Exploring the landscape of compact magic-state distillation factories

Producing high-fidelity magic states using the smallest possible amount of physical qubits and operations stands as a very important challenge to achieve fault-tolerant quantum computation at scale. Besides emerging proposals for alternative methods such as cultivation, magic state distillation remains essential for achieving very low error rates. Known distillation protocols are usually built through quantum codes derived from triorthogonal matrices. Here, exploiting the specific noise structure present in magic state distillation protocols, we show that classical error-correcting codes offer a simpler framework for deriving these protocols. This formulation is particularly well suited to systematic numerical and analytical studies of distillation protocols involving a fixed number of qubits. Specifically, we use a SAT solver to derive a series of no-go theorems that relate key figures of merit, including the number of qubits, the protocol depth, the factory distance, and the prefactor in the output error rate. For instance, we prove that any $T$-to-$T$ state distillation protocol using fewer than eight qubits can detect at most three errors, while any $T$-to-$\mathrm{CC}Z$ state distillation protocol using fewer than eight qubits can detect at most two errors. Our results also include new distillation protocols with the smallest number of qubits for a given distance in the literature, namely distance 4 and 5 $T$-to-$T$ state protocols supported on 10 and 11 qubits, as well as distance 3 and 4 $T$-to-$\mathrm{CC}Z$ state distillation protocols supported on 9 and 10 qubits.