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Breaking the cubic barrier in the Solovay-Kitaev algorithm

Greg Kuperberg

TL;DR

This work upgrades the algorithmic Solovay–Kitaev theorem by achieving a gate-approximation length ℓ = O($n^α$) with α > $\log_φ(2) = 1.44042...$ for any finite symmetric gate set that densely generates a connected semisimple Lie group, along with a compressed length ŷℓ = O($n^{1+δ}$). The authors blend a multiscale framework that pairs roughly exponential steps with higher commutators inspired by Elkasapy–Thom and a zigzag golf technique to reach precise targets efficiently, reducing the exponent below 3 and approaching the doubly-exponential convergence of prior methods without incurring their cost. They generalize the results to arbitrary connected semisimple real Lie groups, with an additive distance term R in the noncompact case and corresponding bounds when group elements have bounded norm. The findings have direct implications for fault-tolerant quantum gate synthesis, providing tighter, scalable bounds on the resources required to approximate arbitrary gates across broad Lie-group gate sets. Overall, the paper delivers a principled, scalable framework that tightens the experiential and theoretical limits of efficient gate synthesis in quantum computing and group-theoretic approximation problems.

Abstract

We improve the Solovay--Kitaev theorem and algorithm for a general finite, inverse-closed generating set acting on a qudit. Prior versions of the algorithm efficiently find a word of length $O(n^{3+δ})$ to approximate an arbitrary target gate to $n$ bits of precision. Using two new ideas, each of which reduces the exponent separately, our new bound on the word length is $O(n^{1.44042\ldots+δ})$. Our result holds more generally for any finite set that densely generates any connected, semisimple real Lie group, with an extra length term in the noncompact case to reach group elements far away from the identity.

Breaking the cubic barrier in the Solovay-Kitaev algorithm

TL;DR

This work upgrades the algorithmic Solovay–Kitaev theorem by achieving a gate-approximation length ℓ = O() with α > for any finite symmetric gate set that densely generates a connected semisimple Lie group, along with a compressed length ŷℓ = O(). The authors blend a multiscale framework that pairs roughly exponential steps with higher commutators inspired by Elkasapy–Thom and a zigzag golf technique to reach precise targets efficiently, reducing the exponent below 3 and approaching the doubly-exponential convergence of prior methods without incurring their cost. They generalize the results to arbitrary connected semisimple real Lie groups, with an additive distance term R in the noncompact case and corresponding bounds when group elements have bounded norm. The findings have direct implications for fault-tolerant quantum gate synthesis, providing tighter, scalable bounds on the resources required to approximate arbitrary gates across broad Lie-group gate sets. Overall, the paper delivers a principled, scalable framework that tightens the experiential and theoretical limits of efficient gate synthesis in quantum computing and group-theoretic approximation problems.

Abstract

We improve the Solovay--Kitaev theorem and algorithm for a general finite, inverse-closed generating set acting on a qudit. Prior versions of the algorithm efficiently find a word of length to approximate an arbitrary target gate to bits of precision. Using two new ideas, each of which reduces the exponent separately, our new bound on the word length is . Our result holds more generally for any finite set that densely generates any connected, semisimple real Lie group, with an extra length term in the noncompact case to reach group elements far away from the identity.
Paper Structure (12 sections, 6 theorems, 53 equations, 1 figure, 2 algorithms)

This paper contains 12 sections, 6 theorems, 53 equations, 1 figure, 2 algorithms.

Table of Contents

  1. Introduction
  2. Ideas in the algorithm and proof
  3. Context and history
  4. Motivation for this work
  5. Modes of approximation
  6. Preliminaries
  7. Models of numbers, words, and computation
  8. Matrix Lie groups
  9. The algorithm for $\operatorname{SU}(2)$
  10. Geometry
  11. Roughly exponential steps
  12. Zigzag golf

Key Result

Theorem 1.1

Let $A = A^{-1} \subset \operatorname{SU}(d)$ be a finite set of matrices that densely generate $\operatorname{SU}(d)$ for some fixed $d$. Let where $\phi = (1+\sqrt5)/2$ is the golden ratio, and let $\delta > 0$. Then there is a polynomial time classical algorithm to approximate any $g \in \operatorname{SU}(d)$ to $n$ bits of precision by a product $w$ in $A$ with word length $\ell = O(n^\alpha)

Figures (1)

  • Figure :

Theorems & Definitions (7)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Proposition 4.1
  • Lemma 4.2
  • proof