The Solovay-Kitaev algorithm

TL;DR

The paper addresses the problem of efficiently compiling an arbitrary single-qubit gate into a fixed finite universal gate set, with extensions to qudits and SU(d). It introduces the Solovay-Kitaev algorithm, a recursive construction that uses balanced group commutators to iteratively improve approximation accuracy, achieving an error scaling $oldsymbol{psilon_n} obreak= obreak c_{ m approx}oldsymbol{psilon_{n-1}}^{3/2}$ and polylogarithmic classical runtime with a gate-sequence length of $O(oldsymbol{.97}(1/oldsymbol{psilon}))$. The framework is developed first for qubits and then generalized to higher dimensions, with explicit constants and complexity analyses, including the dependence on the dimension $d$ for balanced commutators in SU(d). The work situates these results among prior constructive and non-constructive approaches, highlighting practical implications for fault-tolerant quantum computation and the efficient compilation of algorithms such as Shor's into fault-tolerant gate sets.

Abstract

This pedagogical review presents the proof of the Solovay-Kitaev theorem in the form of an efficient classical algorithm for compiling an arbitrary single-qubit gate into a sequence of gates from a fixed and finite set. The algorithm can be used, for example, to compile Shor's algorithm, which uses rotations of $π/ 2^k$, into an efficient fault-tolerant form using only Hadamard, controlled-{\sc not}, and $π/ 8$ gates. The algorithm runs in $O(\log^{2.71}(1/ε))$ time, and produces as output a sequence of $O(\log^{3.97}(1/ε))$ quantum gates which is guaranteed to approximate the desired quantum gate to an accuracy within $ε> 0$. We also explain how the algorithm can be generalized to apply to multi-qubit gates and to gates from $SU(d)$.

Theorems & Definitions (4)

• Definition 1
• Lemma 1
• Lemma 2
• Lemma 3