One-shot distillation with constant overhead using catalysts

TL;DR

This paper addresses the prohibitive one-shot overhead in quantum resource distillation by introducing quantum catalysts. It proves a general overhead-reduction result, showing that any multi-shot distillation can be emulated catalytically in a single shot with overhead $\widetilde{C}_{\varepsilon,p}(\rho,\sigma)\le C_{\varepsilon,p}(\rho,\sigma)$ and catalyst reuse; in magic state distillation this enables constant-overhead, arbitrarily precise protocols and a tunable spacetime trade-off. It also develops a spacetime-catalysis framework to trade success probability for overhead, and extends catalytic methods to dynamical resources, establishing that channel mutual information $I({\mathcal N})$ governs catalytic convertibility of quantum channels. Overall, the work provides a versatile, experimentally friendly approach to optimize distillation costs and reveals an operational interpretation of channel mutual information in one-shot catalytic transformations, with practical implications for fault-tolerant quantum computing and quantum communication.

Abstract

Quantum resource distillation is a fundamental task in quantum information science and technology. Minimizing the overhead of distillation is crucial for the realization of quantum computation and other technologies. Here we explicitly demonstrate how, for general quantum resources, suitably designed quantum catalysts (i.e., auxiliary systems that remain unchanged before and after the process) enable distillation with constant overhead in the practical one-shot setting, thereby overcoming the established logarithmic lower bound for one-shot distillation overhead. In particular, for magic state distillation, our catalysis method paves a path for tackling the diverging batch size problem associated with code-based low-overhead protocols by enabling arbitrary reduction of the protocol size for any desired accuracy. Notably, this first yields constant-overhead magic state distillation methods with controllable protocol size. Furthermore, we demonstrate a tunable spacetime trade-off between overhead and success probability enabled by catalysts which offers significant versatility for practical implementation. Finally, we extend catalysis techniques to dynamical quantum resources and show that channel mutual information determines one-shot catalytic channel transformation, thereby advancing our understanding for both dynamical catalysis and information theory.

Figures (3)

• Figure 1: Comparison of distillation overhead $\log^\gamma(1/\varepsilon)$ across different settings. A fundamental no-go limit of $\gamma \geq 1$ has been shown to exist for one-shot (unassisted) distillation Fang2020PRLFangLiuPRXQ. We show in this work that this limit can be surpassed with the aid of quantum catalysts.
• Figure 2: An illustration for the steps of the procedure used to prove Theorem \ref{['thm: overheads']}, for the case where $n=15$, $m=5$, and therefore $k = 3$. Each dot represents a quantum state on the system $S^3$. Red dots correspond to groups of states $\zeta = \rho^{\otimes 3} \in \mathscr{D}(S^3)$, while blue dots represent embedded states $\hat{\sigma} = \sigma \otimes \pi^{\otimes 2} \in \mathscr{D}(S^3)$ that match the system of $\zeta$. The overall quantum state is a mixture of rows, each labeled with a classical register. In the first step, a classically controlled free operation is applied, transforming the last row from $\zeta^{\otimes 5} = \rho^{\otimes 15}$ to $\hat{\sigma}^{\otimes 5}$. The second step involves cyclically permuting the classical registers (i.e., the rows of dots), and the third step involves cyclically permuting the quantum registers (i.e., the columns of dots). The dashed boxes highlight the catalyst state, showing that it remains unchanged before and after the transformation.
• Figure 3: An illustration for the steps of the procedure used to prove Theorem \ref{['thm: reduce the overhead']}, for the case where $n=5$, $m=2$, and $k = 1$. Each dot represents a quantum state on the system $T$. Red dots correspond to the embedded source state $\tilde{\rho} \in \mathscr{D}(T)$, while blue dots represent the embedded target state $\tilde{\sigma} \in \mathscr{D}(T)$. Grey dots represent the embedded free state $\tilde{\theta} \in \mathscr{D}(T)$, which is orthogonal to $\tilde{\sigma}$. The overall quantum state is a mixture of rows, each labeled with a classical register. In the first step, a classically controlled free operation is applied, transforming the last row from $\tilde{\rho}^{\otimes 5}$ to $\tilde{\sigma}^{\otimes 2} \otimes \tilde{\theta}^{\otimes 3}$. The second step involves cyclically permuting the classical registers (i.e., the rows of dots), and the third step involves cyclically permuting the quantum registers (i.e., the columns of dots). The dashed boxes highlight the catalyst state, showing that it remains unchanged before and after the transformation.

Theorems & Definitions (19)

• Definition 1: Unassisted overhead
• Definition 2: One-shot catalytic distillation overhead
• Theorem 3
• proof
• Theorem 4
• proof
• Lemma 5
• proof
• Corollary 6
• Theorem 7