对称性下精确酉设计的随机性
Randomness of exact unitary designs under symmetry
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中文总结 AI 辅助
本文定义并研究了有限酉系综在对称性下的设计强度，证明在广泛对称性下存在对称设计强度高于酉设计强度的系综，并给出了群结构系综的对称设计强度上界。
中文摘要 AI 辅助
我们将有限酉系综的酉设计强度定义为其第$t$阶统计矩与酉群相同的最大整数$t$。在存在物理对称性的情况下，系综中的并非所有算子都与对称性相容。这促使我们将有限酉系综的对称设计强度定义为其对称相容子系综的第$t$阶矩与对称相容酉群相匹配的最大整数$t$。我们证明，在足够高的维度下，对于包括全局在位$\u2009\mathrm{U}(1)$和$\u2009\mathrm{SU}(d)$对称性在内的广泛对称性族，存在任意高设计强度的有限酉系综，其对称设计强度严格更高。具有此性质的系综在依赖酉随机化的协议中，在对称性约束下可能比没有对称性时更具通用性。我们的结果扩展了先前的工作[Mitsuhashi和Yoshioka，PRX Quantum 4.4（2023年11月）]，该工作表明对于不平凡化对称相容酉算子的对称性，Clifford群的对称设计强度严格受限于其酉设计强度。由于具有群结构的酉设计具有特殊重要性，我们还推导了算子构成群的有限酉系综的酉设计强度和对称设计强度的多种界限。特别地，我们提供了一个简单的证明：在全局在位$\u2009\mathrm{U}(1)$和$\u2009\mathrm{SU}(d)$对称性下，任意由有限群算子构成的系综的对称设计强度至多为2。在此过程中，我们证明了在任意维度下存在任意高对称设计强度的均匀加权群，并分析了它们的结构。
英文摘要
We define the unitary design strength of a finite unitary ensemble to be the maximal integer $t$ for which its $t$-th statistical moment is identical to that of the unitary group. In the presence of physical symmetry, not all operators of an ensemble are symmetry-compatible. This motivates us to define the symmetric design strength of a finite unitary ensemble to be the maximal integer $t$ for which the $t$-th moments of its symmetry-compatible sub-ensemble match those of the group of symmetry-compatible unitaries. We show that in sufficiently high dimension and for a broad family of symmetries, which includes the global on-site $\mathrm{U}(1)$ and $\mathrm{SU}(d)$ symmetries, there exist finite unitary ensembles of arbitrarily high design strength whose symmetric design strength is strictly higher. Ensembles with this property may be more versatile under symmetry constraints for protocols that rely on unitary randomization than they are without symmetry. Our results extend the previous work [Mitsuhashi and Yoshioka, PRX Quantum 4.4 (Nov. 2023)] showing that the symmetric design strength of the Clifford group is strictly upper bounded by its unitary design strength for symmetries that do not trivialize the symmetry-compatible unitaries. Due to the special significance of unitary designs with a group structure, we also derive a variety of bounds on the unitary and symmetric design strengths of finite unitary ensembles whose operators form a group. In particular, we provide a simple proof that an arbitrary ensemble of operators forming a finite group must have a symmetric design strength of at most two under the global on-site $\mathrm{U}(1)$ and $\mathrm{SU}(d)$ symmetries. Along the way, we show that there exist uniformly weighted groups of arbitrarily high symmetric design strength in arbitrary dimension and analyze their structure.
发表机构
- University of Illinois Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
- Illinois Quantum Information Science and Technology Center (IQUIST)（伊利诺伊量子信息与科技中心）
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