A Framework for Approximating Qubit Unitaries (1510.03888v1)

Abstract: We present an algorithm for efficiently approximating of qubit unitaries over gate sets derived from totally definite quaternion algebras. It achieves $\varepsilon$-approximations using circuits of length $O(\log(1/\varepsilon))$, which is asymptotically optimal. The algorithm achieves the same quality of approximation as previously-known algorithms for Clifford+T [arXiv:1212.6253], V-basis [arXiv:1303.1411] and Clifford+$\pi/12$ [arXiv:1409.3552], running on average in time polynomial in $O(\log(1/\varepsilon))$ (conditional on a number-theoretic conjecture). Ours is the first such algorithm that works for a wide range of gate sets and provides insight into what should constitute a "good" gate set for a fault-tolerant quantum computer.

• The paper presents an efficient algorithm that achieves ε-approximations with circuit lengths scaling as O(log(1/ε)).
• The framework generalizes unitary approximation across diverse gate sets derived from totally definite quaternion algebras.
• Extensive numerical experiments validate the approach, highlighting reduced compilation overhead for fault-tolerant quantum computing.

A Framework for Approximating Qubit Unitaries: A Critical Overview

The paper presents a sophisticated algorithmic framework aimed at efficiently approximating single-qubit unitaries over gate sets that arise from totally definite quaternion algebras. This is crucial for quantum computing, particularly in constructing circuits composed of a finite set of gates supported by quantum architectures, since most architectures, especially those based on fault-tolerant quantum computing, support only a finite set of unitary gates. The innovation provided by this framework is its generality—it is the first algorithm enabling such approximation for a wide range of gate sets beyond known exemplars like Clifford+T, V-basis, and Clifford+π/12.

Key Contributions

1. Efficient Approximation Algorithm: The algorithm achieves ε-approximations using circuit lengths that scale optimally with O(log(1/ε)), which aligns with theoretical bounds on circuit lengths derived from volume arguments in SU(2). The ability to handle approximations efficiently in logarithmic time is a substantial achievement, especially given the complexities of underlying number-theoretic structures.
2. Wide Applicability: This work generalizes specific known cases to broader classes of gate sets derived from totally definite quaternion algebras. By doing this, it provides insights into the criteria that make a gate set beneficial for quantum computing, which incorporates an algebraic understanding of quaternions mapped to special unitaries.
3. Exact Synthesis and Cost Optimization: The framework utilizes the results on exact synthesis from previous works, providing insight into unitary approximation in quantum circuits and related cost functions. The cost functions focus on the specific gate set, allowing optimization in constructing approximating circuits.
4. Conjecture reliance: The termination and average-case runtime polynomial in log(1/ε) depend on a number-theoretic conjecture regarding distribution properties of solutions to certain norm equations in number fields. This aligns it with similar conjectural aspects in known algorithms for other gate sets.

Numerical and Experimental Results

The paper includes comprehensive numerical experiments benchmarking the framework's performance across different gate sets like Clifford+T, V-basis, and newly proposed sets, providing practical validation of the theoretical promises. These experiments reveal both the efficiency gains and the precision limits achievable with the new method compared to earlier algorithms.

Potential Implications

The implications of this research stretch both theoretical and practical fields. Theoretically, it showcases the potential of number-theoretic methods in practical quantum computation, especially in efficient gate synthesis. Practically, this framework could substantially reduce the overheads in compiling quantum algorithms to fault-tolerant gates, thus bringing practical quantum computation closer to reality.

Speculation and Future Developments

Looking ahead, the critical steps involve addressing the conjectural gaps in the distribution properties of possible solutions to the norm equations, potentially through advancements in algebraic number theory. Further expansion to quaternion algebras that are not totally definite could widen the scope, influencing error-correcting codes and architecture-specific quantum gates.

Overall, this work represents a significant step forward in quantum computer realization, offering both the robust mathematical underpinning and practical algorithms necessary to tackle unitary approximations across various quantum computing platforms. This research elucidates the algebraic structures underlying quantum gates, paving the way for more advanced quantum compilation techniques capable of meeting the demands of scalable quantum computing architectures.