Asymptotically Good Quantum Codes with Transversal Non-Clifford Gates (2408.09254v1)

Abstract: We construct quantum codes that support transversal $CCZ$ gates over qudits of arbitrary prime power dimension $q$ (including $q=2$) such that the code dimension and distance grow linearly in the block length. The only previously known construction with such linear dimension and distance required a growing alphabet size $q$ (Krishna & Tillich, 2019). Our codes imply protocols for magic state distillation with overhead exponent $\gamma=\log(n/k)/\log(d)\rightarrow 0$ as the block length $n\rightarrow\infty$, where $k$ and $d$ denote the code dimension and distance respectively. It was previously an open question to obtain such a protocol with a contant alphabet size $q$. We construct our codes by combining two modular components, namely, (i) a transformation from classical codes satisfying certain properties to quantum codes supporting transversal $CCZ$ gates, and (ii) a concatenation scheme for reducing the alphabet size of codes supporting transversal $CCZ$ gates. For this scheme we introduce a quantum analogue of multiplication-friendly codes, which provide a way to express multiplication over a field in terms of a subfield. We obtain our asymptotically good construction by instantiating (i) with algebraic-geometric codes, and applying a constant number of iterations of (ii). We also give an alternative construction with nearly asymptotically good parameters ($k,d=n/2^{O(\log^*n)}$) by instantiating (i) with Reed-Solomon codes and then performing a superconstant number of iterations of (ii).

• The paper introduces quantum codes with transversal CCZ gates that achieve linear growth in both dimension and distance.
• The construction leverages a novel transformation from classical AG codes and a multiplication-friendly concatenation scheme to maintain a constant alphabet size.
• The work enables efficient magic state distillation with overhead exponent approaching zero, advancing fault-tolerant quantum computing.

Analyzing Asymptotically Good Quantum Codes with Transversal Non-Clifford Gates

This paper, authored by Louis Golowich and Venkatesan Guruswami, addresses the construction of quantum codes that support transversal $CCZ$ gates over qudits of arbitrary prime power dimension $q$, including the binary case. These codes exhibit a critical property where both the code dimension and distance grow linearly with the block length. The construction is significant because it overcomes previous limitations where achieving such parameters required an increasing alphabet size. Here, a breakthrough is achieved with a constant alphabet size $q$.

Main Contributions

The primary contributions of this paper include:

1. New Quantum Code Construction: The authors introduce quantum codes that support transversal $CCZ$ gates over qudits of constant dimension $q$ such that the code dimension ($k$) and distance ($d$) grow linearly with the block length ($n$). The construction involves two modular components:
   1. Transformation from Classical to Quantum Codes: This part of the construction involves converting classical codes satisfying certain properties to quantum codes supporting transversal $CCZ$ gates.
   2. Concatenation Scheme for Alphabet Reduction: This reduction minimizes the alphabet size of codes supporting transversal $CCZ$ gates. It introduces quantum analogues of multiplication-friendly codes, which facilitate the expression of multiplication over a field in terms of a subfield.

2. Magic State Distillation Protocols: The research implies protocols for magic state distillation with overhead exponent $\gamma=\log(n/k)/\log(d)\rightarrow 0$ as $n \rightarrow \infty$. Notably, prior to this work, achieving such a protocol with a constant alphabet size $q$ remained an open question.

Technical Details

The construction of the quantum codes is instantiated via classical algebraic-geometric (AG) codes, and a constant number of iterations of the concatenation method with multiplication-friendly codes are applied. An alternative construction with near-optimal parameters is offered, employing Reed-Solomon codes followed by a superconstant number of concatenation iterations.

Key Theoretical Insights

1. Classical to Quantum Code Transformation: The authors generalize prior work on code construction leveraging product spectral techniques. By using classical AG codes—a family known for optimal asymptotic properties—the authors ensure the target quantum codes have robust parameters while remaining within a constant alphabet size.
2. Multiplication-Friendly Concatenation: By defining a concatenation scheme employing multiplication-friendly codes, the authors adeptly maintain the algebraic structure necessary to support transversal $CCZ$ and $U$ gates.
3. Encoding and Distance Analysis: Detailed attention is given to maintaining linear code dimension and distance in the resulting quantum codes. Through novel encoding functions and compatibility constraints, the constructed codes meet fault-tolerant requirements while enabling non-Clifford gate implementation.

Practical Implications

• Quantum Computation: The constructed codes play a crucial role in optimizing the resource overhead for fault-tolerant quantum computing, specifically in magic state distillation, which is pivotal for implementing universal quantum computation.
• Algorithm Performance: The efficient design and reduced overhead of the proposed magic state distillation protocols highlight significant practical advancements in quantum computational systems, especially pertinent in the context of near-term quantum devices.

Future Directions

While the presented work achieves substantial progress, several open questions remain:

• Low-Density Parity-Check (LDPC) Codes: Exploring methods to construct LDPC codes with similar properties could enhance fault tolerance by simplifying the measurement of stabilizers.
• Further Optimizations and Practical Implementations: Future efforts could investigate the optimization of constant factors in the presented constructions for specific practical applications in quantum hardware currently under development.

Conclusion

Golowich and Guruswami provide a landmark contribution to quantum error correction by developing asymptotically good quantum codes supporting transversal non-Clifford gates with constant alphabet sizes. These findings offer substantial theoretical insights and set the stage for practical advancements in fault-tolerant quantum computing.