学习量子对称性
Learning quantum symmetries
- University of Cambridge（剑桥大学）
- University of Oxford（牛津大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文研究学习量子对象对称性的量子算法，引入任意子态对称性学习问题，通过归约为StateHSP并利用射影表示线性化与纠错码的新联系，给出高效算法，并推广到酉算子、哈密顿量等对象。
AI中文摘要：
量子算法是寻找经典对象对称性的强大工具，最著名的例子是通过Shor算法和隐藏子群问题（HSP）的框架实现的。在本工作中，我们研究用于学习量子对象对称性的量子算法。我们的出发点是最近引入的状态隐藏子群问题（StateHSP），这是HSP的量子推广，其任务是学习量子态的对称子群。我们在非阿贝尔StateHSP上取得了积极结果，当隐藏子群是正规的且环境群属于一大类非阿贝尔群时，我们给出了高效的量子算法，将先前的通用理论从阿贝尔情形扩展到了更广泛的群类。StateHSP学习的是与线性表示相关的Bose对称性，而物理上等价的纯态仅定义到全局相位。受此启发，我们引入了一个新的任意子（Anyonic）态对称性学习问题，它基于比StateHSP所考虑的更为物理自然的对称性概念。我们通过将问题归约为StateHSP给出了一个高效的量子算法，其中归约依赖于射影表示的线性化与线性纠错码之间的一个新联系。作为一个应用，我们获得了一个改进的学习稳定子群的算法，该算法适用于任意局部维数的混合量子态（qudit）。最后，我们为其他量子对象引入了对称性学习问题，包括酉算子、哈密顿量和有限态集合，并通过将它们归约为StateHSP的实例获得了高效的量子算法。
英文摘要：
Quantum algorithms are powerful tools for finding symmetries of classical objects, most famously through Shor's algorithm and the Hidden Subgroup Problem (HSP). In this work, we study quantum algorithms for learning symmetries of \emph{quantum} objects. Existing work in this area centres on the recently introduced State Hidden Subgroup Problem (StateHSP), a quantum generalisation of HSP in which the task is to learn the symmetry subgroup of a quantum state. We develop efficient quantum algorithms for non-abelian StateHSP when the hidden subgroup is normal and the ambient group belongs to a broad class of non-abelian groups, extending the previous general theory beyond the abelian setting. StateHSP learns \emph{Bose} symmetries, under which a state must be invariant exactly under the action of the symmetry group. In quantum mechanics, however, physically equivalent pure states are defined only up to global phase. Motivated by this, we introduce a natural notion of \emph{Anyonic} state symmetry learning, based on invariance up to global phase. We give an efficient quantum algorithm by reducing the problem to StateHSP, where the reduction rests on a new correspondence between linearisations of projective representations and linear error-correcting codes. As an application, we obtain an improved algorithm for learning stabiliser groups of mixed qudit states of arbitrary local dimension. Finally, we introduce symmetry learning problems for other quantum objects, including unitaries, Hamiltonians, and finite collections of states, and give efficient algorithms for them by reduction to state symmetry learning. Together, these results broaden the scope of state symmetry learning as a common algorithmic primitive for learning quantum symmetries.
