Surpassing the fundamental limits of distillation with catalysts

2024-10-18

arXiv.org

AI Summary

The one-shot overhead of quantum resource distillation is fundamentally bounded below by a logarithmic function, impeding practical deployment. Method: We propose a novel distillation framework leveraging quantum catalysts, achieving— for the first time—constant-overhead one-shot magic-state distillation (with optimal constant equal to 1), thereby breaking the established logarithmic lower bound. Our approach introduces: (i) a general conversion mechanism from multi-round to one-shot protocols; (ii) asymptotically optimal catalytic codes with arbitrarily small batch sizes; (iii) a quantitative space–time tradeoff model linking overhead and success probability; and (iv) an extension of catalysis to dynamic quantum resources, endowing channel mutual information with operational meaning in the one-shot regime. Contributions: We resolve Wilming’s open problem on catalytic distillation; eliminate the batch-size divergence bottleneck inherent in conventional low-overhead code-based protocols; enable controllable output-size distillation at arbitrary precision; and establish a new paradigm for high-fidelity non-Clifford gate implementation.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Distributed Search
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• Responsible Web: Measurement, analysis, and circumvention of Web censorship
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

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.

Problem

• Overcoming logarithmic lower bound for quantum distillation overhead
• Solving diverging batch size problem in magic state distillation
• Enabling constant-overhead distillation with controllable protocol size

Innovation

• Catalysts enable constant-overhead quantum resource distillation
• Method allows arbitrary protocol size reduction for accuracy
• Catalysts provide tunable spacetime trade-off between overhead and success