[Paper Review] Breaking the cubic barrier in the Solovay-Kitaev algorithm

Greg Kuperberg|arXiv (Cornell University)|Jun 22, 2023

Scientific Computing and Data Management4 citations

TL;DR

This paper improves the Solovay-Kitaev algorithm by breaking the long-standing cubic barrier in gate approximation complexity, achieving a word length of $O((\log 1/\epsilon)^{1.44042\ldots+\delta})$ for approximating unitary gates in $\mathrm{SU}(d)$ and more general semisimple Lie groups. The improvement stems from a novel multiscale framework combining exponential and doubly exponential scaling with commutator-based group elements, reducing the exponent below 1.44 via the golden ratio $\phi$. This result significantly enhances efficiency in quantum circuit synthesis and group generation.

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.

Motivation & Objective

• To overcome the longstanding $O((\log 1/\epsilon)^3)$ complexity bound in the Solovay-Kitaev algorithm for quantum gate approximation.
• To generalize the improved bound to any connected, semisimple real Lie group, including non-compact cases.
• To develop a multiscale algorithm that combines the efficiency of exponential scaling with the precision of doubly exponential convergence.
• To establish a tight bound on the nilpotency class of group words generated by commutator recurrences, linking it to the golden ratio.
• To provide a constructive algorithm with explicit word length bounds for arbitrary finite, inverse-closed gate sets generating $\mathrm{SU}(d)$.

Proposed method

• Introduces a two-stage multiscale framework: first generating roughly exponential steps via commutator recurrences, then refining them into precise approximations.
• Uses a recurrence $\omega_{n+2} = [\omega_{n+1}^{-1}, \omega_n]$ to generate group elements with word length $O(2^n)$ and nilpotency class $\Omega(\phi^n)$, where $\phi$ is the golden ratio.
• Applies the commutator identity $[a,b] = aba^{-1}b^{-1}$ to generate non-abelian free subgroups, ensuring sufficient non-commutativity for approximation.
• Employs a recursive improvement process where coarse approximations are refined using shorter, higher-precision words from earlier stages.
• Uses matrix representations in $\mathrm{SU}(2)$ with Pauli matrices to evaluate the canonical nilpotency class and prove tight bounds on word complexity.
• Establishes that the word length $\ell = O((\log 1/\epsilon)^\alpha)$ with $\alpha > \log_\phi 2 \approx 1.44042$, using the golden ratio to minimize the exponent.

Experimental results

Research questions

• RQ1 Can the exponent in the Solovay-Kitaev word length bound be reduced below 3 for general gate sets?
• RQ2 What is the optimal trade-off between step size and refinement cost in multiscale group approximation?
• RQ3 How does the nilpotency class of commutator-generated words relate to their word length and approximation power?
• RQ4 Can the Solovay-Kitaev theorem be generalized to non-compact semisimple Lie groups with controlled distance from identity?
• RQ5 Is the bound $\alpha = \log_\phi 2$ tight for the minimal achievable exponent in the word length?

Key findings

• The paper achieves a word length of $O((\log 1/\epsilon)^{1.44042\ldots+\delta})$ for approximating any gate in $\mathrm{SU}(d)$, breaking the previous cubic barrier of $O((\log 1/\epsilon)^{3+\delta})$.
• The exponent $\log_\phi 2 \approx 1.44042$ is proven to be optimal for the given multiscale construction, arising from the Fibonacci recurrence in the nilpotency class of commutator words.
• For any connected, semisimple real Lie group $G$, the algorithm achieves $\ell = O((\log 1/\epsilon)^\alpha + d_G(g,1))$, where $d_G$ is a left-invariant Finsler metric.
• The nilpotency class of the word $\omega_n$ generated by the recurrence $\omega_{n+2} = [\omega_{n+1}^{-1}, \omega_n]$ is exactly $f_n$, the $n$th Fibonacci number, and this is tight in $\mathrm{SU}(2)$.
• The word length is precisely $\operatorname{len}(\omega_n) = (13 \cdot 2^{n-2} + c)/7$ for $n \geq 3$, with $c$ depending on $n \mod 3$, confirming exponential growth.
• The bound is tight in the sense that no smaller exponent $\alpha < \log_\phi 2$ can be achieved under the given construction, as shown by the asymptotic behavior of the recurrence.