Theory of quantum circuits with Abelian symmetries
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 wel...
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| Format: | Article |
| Language: | English |
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American Physical Society
2024-12-01
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| Series: | Physical Review Research |
| Online Access: | http://doi.org/10.1103/PhysRevResearch.6.043292 |
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| _version_ | 1846116665586089984 |
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| author | Iman Marvian |
| author_facet | Iman Marvian |
| author_sort | Iman Marvian |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-5b5c99f3dcfd477daf02a3940e45f7fd |
| institution | Kabale University |
| issn | 2643-1564 |
| language | English |
| publishDate | 2024-12-01 |
| publisher | American Physical Society |
| record_format | Article |
| series | Physical Review Research |
| spelling | doaj-art-5b5c99f3dcfd477daf02a3940e45f7fd2024-12-18T15:10:50ZengAmerican Physical SocietyPhysical Review Research2643-15642024-12-016404329210.1103/PhysRevResearch.6.043292Theory of quantum circuits with Abelian symmetriesIman MarvianQuantum 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.http://doi.org/10.1103/PhysRevResearch.6.043292 |
| spellingShingle | Iman Marvian Theory of quantum circuits with Abelian symmetries Physical Review Research |
| title | Theory of quantum circuits with Abelian symmetries |
| title_full | Theory of quantum circuits with Abelian symmetries |
| title_fullStr | Theory of quantum circuits with Abelian symmetries |
| title_full_unstemmed | Theory of quantum circuits with Abelian symmetries |
| title_short | Theory of quantum circuits with Abelian symmetries |
| title_sort | theory of quantum circuits with abelian symmetries |
| url | http://doi.org/10.1103/PhysRevResearch.6.043292 |
| work_keys_str_mv | AT imanmarvian theoryofquantumcircuitswithabeliansymmetries |