Efficient Universal Quantum Compilation: An Inverse-free Solovay-Kitaev Algorithm

• URL: http://arxiv.org/abs/2112.02040v1
• Date: Fri, 3 Dec 2021 17:39:41 GMT
• Title: Efficient Universal Quantum Compilation: An Inverse-free Solovay-Kitaev Algorithm
• Authors: Adam Bouland, Tudor Giurgica-Tiron
• Abstract summary: Solovay-Kitaev algorithm is a fundamental result in quantum computation. It gives an algorithm for efficiently compiling arbitrary unitaries using universal gate sets. We provide the first inverse-free Solovay-Kitaev algorithm, which makes no assumption on the structure within a gate set beyond universality.
• Score: 0.8594140167290096
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: The Solovay-Kitaev algorithm is a fundamental result in quantum computation. It gives an algorithm for efficiently compiling arbitrary unitaries using universal gate sets: any unitary can be approximated by short gates sequences, whose length scales merely poly-logarithmically with accuracy. As a consequence, the choice of gate set is typically unimportant in quantum computing. However, the Solovay-Kitaev algorithm requires the gate set to be inverse-closed. It has been a longstanding open question if efficient algorithmic compilation is possible without this condition. In this work, we provide the first inverse-free Solovay-Kitaev algorithm, which makes no assumption on the structure within a gate set beyond universality, answering this problem in the affirmative, and providing an efficient compilation algorithm in the absence of inverses for both $\text{SU}(d)$ and $\text{SL}(d, \mathbb{C})$. The algorithm works by showing that approximate gate implementations of the generalized Pauli group can self-correct their errors.

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