scaling dimension in the fusion algebra generated by the jump operator of the channel is larger than 1/2 and (ii) the number of protected logical qubits k≥Ω(loglogn), where n is the number of physical qubits. As an application, we show that the one-dimensional quantum critical Ising model has a finite threshold for certain types of dephasing noise. Our general results also imply that a CFT code with continuous symmetry saturates a bound on the recovery fidelity for covariant codes.
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