Exact Calculations of Coherent Information for Toric Codes under Decoherence: Identifying the Fundamental Error Threshold
arxiv(2024)
摘要
The toric code is a canonical example of a topological error-correcting code.
Two logical qubits stored within the toric code are robust against local
decoherence, ensuring that these qubits can be faithfully retrieved as long as
the error rate remains below a certain threshold. Recent studies have explored
such a threshold behavior as an intrinsic information-theoretic transition,
independent of the decoding protocol. These studies have shown that
information-theoretic metrics, calculated using the Renyi (replica)
approximation, demonstrate sharp transitions at a specific error rate. However,
an exact analytic expression that avoids using the replica trick has not been
shown, and the connection between the transition in information-theoretic
capacity and the random bond Ising model (RBIM) has only been indirectly
established. In this work, we present the first analytic expression for the
coherent information of a decohered toric code, thereby establishing a rigorous
connection between the fundamental error threshold and the criticality of the
RBIM.
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