广义耦合与量子 Schur 变换的一种基本算法
Generalised Coupling and An Elementary Algorithm for the Quantum Schur Transform
AI总结：
本文提出一种使用 $O(\log(n))$ 辅助比特、分解为 $O(n^3\log(n)\log(n/ε))$ 个 Clifford+T 门的量子 Schur 变换算法，并引入广义耦合量子态概念，给出六个条件确保其高效制备或经典可模拟性。
摘要：
量子 Schur 变换是一种基本构建块，它将计算基映射到由酉群和对称群的不可约表示构成的耦合基。等价地，它可以被视为从计算基到置换量子计算（PQC）[Quantum Inf. Comput., 10, 470-497 (2010)] 的同时自旋本征基的基变换。通过采用后一种视角，我们提出了一种透明的算法来实现量子比特量子 Schur 变换，该算法使用 $O(\log(n))$ 个辅助量子比特，并可分解为 $O(n^3\log(n)\log(\frac{n}{ε}))$ 个 Clifford + T 门序列，其中 $ε$ 是算法在迹范数意义上的精度。我们讨论了在某些应用中，将此操作实现为酉算子而非等距算子的必要性，后者是通常的呈现方式。通过研究相关的 Schur 态（由通过 Clebsch-Gordan 系数耦合的量子比特组成），我们引入了广义耦合量子态的概念。我们提出了六个条件，这些条件的不同组合确保了这些态在量子计算机上的高效制备或其经典可模拟性（在计算可处理性的意义上）。结果表明，Wigner 6-j 符号和 SU(N) Clebsch-Gordan 系数自然地适合我们的框架。最后，我们研究了保持计算可处理态类别的酉变换。
英文摘要：
The quantum Schur transform is a fundamental building block that maps the computational basis to a coupled basis consisting of irreducible representations of the unitary and symmetric groups. Equivalently, it may be regarded as a change of basis from the computational basis to a simultaneous spin eigenbasis of Permutational Quantum Computing (PQC) [Quantum Inf. Comput., 10, 470-497 (2010)]. By adopting the latter perspective, we present a transparent algorithm for implementing the qubit quantum Schur transform which uses $O(\log(n))$ ancillas and can be decomposed into a sequence of $O(n^3\log(n)\log(\frac{n}ε))$ Clifford + T gates, where $ε$ is the accuracy of the algorithm in terms of the trace norm. We discuss the necessity for some applications of implementing this operation as a unitary rather than an isometry, as is often presented. By studying the associated Schur states, which consist of qubits coupled via Clebsch-Gordan coefficients, we introduce the notion of generally coupled quantum states. We present six conditions, which in different combinations ensure the efficient preparation of these states on a quantum computer or their classical simulability (in the sense of computational tractability). It is shown that Wigner 6-j symbols and SU(N) Clebsch-Gordan coefficients naturally fit our framework. Finally, we investigate unitary transformations which preserve the class of computationally tractable states.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
