[Paper Review] A Framework for Approximating Qubit Unitaries

Vadym Kliuchnikov, Alex Bocharov+2 more|arXiv (Cornell University)|Oct 13, 2015

Quantum Computing Algorithms and Architecture12 citations

TL;DR

This paper presents a general framework for efficiently approximating single-qubit unitaries over gate sets derived from totally definite quaternion algebras, achieving ε-approximations using circuits of length O(log(1/ε))—asymptotically optimal. The algorithm runs in average polynomial time in log(1/ε), conditional on a number-theoretic conjecture, and applies to a wide range of fault-tolerant gate sets including Clifford+T, V-basis, and Clifford+√T.

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+$π/12$ [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.

Motivation & Objective

• To develop a general, efficient algorithm for approximating single-qubit unitaries over a broad class of universal gate sets used in fault-tolerant quantum computing.
• To address the open question of what mathematical properties make a gate set amenable to efficient approximation algorithms.
• To unify and generalize existing algorithms for specific gate sets such as Clifford+T, V-basis, and Clifford+π/12 into a single mathematical framework.
• To provide theoretical and empirical evidence that the efficiency of approximation is tied to number-theoretic properties of the underlying gate set.
• To demonstrate the framework's applicability to new gate sets, including Clifford+√T, and reproduce known results for established sets.

Proposed method

• Leverages the algebraic structure of totally definite quaternion algebras to model universal gate sets in SU(2), enabling number-theoretic analysis of unitary approximation.
• Reduces the problem of approximating a unitary to finding short factorizations in the multiplicative group of a maximal order in a quaternion algebra.
• Uses the reduced norm and unit group structure to classify and group equivalent unitaries (up to Clifford equivalence), minimizing circuit cost.
• Employs a recursive approximation procedure that iteratively reduces the distance to the target unitary using generators from the gate set.
• Relies on a number-theoretic conjecture (generalizing prior conjectures in [63, 8, 9]) to prove average-case polynomial runtime in log(1/ε).
• Implements the algorithm in the Magma computer algebra system and validates it across multiple gate sets, including Clifford+T+V and Clifford+√T.

Experimental results

Research questions

• RQ1 What structural properties of a gate set enable efficient approximation of arbitrary single-qubit unitaries using circuits of length O(log(1/ε))?
• RQ2 Can a unified mathematical framework be developed to explain why efficient approximation algorithms exist for specific gate sets like Clifford+T and V-basis?
• RQ3 How do the number-theoretic properties of the entries in the gate set’s generators relate to the efficiency of the approximation algorithm?
• RQ4 To what extent can the framework be generalized beyond known gate sets, such as Clifford+√T or other 'Golden Gate' constructions?
• RQ5 What evidence supports the conjecture that the algorithm runs in average polynomial time in log(1/ε), and how does this relate to existing results in computational number theory?

Key findings

• The algorithm achieves ε-approximations of any single-qubit unitary using circuits of length O(log(1/ε)), which is asymptotically optimal.
• The algorithm runs in average time polynomial in log(1/ε), conditional on a number-theoretic conjecture, matching the efficiency of prior specialized algorithms.
• The framework successfully reproduces known results for Clifford+T, V-basis, and Clifford+π/12 gate sets, validating its generality.
• Numerical experiments on Clifford+T+V show that the algorithm produces circuits with low cost vectors, with average runtime under 1 second per approximation for ε = 10⁻⁶.
• The algorithm identifies 26 distinct gates from the Clifford+T+V set corresponding to quaternions of reduced norm 5, with 20 of them expressible via combinations of T and V gates, suggesting potential cost savings.
• The results provide indirect evidence that number-theoretic conjectures underlying efficient synthesis may extend beyond known gate sets to a broader class of 'Golden Gate' constructions.