Efficient Clifford+T approximation of single-qubit operators (1212.6253v2)

Abstract: We give an efficient randomized algorithm for approximating an arbitrary element of $SU(2)$ by a product of Clifford+$T$ operators, up to any given error threshold $\epsilon>0$. Under a mild hypothesis on the distribution of primes, the algorithm's expected runtime is polynomial in $\log(1/\epsilon)$. If the operator to be approximated is a $z$-rotation, the resulting gate sequence has $T$-count $K+4\log_2(1/\epsilon)$, where $K$ is approximately equal to $10$. We also prove a worst-case lower bound of $K+4\log_2(1/\epsilon)$, where $K=-9$, so that our algorithm is within an additive constant of optimal for certain $z$-rotations. For an arbitrary member of $SU(2)$, we achieve approximations with $T$-count $K+12\log_2(1/\epsilon)$. By contrast, the Solovay-Kitaev algorithm achieves $T$-count $O(\log^c(1/\epsilon))$, where $c$ is approximately $3.97$.

• The paper presents a randomized algorithm that approximates any SU(2) operator with Clifford+T gates in expected polynomial time relative to log(1/ε).
• The paper optimizes T-count for z-rotations, achieving a bound of K + 4log2(1/ε) and outperforming the traditional Solovay-Kitaev algorithm.
• The paper leverages advanced algebraic number theory and empirical validation to ensure efficient, high-precision quantum gate synthesis for scalable quantum computing.

Efficient Clifford+$T$ approximation of single-qubit operators

The presented paper explores the field of quantum information theory, particularly focusing on the problem of decomposing single-qubit operators into a product of Clifford+$T$ operators. This research is crucial as the Clifford+$T$ gate set is well-established for fault-tolerant quantum computation. The paper introduces an efficient randomized algorithm that approximates any element of $SU(2)$ with a product of these operators, achieving a precinct error threshold, denoted as $\epsilon > 0$.

Key Contributions and Findings

1. Randomized Algorithm with Polynomial Complexity: The core contribution is a randomized algorithm that, under a certain mild hypothesis on the distribution of primes, offers an expected polynomial runtime concerning $\log(1/\epsilon)$. This conjecture aligns well with efficient quantum computing, as achieving fast and accurate approximations is critical.
2. $T$-Count Optimization: A critical metric for efficiency in these methodologies is the $T$-count, which denotes the number of $T$ gates in the decomposition. The paper reports a $T$-count for $z$-rotations according to $K + 4\log_2(1/\epsilon)$, with $K$ being approximately 10. Moreover, the paper demonstrates a lower bound with $K = -9$, suggesting that the algorithm is close to optimal for certain $z$-rotations.
3. Comparison with Solovay-Kitaev Algorithm: The canonical Solovay-Kitaev algorithm previously set the standard for this problem, achieving circuit decompositions with $T$-count $O(\log^c(1/\epsilon))$, where $c$ is approximately 3.97. The new algorithm significantly lowers this exponent to 1, marking a substantial theoretical improvement.
4. Technical Machinery and Proofs: The paper is rigorous in its approach, utilizing complex algebraic number theory to underpin its algorithms. It lays out detailed proofs for Diophantine equations relevant for the synthesis of unitary operators over $Z[\omega]$ and applies advanced concepts like integer-based norm calculations and Euclidean domains.
5. Experimental Validation: The authors provide empirical performance data, showcasing the algorithm's practical viability for high precision decompositions, i.e., for $\epsilon$ as small as $10^{-100}$ with moderate computational effort.

Theoretical and Practical Implications

The research holds significant implications. Theoretically, it strengthens the landscape of quantum gate synthesis by narrowing down the optimality space concerning $T$-count metrics. Practically, the direct application is in the preparation and manipulation of quantum states with high fidelity, critical for building scalable quantum computers.

Speculations for Future AI Developments

The intersection of fault-tolerant quantum computing and advanced decomposition methods, as illustrated in this paper, paves the way for innovations in quantum algorithm design and potentially improving error correction techniques. If extended further, similar approaches might benefit streamlined quantum machine learning models, optimizing resource-intensive quantum state transformations.

Overall, this paper presents a detailed and impactful paper that advances both theoretical and practical aspects of single-qubit gate decomposition, promising potential improvements in quantum computational efficiency and complexity.