Block encoding of sparse matrices with a periodic diagonal structure
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arXiv
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| Autori principali: | , , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866917267059507200 |
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| author | Zecchi, Alessandro Andrea Sanavio, Claudio Cappelli, Luca Perotto, Simona Roggero, Alessandro Succi, Sauro |
| author_facet | Zecchi, Alessandro Andrea Sanavio, Claudio Cappelli, Luca Perotto, Simona Roggero, Alessandro Succi, Sauro |
| contents | Block encoding is a successful technique used in several powerful quantum algorithms. In this work we provide an explicit quantum circuit for block encoding a sparse matrix with a periodic diagonal structure. The proposed methodology is based on the linear combination of unitaries (LCU) framework and on an efficient unitary operator used to project the complex exponential at a frequency $ω$ multiplied by the computational basis into its real and imaginary components. We demonstrate a distinct computational advantage with a $\mathcal{O}(\text{poly}(n))$ gate complexity, where $n$ is the number of qubits, in the worst-case scenario used for banded matrices, and $\mathcal{O}(n)$ when dealing with a simple diagonal matrix, compared to the exponential scaling of general-purpose methods for dense matrices. Various applications for the presented methodology are discussed in the context of solving differential problems such as the advection-diffusion-reaction (ADR) dynamics, using quantum algorithms with optimal scaling, e.g., quantum singular value transformation (QSVT). Numerical results are used to validate the analytical formulation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_10589 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Block encoding of sparse matrices with a periodic diagonal structure Zecchi, Alessandro Andrea Sanavio, Claudio Cappelli, Luca Perotto, Simona Roggero, Alessandro Succi, Sauro Quantum Physics Numerical Analysis Block encoding is a successful technique used in several powerful quantum algorithms. In this work we provide an explicit quantum circuit for block encoding a sparse matrix with a periodic diagonal structure. The proposed methodology is based on the linear combination of unitaries (LCU) framework and on an efficient unitary operator used to project the complex exponential at a frequency $ω$ multiplied by the computational basis into its real and imaginary components. We demonstrate a distinct computational advantage with a $\mathcal{O}(\text{poly}(n))$ gate complexity, where $n$ is the number of qubits, in the worst-case scenario used for banded matrices, and $\mathcal{O}(n)$ when dealing with a simple diagonal matrix, compared to the exponential scaling of general-purpose methods for dense matrices. Various applications for the presented methodology are discussed in the context of solving differential problems such as the advection-diffusion-reaction (ADR) dynamics, using quantum algorithms with optimal scaling, e.g., quantum singular value transformation (QSVT). Numerical results are used to validate the analytical formulation. |
| title | Block encoding of sparse matrices with a periodic diagonal structure |
| topic | Quantum Physics Numerical Analysis |
| url | https://arxiv.org/abs/2602.10589 |