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Large-scale variational quantum algorithms are widely recognized as a potential pathway to achieve practical quantum advantages.
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The superoperator 𝒮 i \mathcal{S}_{i} , as defined in Wood et al. 2011 , is denoted as the operator 𝒰 ¯ i ⊗ 𝒰 i \mathcal{\overline{U}}_{i}\otimes\mathcal{U}_{i} , and the symbol | ⋅ ⟩ ⟩ |\cdot\rangle\!\rangle indicates the vectorization of a matrix. The expected value ℒ ( 𝜽 ) \mathcal{L}(\bm{\theta}) is given by the expression: ⟨ ⟨ H | ∏ i = 1 L 𝒮 i | ρ ⟩ ⟩ = ∑ s ∈ 𝑷 n L + 1 ⟨ ⟨ H | s L ⟩ ⟩ ( ∏ i = 1 L ⟨ ⟨ s i | 𝒮 i | s i − 1 ⟩ ⟩ ) ⟨ ⟨ s 0 | ρ ⟩ ⟩ \langle\!\langle H|\prod\displaylimits_{i=1}^{L}\mathcal{S}_{i}|\rho\rangle\!\rangle=\sum\displaylimits_{s\in\bm{P}^{L+1}_{n}}\langle\!\langle H|s_{L}\rangle\!\rangle\left(\prod\displaylimits_{i=1}^{L}\langle\!\langle s_{i}|\mathcal{S}_{i}|s_{i-1}\rangle\!\rangle\right)\langle\!\langle s_{0}|\rho\rangle\!\rangle . Here, each term of the summation is an alternative representation of the expression in Eq. ( 4
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