Binary Tree Block Encoding of Classical Matrix
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866915233318043648 |
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| author | Li, Zexian Zhang, Xiao-Ming Yang, Chunlin Zhang, Guofeng |
| author_facet | Li, Zexian Zhang, Xiao-Ming Yang, Chunlin Zhang, Guofeng |
| contents | Block-encoding is a critical subroutine in quantum computing, enabling the transformation of classical data into a matrix representation within a quantum circuit. The resource trade-offs in simulating a block-encoding can be quantified by the circuit size, the normalization factor, and the time and space complexity of parameter computation. Previous studies have primarily focused either on the time and memory complexity of computing the parameters, or on the circuit size and normalization factor in isolation, often neglecting the balance between these trade-offs. In early fault-tolerant quantum computers, the number of qubits is limited. For a classical matrix of size $2^{n}\times 2^{n}$, our approach not only improves the time of decoupling unitary for block-encoding with time complexity $\mathcal{O}(n2^{2n})$ and memory complexity $Θ(2^{2n})$ using only a few ancilla qubits, but also demonstrates superior resource trade-offs. Our proposed block-encoding protocol is named Binary Tree Block-encoding (\texttt{BITBLE}). Under the benchmark, \textit{size metric}, defined by the product of the number of gates and the normalization factor, numerical experiments demonstrate the improvement of both resource trade-off and classical computing time efficiency of the \texttt{BITBLE} protocol. The algorithms are all open-source. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_05624 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Binary Tree Block Encoding of Classical Matrix Li, Zexian Zhang, Xiao-Ming Yang, Chunlin Zhang, Guofeng Quantum Physics Data Structures and Algorithms 68Q12, 68Q06 Block-encoding is a critical subroutine in quantum computing, enabling the transformation of classical data into a matrix representation within a quantum circuit. The resource trade-offs in simulating a block-encoding can be quantified by the circuit size, the normalization factor, and the time and space complexity of parameter computation. Previous studies have primarily focused either on the time and memory complexity of computing the parameters, or on the circuit size and normalization factor in isolation, often neglecting the balance between these trade-offs. In early fault-tolerant quantum computers, the number of qubits is limited. For a classical matrix of size $2^{n}\times 2^{n}$, our approach not only improves the time of decoupling unitary for block-encoding with time complexity $\mathcal{O}(n2^{2n})$ and memory complexity $Θ(2^{2n})$ using only a few ancilla qubits, but also demonstrates superior resource trade-offs. Our proposed block-encoding protocol is named Binary Tree Block-encoding (\texttt{BITBLE}). Under the benchmark, \textit{size metric}, defined by the product of the number of gates and the normalization factor, numerical experiments demonstrate the improvement of both resource trade-off and classical computing time efficiency of the \texttt{BITBLE} protocol. The algorithms are all open-source. |
| title | Binary Tree Block Encoding of Classical Matrix |
| topic | Quantum Physics Data Structures and Algorithms 68Q12, 68Q06 |
| url | https://arxiv.org/abs/2504.05624 |