Measurable Krylov spaces and eigenenergy count in quantum state dynamics
Abstract In this work, we propose a quantum-mechanically measurable basis for the computation of spread complexity. Current literature focuses on computing different powers of the Hamiltonian to construct a basis for the Krylov state space and the computation of the spread complexity. We show, throu...
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| Language: | English |
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2024-10-01
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| Series: | Journal of High Energy Physics |
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| Online Access: | https://doi.org/10.1007/JHEP10(2024)083 |
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| author | Saud Čindrak Adrian Paschke Lina Jaurigue Kathy Lüdge |
| author_facet | Saud Čindrak Adrian Paschke Lina Jaurigue Kathy Lüdge |
| author_sort | Saud Čindrak |
| collection | DOAJ |
| description | Abstract In this work, we propose a quantum-mechanically measurable basis for the computation of spread complexity. Current literature focuses on computing different powers of the Hamiltonian to construct a basis for the Krylov state space and the computation of the spread complexity. We show, through a series of proofs, that time-evolved states with different evolution times can be used to construct an equivalent space to the Krylov state space used in the computation of the spread complexity. Afterwards, we introduce the effective dimension, which is upper-bounded by the number of pairwise distinct eigenvalues of the Hamiltonian. The computation of the spread complexity requires knowledge of the Hamiltonian and a classical computation of the different powers of the Hamiltonian. The computation of large powers of the Hamiltonian becomes increasingly difficult for large systems. The first part of our work addresses these issues by defining an equivalent space, where the original basis consists of quantum-mechanically measurable states. We demonstrate that a set of different time-evolved states can be used to construct a basis. We subsequently verify the results through numerical analysis, demonstrating that every time-evolved state can be reconstructed using the defined vector space. Based on this new space, we define an upper-bounded effective dimension and analyze its influence on finite-dimensional systems. We further show that the Krylov space dimension is equal to the number of pairwise distinct eigenvalues of the Hamiltonian, enabling a method to determine the number of eigenenergies the system has experimentally. Lastly, we compute the spread complexities of both basis representations and observe almost identical behavior, thus enabling the computation of spread complexities through measurements. |
| format | Article |
| id | doaj-art-0faaccae3b3e4a4fbf2c7a4a74a53f03 |
| institution | Kabale University |
| issn | 1029-8479 |
| language | English |
| publishDate | 2024-10-01 |
| publisher | SpringerOpen |
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| series | Journal of High Energy Physics |
| spelling | doaj-art-0faaccae3b3e4a4fbf2c7a4a74a53f032024-12-08T12:11:02ZengSpringerOpenJournal of High Energy Physics1029-84792024-10-0120241012110.1007/JHEP10(2024)083Measurable Krylov spaces and eigenenergy count in quantum state dynamicsSaud Čindrak0Adrian Paschke1Lina Jaurigue2Kathy Lüdge3Institute of Physics, Technische Universität IlmenauInstitute of Informatics, Freie Universität BerlinInstitute of Physics, Technische Universität IlmenauInstitute of Physics, Technische Universität IlmenauAbstract In this work, we propose a quantum-mechanically measurable basis for the computation of spread complexity. Current literature focuses on computing different powers of the Hamiltonian to construct a basis for the Krylov state space and the computation of the spread complexity. We show, through a series of proofs, that time-evolved states with different evolution times can be used to construct an equivalent space to the Krylov state space used in the computation of the spread complexity. Afterwards, we introduce the effective dimension, which is upper-bounded by the number of pairwise distinct eigenvalues of the Hamiltonian. The computation of the spread complexity requires knowledge of the Hamiltonian and a classical computation of the different powers of the Hamiltonian. The computation of large powers of the Hamiltonian becomes increasingly difficult for large systems. The first part of our work addresses these issues by defining an equivalent space, where the original basis consists of quantum-mechanically measurable states. We demonstrate that a set of different time-evolved states can be used to construct a basis. We subsequently verify the results through numerical analysis, demonstrating that every time-evolved state can be reconstructed using the defined vector space. Based on this new space, we define an upper-bounded effective dimension and analyze its influence on finite-dimensional systems. We further show that the Krylov space dimension is equal to the number of pairwise distinct eigenvalues of the Hamiltonian, enabling a method to determine the number of eigenenergies the system has experimentally. Lastly, we compute the spread complexities of both basis representations and observe almost identical behavior, thus enabling the computation of spread complexities through measurements.https://doi.org/10.1007/JHEP10(2024)083Field Theories in Lower DimensionsLattice Integrable ModelsNonperturbative Effects |
| spellingShingle | Saud Čindrak Adrian Paschke Lina Jaurigue Kathy Lüdge Measurable Krylov spaces and eigenenergy count in quantum state dynamics Journal of High Energy Physics Field Theories in Lower Dimensions Lattice Integrable Models Nonperturbative Effects |
| title | Measurable Krylov spaces and eigenenergy count in quantum state dynamics |
| title_full | Measurable Krylov spaces and eigenenergy count in quantum state dynamics |
| title_fullStr | Measurable Krylov spaces and eigenenergy count in quantum state dynamics |
| title_full_unstemmed | Measurable Krylov spaces and eigenenergy count in quantum state dynamics |
| title_short | Measurable Krylov spaces and eigenenergy count in quantum state dynamics |
| title_sort | measurable krylov spaces and eigenenergy count in quantum state dynamics |
| topic | Field Theories in Lower Dimensions Lattice Integrable Models Nonperturbative Effects |
| url | https://doi.org/10.1007/JHEP10(2024)083 |
| work_keys_str_mv | AT saudcindrak measurablekrylovspacesandeigenenergycountinquantumstatedynamics AT adrianpaschke measurablekrylovspacesandeigenenergycountinquantumstatedynamics AT linajaurigue measurablekrylovspacesandeigenenergycountinquantumstatedynamics AT kathyludge measurablekrylovspacesandeigenenergycountinquantumstatedynamics |