Quantum circuits with gates (local unitaries) respecting a global symmetry have broad applications in quantum information science and related fields, such as condensed matter theory and quantum thermodynamics. However, despite their widespread use, fundamental properties of such circuits are not well-understood. Recently, it was found that generic unitaries respecting a global symmetry cannot be realized, even approximately, using gates that respect the same symmetry. This observation raises important open questions: What unitary transformations can be realized with k-local gates that respect a global symmetry? In other words, in the presence of a global symmetry, how does the locality of interactions constrain the possible time evolution of a composite system? In this work, we address these questions for the case of Abelian (commutative) symmetries and develop constructive methods for synthesizing circuits with such symmetries. Remarkably, as a corollary, we find that, while the locality of interactions still imposes additional constraints on realizable unitaries, certain restrictions observed in the case of non-Abelian symmetries do not apply to circuits with Abelian symmetries. For instance, in circuits with a general non-Abelian symmetry such as SU($d$), the unitary realized in a subspace with one irreducible representation (charge) of the symmetry dictates the realized unitaries in multiple other sectors with inequivalent representations of the symmetry. Furthermore, in certain sectors, rather than all unitaries respecting the symmetry, the realizable unitaries are the symplectic or orthogonal subgroups of this group. We prove that none of these restrictions appears in the case of Abelian symmetries. This result suggests that global non-Abelian symmetries may affect the thermalization of quantum systems in ways not possible under Abelian symmetries.
Invariants of Sequential Circuits and Generalized Non-Abelian Statistics
The University of Tokyo（东京大学）；Yukawa Institute for Theoretical Physics, Kyoto University（京都大学汤川理论物理研究所）；RIKEN Center for Interdisciplinary Theoretical and Mathematical Sciences (iTHEMS), RIKEN（理化学研究所交叉理论与数学科学中心（iTHEMS））；Department of Applied Physics, The University of Tokyo（东京大学应用物理系）
Statistical Mechanics and Symmetries of Non-Abelian Anyon Proliferation: From Deformation to Decoherence
California Institute of Technology（加州理工学院）；Ghent university（根特大学）；Pritzker School of Molecular Engineering, University of Chicago（芝加哥大学普里茨克分子工程学院）；University of California, Berkeley（加利福尼亚大学伯克利分校）
Gauging Modulated Symmetries via Multiple Gauge Symmetry Operators and Adaptive Quantum Circuits
Sungkyunkwan University（成均馆大学）；Institute of Basic Science, Sungkyunkwan University（成均馆大学基础科学研究院）；University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
Implementing non-Abelian Hatano-Nelson model in electric circuits
Key Laboratory of Artificial Micro- and Nanostructures of Ministry of Education and School of Physics and Technology, Wuhan University（教育部人工微纳结构重点实验室和武汉大学物理科学与技术学院）；School of Physics and Electronic Science, Changsha University of Science and Technology（长沙理工大学物理与电子科学学院）；Institute for Advanced Studies, Wuhan University（武汉大学高等研究院）
本文提出非阿贝尔Hatano-Nelson模型，利用非 reciprocal U(2) gauge field 实现Hopf-link状复能量辫形和双极皮肤效应，通过电路实验验证了非赫米特物理的新现象。
