Low-overhead Magic State Circuits with Transversal CNOTs (2501.10291v1)

Abstract: With the successful demonstration of transversal CNOTs in many recent experiments, it is the right moment to examine its implications on one of the most critical parts of fault-tolerant computation -- magic state preparation. Using an algorithm that can recompile and simplify a circuit of consecutive multi-qubit phase rotations, we manage to construct fault-tolerant circuits for CCZ, CS and T states with minimal T-depth and also much lower CNOT depths and qubit counts than before. These circuits can play crucial roles in fault-tolerant computation with transversal CNOTs, and we hope that the algorithms and methods developed in this paper can be used to further simplify other protocols in similar contexts.

• The paper introduces a recompiled circuit design that reduces CNOT depth, T-depth, and qubit counts for efficient magic state preparation.
• The paper presents fault-tolerant implementations of CCZ and CS gates by adapting Jones constructions through transversal CNOTs.
• The paper optimizes T-gate distillation circuits, lowering computational overhead for scalable fault-tolerant quantum operations.

Low-overhead Magic State Circuits with Transversal CNOTs

The paper "Low-overhead Magic State Circuits with Transversal CNOTs" by Nicholas Fazio, Mark Webster, and Zhenyu Cai addresses a key challenge in quantum computing—optimizing the process of magic state preparation, particularly within the framework of fault-tolerant quantum computation. The research leverages recent developments in quantum architectures that support transversal CNOT gates, offering new approaches to reducing computational overhead, specifically CNOT depths, qubit counts, and $T$-depth.

Overview of Contributions

The primary focus of the paper is on devising algorithms that recompile and simplify circuits based on consecutive multi-qubit phase rotations. By doing so, the authors have constructed circuits for $\ket{CCZ}$, $\ket{CS}$, and $\ket{T}$ states with reduced $T$-depth while also minimizing the required CNOT depths and qubit counts compared to previous methodologies. These improvements have critical implications for fault-tolerant quantum computing where overhead is a significant concern.

Key Results

1. Circuit Optimization: The paper proposes a method for optimizing CNOT+$T$ circuits essential for magic state preparation. The approach involves an algorithm that systematically reduces the complexity of these circuits without the need for auxiliary qubits, primarily through a series of phase rotation and CNOT gate transformations.
2. Fault-tolerant $\ket{CCZ}$ and $\ket{CS}$ Gates: The authors present constructions for fault-tolerant implementation of $\ket{CCZ}$ and $\ket{CS}$ gates. They achieve this by reinterpreting the Jones fault-tolerant construction within a framework that considers transversal CNOT availability, demonstrating reductions in CNOT depth and qubit requirements.
3. $T$-gate Distillation: Beyond $\ket{CCZ}$ and $\ket{CS}$, the work explores optimizations in distillation circuits for the $T$ gate, a critical component in quantum algorithms that require non-Clifford operations. The methodologies promise lower qubit overhead through enhanced circuit design.

Implications and Future Directions

The implications of this research are particularly relevant for currently evolving hardware platforms such as trapped ions and neutral atoms, which have demonstrated enhanced qubit connectivity and transversal CNOT capabilities. The authors convincingly illustrate that utilizing transversal operations can lead to substantial reductions in the spacetime overhead necessary for fault-tolerant quantum computations.

Theoretical formulations presented in the paper may refine existing models and calculations for fault-tolerant quantum circuits, reducing costs associated with error rates and state synthesis. Practically, these advances might accelerate advancements in quantum technology, making large-scale, fault-tolerant quantum computers more feasible.

Future work could explore further generalization of the described algorithms to account for more diverse quantum gate sets and expand investigations into other bases beyond phase rotations. Additionally, implementations in real-world architectures could provide valuable feedback for refining and validating the proposed methodologies.

In summary, this paper presents a comprehensive approach to minimizing the overhead of magic state preparation circuits, addressing a pivotal aspect of quantum computing that holds significant potential for innovation in the fault-tolerant domain.