Nonlinear transformation of complex amplitudes via quantum singular value transformation
Due to the linearity of quantum operations, it is not straightforward to implement nonlinear transformations on a quantum computer, making some practical tasks like a neural network hard to achieve. In this paper, we define a task called nonlinear transformation of complex amplitudes and provide an...
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| Main Authors: | , , |
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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.6.043227 |
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| _version_ | 1846143370179641344 |
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| author | Naixu Guo Kosuke Mitarai Keisuke Fujii |
| author_facet | Naixu Guo Kosuke Mitarai Keisuke Fujii |
| author_sort | Naixu Guo |
| collection | DOAJ |
| description | Due to the linearity of quantum operations, it is not straightforward to implement nonlinear transformations on a quantum computer, making some practical tasks like a neural network hard to achieve. In this paper, we define a task called nonlinear transformation of complex amplitudes and provide an algorithm to achieve this task. Specifically, we construct a block encoding of complex amplitudes from a state preparation unitary. This allows us to transform the complex amplitudes by using quantum singular value transformation. We evaluate the required overhead in terms of input dimension and precision, which reveals that the algorithm depends on the roughly square root of input dimension and achieves an exponential speedup on precision compared with previous work. We also discuss its possible applications to quantum machine learning, where complex amplitudes encoding classical or quantum data are processed by the proposed method. In this paper, we provide a promising way to introduce the highly complex nonlinearity of the quantum states, which is essentially missing in quantum mechanics. |
| format | Article |
| id | doaj-art-9f12b6e961f8492898e1b4dc8c65b06a |
| 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-9f12b6e961f8492898e1b4dc8c65b06a2024-12-02T16:13:51ZengAmerican Physical SocietyPhysical Review Research2643-15642024-12-016404322710.1103/PhysRevResearch.6.043227Nonlinear transformation of complex amplitudes via quantum singular value transformationNaixu GuoKosuke MitaraiKeisuke FujiiDue to the linearity of quantum operations, it is not straightforward to implement nonlinear transformations on a quantum computer, making some practical tasks like a neural network hard to achieve. In this paper, we define a task called nonlinear transformation of complex amplitudes and provide an algorithm to achieve this task. Specifically, we construct a block encoding of complex amplitudes from a state preparation unitary. This allows us to transform the complex amplitudes by using quantum singular value transformation. We evaluate the required overhead in terms of input dimension and precision, which reveals that the algorithm depends on the roughly square root of input dimension and achieves an exponential speedup on precision compared with previous work. We also discuss its possible applications to quantum machine learning, where complex amplitudes encoding classical or quantum data are processed by the proposed method. In this paper, we provide a promising way to introduce the highly complex nonlinearity of the quantum states, which is essentially missing in quantum mechanics.http://doi.org/10.1103/PhysRevResearch.6.043227 |
| spellingShingle | Naixu Guo Kosuke Mitarai Keisuke Fujii Nonlinear transformation of complex amplitudes via quantum singular value transformation Physical Review Research |
| title | Nonlinear transformation of complex amplitudes via quantum singular value transformation |
| title_full | Nonlinear transformation of complex amplitudes via quantum singular value transformation |
| title_fullStr | Nonlinear transformation of complex amplitudes via quantum singular value transformation |
| title_full_unstemmed | Nonlinear transformation of complex amplitudes via quantum singular value transformation |
| title_short | Nonlinear transformation of complex amplitudes via quantum singular value transformation |
| title_sort | nonlinear transformation of complex amplitudes via quantum singular value transformation |
| url | http://doi.org/10.1103/PhysRevResearch.6.043227 |
| work_keys_str_mv | AT naixuguo nonlineartransformationofcomplexamplitudesviaquantumsingularvaluetransformation AT kosukemitarai nonlineartransformationofcomplexamplitudesviaquantumsingularvaluetransformation AT keisukefujii nonlineartransformationofcomplexamplitudesviaquantumsingularvaluetransformation |