Locality vs Quantum Codes (2409.15203v1)

Abstract: This paper proves optimal tradeoffs between the locality and parameters of quantum error-correcting codes. Quantum codes give a promising avenue towards quantum fault tolerance, but the practical constraint of locality limits their quality. The seminal Bravyi-Poulin-Terhal (BPT) bound says that a $[[n,k,d]]$ quantum stabilizer code with 2D-locality must satisfy $kd^2\le O(n)$. We answer the natural question: for better code parameters, how much "non-locality" is needed? In particular, (i) how long must the long-range interactions be, and (ii) how many long-range interactions must there be? We give a complete answer to both questions for all $n,k,d$: above the BPT bound, any 2D-embedding must have at least $\Omega(#^*)$ interactions of length $\Omega(\ell^*)$, where $#^*= \max(k,d)$ and $\ell^*=\max\big(\frac{d}{\sqrt{n}}, \big( \frac{kd^2}{n} \big)^{1/4} \big)$. Conversely, we exhibit quantum codes that show, in strong ways, that our interaction length $\ell^*$ and interaction count $#^*$ are asymptotically optimal for all $n,k,d$. Our results generalize or improve all prior works on this question, including the BPT bound and the results of Baspin and Krishna. One takeaway of our work is that, for any desired distance $d$ and dimension $k$, the number of long-range interactions is asymptotically minimized by a good qLDPC code of length $\Theta(\max(k,d))$. Following Baspin and Krishna, we also apply our results to the codes implemented in the stacked architecture and obtain better bounds. In particular, we rule out any implementation of hypergraph product codes in the stacked architecture.

• The paper demonstrates that exceeding the BPT bound requires any 2D-local stabilizer code to incorporate at least Ω(max(k, d)) long-range interactions spanning distances of at least Ω(max(d/√n, (kd²/n)^(1/4))).
• It generalizes and improves previous bounds by analyzing worst-case scenarios for interaction lengths and counts in quantum error-correcting codes.
• It shows that optimal qLDPC codes can minimize long-range interactions, providing actionable insights for designing robust fault-tolerant quantum computing architectures.

Locality vs Quantum Codes

This paper establishes new, optimal tradeoffs between locality and parameters for quantum error-correcting codes (QECCs). Quantum codes are vital for achieving fault tolerance in quantum computation. However, practical implementations are constrained by locality, impacting their performance. The seminal Bravyi-Poulin-Terhal (BPT) bound, which limits the performance of 2D-local quantum stabilizer codes, is $kd^2 \leq O(n)$. This paper addresses the question of how much non-locality is needed to exceed the BPT bound. Specifically, it examines (i) the extent of long-range interactions and (ii) the quantity required to improve code parameters.

Main Contributions

The primary contribution is a detailed answer to the above questions for all $n, k, d$. The paper proves that for $kd^2 \geq O(n)$, any 2D-embedded stabilizer code must have at least $\Omega(\#^*)$ interactions of length $\Omega(\ell^*)$, where $\#^* = \max(k, d)$ and $$\ell^* = \max(\frac{d}{\sqrt{n}}, (\frac{kd^2}{n})^{1/4})$$. Additionally, the authors demonstrate quantum codes showing these bounds are asymptotically optimal for all parameters.

Detailed Findings

• Interaction Length and Count: The paper shows that any 2D-embedded $[[n,k,d]]$ stabilizer code exceeding the BPT bound needs a substantial number of long-range interactions. Specifically, the number of interactions must be at least $\Omega(\max(k, d))$ and these interactions must span distances of at least $$\Omega(\max(\frac{d}{\sqrt{n}}, (\frac{kd^2}{n})^{1/4}))$$.
• Generalization and Improvement: The results generalize and improve upon prior works, including the bounds given by Bravyi-Poulin-Terhal and findings by Baspin and Krishna. This is achieved by exploring the worst-case scenarios for interaction lengths and counts, providing a more comprehensive understanding of locality constraints in quantum codes.
• Stacked Architecture Analysis: Applying these findings to practical architectures, the paper demonstrates superior bounds for codes implemented in the stacked architecture. Specifically, it rules out any implementation of hypergraph product codes, which was previously undetermined.

Implications

The theoretical and practical implications of this research are substantial. By establishing precise bounds on the locality required for quantum error-correcting codes, this paper influences both the design and implementation of future quantum computers. It suggests that for any desired distance and dimension, minimizing the number of long-range interactions is best achieved using an optimal qLDPC code, thus providing a substantial guideline for practical implementations.

Future Directions

Given the optimality of the presented bounds, future research might explore:

• Generalizations to Higher Dimensions: Extending these results to 3D or higher-dimensional embeddings, potentially leading to even stronger coding schemes.
• Improved qLDPC Codes: Further research into qLDPC codes that might offer better tradeoffs for specific applications or fault-tolerant quantum computing models.
• New Architectures: Exploring novel architectures that can leverage the optimal bounds discovered to maximize the performance and feasibility of quantum computers.

Conclusion

This paper makes significant strides in elucidating the tradeoffs between locality and performance in quantum error-correcting codes. By providing a stringent analysis of interaction length and count required for exceeding the BPT bound, it sets a new benchmark for both theoretical and practical advancements in the quantum computing field. These findings have both immediate and long-term implications, guiding future research and development in quantum fault tolerance.