Breaking the cubic barrier for the inverse-free Solovay-Kitaev algorithm

Abstract:The Solovay-Kitaev algorithm states that it is possible to efficiently compile unitaries to accuracy $\epsilon$ in $\text{polylog}(1/\epsilon)$ time using any universal gate set. While Solovay and Kitaev's initial work addressed inverse-closed gate sets, recently Bouland and Giurgicǎ-Tiron showed it is possible to efficiently compile arbitrary gate sets without inverses. However, the exponent of their compilation algorithm is high. For example the inverse-free exponent for a qubit is over $8.62$, while information theoretically it is known that an inverse-free exponent of $3$ is possible in any dimension by a result of Oszmaniec, Sawicki, and Horodecki. This leads to a natural question: can one improve the exponent in Bouland and Giurgicǎ-Tiron's algorithm? And are there any cases where the inverse-free exponent is below 3, thus surpassing existing information-theoretic arguments? To answer this question, we first show that a special case of the inverse-free compiling problem -- namely Pauli gates supplemented by an irrational rotation of a qubit - admits a highly efficient compilation algorithm with an exponent at most $\approx 2.138$. Our algorithm fuses ideas from prior algorithms of Kuperberg and Sardharwalla, Cubitt, Harrow and Linden to construct an efficient ``Pauli golf'' compilation routine. We then extend these ideas to the general inverse-free case for a qubit, where we give a compiling algorithm with exponent at most $2.988$. This result uses a novel form of approximate ``aiming'' of unitary corrections to improve the compilation efficiency. Our results give two examples of quantum compiling algorithms surpassing information-theoretic arguments, and demonstrate that there are many efficiency gains possible for inverse-free compilation.

arXiv:2610.03655 [quant-ph]

(or arXiv:2610.03655v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2610.03655 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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