Surpassing the fundamental limits of distillation with catalysts

Abstract: Quantum resource distillation is a fundamental task in quantum information science. Minimizing the distillation overhead, i.e., the amount of noisy source states required to produce some desired output state within some target error, is crucial for the scalability of quantum computation and communication. Here, we show that quantum catalysts -- an additional resource that facilitates the transformation but remains unchanged before and after the process -- can help surpass previously known fundamental limitations on distillation overhead. Specifically, we show that multi-shot distillation protocols can be converted into one-shot catalytic protocols, which hold significant practical benefits, while maintaining the distillation overhead. In particular, in the context of magic state distillation, our result indicates that the code-based low-overhead distillation protocols that rely on divergingly large batches can be promoted to the one-shot setting where the batch volume can be arbitrarily small for any accuracy. Combining with very recent results on asymptotically good quantum codes with transversal non-Clifford gates, we demonstrate that magic state distillation with constant overhead can be achieved with controllable output size using catalytic protocols. Furthermore, we demonstrate that catalysis enables a spacetime trade-off between overhead and success probability. Notably, we show that the optimal constant for constant-overhead catalytic magic state distillation can be reduced to $1$ at the price of compromising the success probability by a constant factor. Finally, we present an illustrative example that extends the catalysis techniques to the study of dynamic quantum resources. This provides the channel mutual information with a one-shot operational interpretation, thereby addressing an open question posed by Wilming.