Efficient Decomposition of Single-Qubit Gates into $V$ Basis Circuits (1303.1411v1)

Abstract: We develop the first constructive algorithms for compiling single-qubit unitary gates into circuits over the universal $V$ basis. The $V$ basis is an alternative universal basis to the more commonly studied ${H,T}$ basis. We propose two classical algorithms for quantum circuit compilation: the first algorithm has expected polynomial time (in precision $\log(1/\epsilon)$) and offers a depth/precision guarantee that improves upon state-of-the-art methods for compiling into the ${H,T}$ basis by factors ranging from 1.86 to $\log_2(5)$. The second algorithm is analogous to direct search and yields circuits a factor of 3 to 4 times shorter than our first algorithm, and requires time exponential in $\log(1/\epsilon)$; however, we show that in practice the runtime is reasonable for an important range of target precisions.

• The paper introduces two novel algorithms for efficiently decomposing single-qubit gates into circuits using the universal V basis, departing from the conventional {H, T} basis.
• The first algorithm offers polynomial time complexity with respect to precision, showing superior depth/precision trade-offs compared to {H, T} methods, while the second yields significantly shorter circuits with practical runtime for relevant precision ranges.
• This work provides theoretical insights into alternative universal bases and has practical implications for enhancing quantum resource management and optimizing circuit design for fault-tolerant quantum computation.

Efficient Decomposition of Single-Qubit Gates into $V$ Basis Circuits

This paper presents a seminal paper in the field of quantum computing, focusing on the decomposition of single-qubit unitary gates into circuits based on the universal $V$ basis. Departing from the conventional $\{H, T\}$ basis, the authors introduce novel algorithms that enhance the efficiency of quantum gate compilation in terms of circuit depth and precision.

Summary of Contributions

The authors develop two algorithms for compiling single-qubit gates over the $V$ basis. The first algorithm is notable for its expected polynomial time complexity with respect to precision, $\log(1/\epsilon)$. This algorithm showcases improved scalability and a depth/precision trade-off that proves superior to the $\{H, T\}$ methods by factors ranging from 1.86 to $\log_2(5)$.

The second algorithm, while more computationally intensive with a time complexity exponential in $\log(1/\epsilon)$, excels in delivering shorter circuits. It achieves this by producing output that is three to four times shorter than the first algorithm, while maintaining practical runtime performance for relevant precision ranges.

Numerical Results and Analysis

The paper provides compelling numeric demonstrations where the $V$ basis allows for significantly reduced circuit lengths. The randomized algorithm yields $\epsilon$-approximations with a circuit depth approximately $\leq 12 \log_5(2/\epsilon)$, while the direct compilation approach often results in depth $\leq 3\log_5(1/\epsilon)$. In edge cases, circuits of depth $4\log_5(2/\epsilon)$ are produced. These results uphold the theoretical lower bounds and challenge the established norms practiced with $\{H, T\}$ decomposition strategies.

Theoretical and Practical Implications

Theoretical implications of this work extend to foundational quantum information science, providing new insights into efficient gate synthesis in alternative bases. This work invigorates the exploration of alternative universal bases, emphasizing a strategic advantage in circuit synthesis and optimization tasks which are vital for fault-tolerant quantum computations.

In practice, the findings suggest meaningful enhancements to the quantum resource management strategies with prospective implications in scalable quantum architectures. The reduced circuit depth has potential applications in reducing error footprints and resource costs within quantum algorithms crucial for advanced technologies.

Future Prospects

Future research may explore identifying lower-cost, fault-tolerant implementations of $V$ basis gates. Such innovations would capitalize on the efficient decomposition algorithms proposed, potentially allowing for direct physical implementations of these gates in quantum processors. Further exploration of other universal bases might yield additional optimization techniques suited for specific quantum computational tasks.

In conclusion, the presented work advances both theoretical and computational aspects of single-qubit gate decomposition. By challenging the conventional approaches and offering reduced circuit depths, the research paves the way for new opportunities within quantum circuit design and resource optimization, priming the field for continuous quantum computational advancements.