The Near-optimal Performance of Quantum Error Correction Codes
arxiv(2024)
摘要
The Knill-Laflamme (KL) conditions distinguish perfect quantum error
correction codes, and it has played a critical role in the discovery of
state-of-the-art codes. However, the family of perfect codes is a very
restrictive one and does not necessarily contain the best-performing codes.
Therefore, it is desirable to develop a generalized and quantitative
performance metric. In this Letter, we derive the near-optimal channel
fidelity, a concise and optimization-free metric for arbitrary codes and noise.
The metric provides a narrow two-sided bound to the optimal code performance,
and it can be evaluated with exactly the same input required by the KL
conditions. We demonstrate the numerical advantage of the near-optimal channel
fidelity through multiple qubit code and oscillator code examples. Compared to
conventional optimization-based approaches, the reduced computational cost
enables us to simulate systems with previously inaccessible sizes, such as
oscillators encoding hundreds of average excitations. Moreover, we analytically
derive the near-optimal performance for the thermodynamic code and the
Gottesman-Kitaev-Preskill (GKP) code. In particular, the GKP code's performance
under excitation loss improves monotonically with its energy and converges to
an asymptotic limit at infinite energy, which is distinct from other oscillator
codes.
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