混合量子比特-转子量子系统：克利福德结构、通用控制及应用
Hybrid Qubit-Rotor Quantum Systems: Clifford Structure, Universal Control, and Applications
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中文总结 AI 辅助
该研究针对混合量子比特-转子寄存器建立克利福德理论，明确其自同构分类与通用控制条件，并将其应用于规范协变物质跳跃、转子相位估计及转子动量编码傅里叶变换等场景。
中文摘要 AI 辅助
U(1)量子转子将周期角与整数共轭动量配对，存在于分子转动、超导相-电荷电路及紧致规范场中。将此类转子与量子比特相干耦合，可得到混合寄存器，其控制结构既非继承自振子-量子比特情况，也非继承自量子位元情况。我们针对包含n个量子比特和r个转子的寄存器，建立了克利福德理论并得出通用控制结果。我们对混合相空间F₂²ⁿ×Zʳ×Tʳ中保持外尔对易关系的所有自同构进行分类，并为每个自同构给出显式有限克利福德电路。该分类具有方向性：转子动量奇偶性可在克利福德群内控制量子比特泡利操作，而每个非零量子比特控制的转子动量位移均为非克利福德操作。该分类还给出了混合量子比特-转子耦合的标准形式，以及合成这些耦合所需的基本混合门的精确最小数量。在局部克利福德操作中添加一个转子余弦势和一个固定的量子比特-转子条件相位，可在强算子拓扑下实现全希尔伯特空间上的通用控制。随后我们将该结构应用于三个场景：规范协变物质跳跃的精确受控位移实现，其必然为非克利福德操作；在动量支撑和能量约束下具有直接角度读出和探针优化的转子相位估计；以及转子动量编码上的有限傅里叶变换，其中d=2ˢ的单转子变换可编译为O(s)个动量选择性和受控位移指令，每个跨寄存器傅里叶因子由一个二次转子克利福德门实现。
英文摘要
A $U(1)$ quantum rotor pairs a periodic angle with an integer-valued conjugate momentum, and occurs in molecular rotation, superconducting phase-charge circuits, and compact gauge fields. Coupling such a rotor coherently to qubits gives a hybrid register whose control structure is not inherited from either the oscillator-qubit or the qudit case. We develop a Clifford theory, together with a universal-control result, for registers of $n$ qubits and $r$ rotors. We classify all automorphisms of the hybrid phase space $\mathbb{F}_2^{2n}\times\mathbb{Z}^r\times\mathbb{T}^r$ that preserve the Weyl commutation relations, and give an explicit finite Clifford circuit for each one. The classification is directional: rotor momentum parity may control qubit Pauli operations within the Clifford group, while every nonzero qubit-controlled rotor momentum shift is non-Clifford. It also yields normal forms for the mixed qubit-rotor couplings and the exact minimum number of elementary mixed gates needed to synthesize them. Adding a rotor cosine potential and one fixed qubit-rotor conditional phase to the local Clifford operations gives universal control on the full Hilbert space in the strong operator topology. We then apply this structure in three settings: an exact controlled-shift realization of gauge-covariant matter hopping, which is necessarily non-Clifford; rotor phase estimation with direct angle readout and probe optimization under momentum-support and energy constraints; and finite Fourier transforms on rotor momentum codes, where the one-rotor transform for $d=2^s$ compiles into $O(s)$ momentum-selective and controlled-shift instructions and each cross-register Fourier factor is implemented by one quadratic rotor Clifford gate.
