For qudit channels, learning is exponentially hard with c < d parallel copies and becomes efficient at c = d with tight ε^(-2d) scaling; access to the complex-conjugate channel gives tight ε^(-4) bounds, and bosonic channels are hard for all c = O(1/ε).
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Quantum channel learning with limited parallel access
For qudit channels, learning is exponentially hard with c < d parallel copies and becomes efficient at c = d with tight ε^(-2d) scaling; access to the complex-conjugate channel gives tight ε^(-4) bounds, and bosonic channels are hard for all c = O(1/ε).