Block encoding of matrix product operators
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866912084289126400 |
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| author | Nibbi, Martina Mendl, Christian B. |
| author_facet | Nibbi, Martina Mendl, Christian B. |
| contents | Quantum signal processing combined with quantum eigenvalue transformation has recently emerged as a unifying framework for several quantum algorithms. In its standard form, it consists of two separate routines: block encoding, which encodes a Hamiltonian in a larger unitary, and signal processing, which achieves an almost arbitrary polynomial transformation of such a Hamiltonian using rotation gates. The bottleneck of the entire operation is typically constituted by block encoding and, in recent years, several problem-specific techniques have been introduced to overcome this problem. Within this framework, we present a procedure to block-encode a Hamiltonian based on its matrix product operator (MPO) representation. More specifically, we encode every MPO tensor in a larger unitary of dimension $D+2$, where $D = \lceil\log(χ)\rceil$ is the number of subsequently contracted qubits that scales logarithmically with the virtual bond dimension $χ$. Given any system of size $L$, our method requires $L+D$ ancillary qubits in total, while the number of one- and two-qubit gates decomposing the block encoding circuit scales as $\mathcal{O}(L\cdotχ^2)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_08861 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Block encoding of matrix product operators Nibbi, Martina Mendl, Christian B. Quantum Physics Quantum signal processing combined with quantum eigenvalue transformation has recently emerged as a unifying framework for several quantum algorithms. In its standard form, it consists of two separate routines: block encoding, which encodes a Hamiltonian in a larger unitary, and signal processing, which achieves an almost arbitrary polynomial transformation of such a Hamiltonian using rotation gates. The bottleneck of the entire operation is typically constituted by block encoding and, in recent years, several problem-specific techniques have been introduced to overcome this problem. Within this framework, we present a procedure to block-encode a Hamiltonian based on its matrix product operator (MPO) representation. More specifically, we encode every MPO tensor in a larger unitary of dimension $D+2$, where $D = \lceil\log(χ)\rceil$ is the number of subsequently contracted qubits that scales logarithmically with the virtual bond dimension $χ$. Given any system of size $L$, our method requires $L+D$ ancillary qubits in total, while the number of one- and two-qubit gates decomposing the block encoding circuit scales as $\mathcal{O}(L\cdotχ^2)$. |
| title | Block encoding of matrix product operators |
| topic | Quantum Physics |
| url | https://arxiv.org/abs/2312.08861 |