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Leakage errors are unwanted transfer of population outside of a defined computational subspace and they occur in almost every platform for quantum computing.
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More explicitly, let us consider an error process with a leakage error probability of p L p_{L} and a seepage probability p S p_{S} . For a circuit with ℓ \ell gates, the probability of one leakage event is p ( one leakage ) = ℓ p L ( 1 − p L ) ℓ − 1 = 𝒪 ( ℓ p L ) p(\textrm{one leakage})=\ell p_{L}(1-p_{L})^{\ell-1}=\mathcal{O}(\ell p_{L}) . If the quantum state starts in the computational space, then a seepage event can happen only after a leakage event has happened. Therefore, the probability of a one seepage event is p ( one seepage ) = \tsum k = 1 ℓ ( ℓ − k ) p S ( 1 − p S ) ℓ − k − 1 p L ( 1 − p L ) ℓ − 1 = 𝒪 ( ℓ 2 p L p S ) p(\textrm{one seepage})=\tsum\nolimits_{k=1}^{\ell}(\ell-k)p_{S}(1-p_{S})^{\ell-k-1}p_{L}(1-p_{L})^{\ell-1}=\mathcal{O}(\ell^{2}p_{L}p_{S}) , which is a second-order effect when p S ≈ p L p_{S}\approx p_{L}
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To show this, consider the non-negative quantity ⟨ k | Λ ( | i ⟩ ⟨ i | ) | k ⟩ ≥ 0 \langle k|\Lambda(|i\rangle\langle i|)|k\rangle\geq 0 , where ⟨ k | i ⟩ = 0 \langle k|i\rangle=0 and | i ⟩ ⟨ i | , | k ⟩ ⟨ k | ∈ χ C |i\rangle\langle i|,|k\rangle\langle k|\in\chi_{C} . This implies t − r ≥ 0 t-r\geq 0 from Eq. ( 8
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M. Kang, W. C. Campbell, and K. R. Brown, Quantum error correction with metastable states of trapped ions using erasure conversion, PRX Quantum 4
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2024
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