Quantum fault tolerance with constant-space and logarithmic-time overheads (2411.03632v1)

Abstract: In a model of fault-tolerant quantum computation with quick and noiseless polyloglog-time auxiliary classical computation, we construct a fault tolerance protocol with constant-space and $\widetilde{O}(\log N)$-time overhead, where $\widetilde{O}(\cdot)$ hides sub-polylog factors. Our construction utilizes constant-rate quantum locally testable codes (qLTC), new fault-tolerant gadgets on qLTCs and qLDPC codes, and a new analysis framework. In particular, 1) we develop a new simple and self-contained construction of magic state distillation for qubits using qudit quantum Reed-Solomon codes with $(\log \frac{1}{\varepsilon})^{\gamma}$ spacetime overhead, where $\gamma \rightarrow 0$. 2) We prove that the recent family of almost-good qLTCs of Dinur-Lin-Vidick admit parallel single-shot decoders against adversarial errors of weight scaling with the code distance. 3) We develop logical state preparation and logical gate procedures with $\widetilde{O}(1)$-spacetime overhead on qLTCs. 4) To combine these ingredients, we introduce a new framework of fault tolerance analysis called the weight enumerator formalism. The framework permits easy formal composition of fault-tolerant gadgets, so we expect it to be of independent interest. Our work gives the lowest spacetime overhead to date, which, for the first time, matches that of classical fault tolerance up to sub-polylog factors. We conjecture this is optimal up to sub-polylog factors.

• The paper introduces a protocol that achieves near-constant space and logarithmic-time overheads through innovative magic state distillation using Reed-Solomon codes.
• It employs quantum locally testable and low-density parity-check codes with single-shot decoders to enhance error resilience against adversarial faults.
• The approach streamlines logical state preparation and resource management, paving the way for practical and scalable fault-tolerant quantum computing.

Quantum Fault Tolerance with Constant-Space and Logarithmic-Time Overheads

The paper introduces a sophisticated quantum fault tolerance protocol built to improve spatial and temporal overheads in quantum computations. The model addresses faults using auxiliary classical computation with a quick, polyloglog-time structure, significantly refining previous protocols by Yamasaki and Koashi, and utilizing constant-rate quantum locally testable codes (qLTC) alongside quantum low-density parity-check (qLDPC) codes.

Key Contributions and Results

The primary accomplishments of this research include:

1. Magic and Stabilizer State Distillation: The paper presents a novel magic state distillation protocol, achieving spacetime overhead $(\log \frac{1}{\varepsilon})^{\gamma}$, where $\gamma \rightarrow 0$ as $\varepsilon \rightarrow 0$. By leveraging Reed-Solomon codes, they construct a method to distill states, minimizing spacetime overhead compared to preceding works. Stabilizer state distillation aligns with this novel approach, ensuring low spacetime overhead remains.
2. Single-Shot Decoders for Quantum Codes: Expanding on cubical complex-based quantum codes, the paper proves the existence of sequential and parallel single-shot decoders, enhancing error-resiliency. This methodology adapts to the Dinur-Lin-Vidick (DLV) almost-good qLTCs and the good qLDPC codes seen in Dinur-Hsieh-Lin-Vidick (DHLV), guaranteeing robustness against adversarial errors.
3. State Preparation and Logical Operations: Through newly developed distillation schemes, robust fault-tolerant logical state preparation is achieved with $\widetilde{O}(1)$ spacetime overhead on qLTCs. This ensures logical gates are effective and efficient on both qLTC and qLDPC platforms.
4. Resource and Logical Operations on qLDPC: Implementation of multiple resource states allows flexible preparation of qLDPC logic, simplifying addressability in logical state preparation and opening avenues for transversal gates.

Theoretical and Practical Implications

From a theoretical standpoint, optimizing spacetime overhead addresses long-standing questions in the field, potentially steering the design of future quantum computers capable of fault-tolerant operations autonomously. Practically, these advancements bring us closer to realization with a constant-space protocol, suggesting feasible implementations within foreseeable technological advancements.

Speculative Future Developments

The methods introduced posit a framework for achieving virtually ideal overheads in quantum computation fault tolerance. Future research could delve into further classically-assisted state preparations, simplifying or bypassing existing distillation schemes. Exploring instantaneous classical error correction might diminish complexities further. Additionally, increasing the robustness of qLTCs may yield consistent overhead reductions or open paths to alternative state-of-the-art quantum error-correcting methods.

Conclusion

This research effectively pioneers a scalable approach to quantum fault tolerance, drastically reducing spacetime overheads. By integrating qLTCs and addressing distinct logical and resource states, this paper lays the groundwork for future theoretical paradigms and practical innovations, promising enhanced reliability and efficiency in quantum computations.