Binary Tree Block Encoding of Classical Matrix

State preparation and block encoding are essential subroutines in quantum computing. The former provides basic encoding of quantum states, while the latter transforms classical data into a matrix representation within a quantum circuit. Some quantum advantages are built on the assumption that the block-encoding subroutine has been compiled in the quantum circuit, and this derives a problem of how to efficiently compile a block encoding. The resource tradeoffs of block encoding, such as circuit size, subnormalization factor, compilation complexity (both time and space), and robustness against errors, are central to its efficiency. In this work, the binary tree block-encoding (<monospace>BITBLE</monospace>) protocol is introduced, which optimizes these tradeoffs. For a classical matrix in <inline-formula><tex-math notation="LaTeX">$\mathbb {C}^{2^{n}\times 2^{n}}$</tex-math></inline-formula>, our approach reduces the compilation time to <inline-formula><tex-math notation="LaTeX">$\mathcal {O}(n2^{2n})$</tex-math></inline-formula> using <inline-formula><tex-math notation="LaTeX">$n$</tex-math></inline-formula> ancilla qubits, achieving superior resource tradeoffs compared to existing methods. Numerical experiments further reveal that the approach outlined in <monospace>BITBLE</monospace> enhances compilation efficiency, resource scalability, and robustness against single-qubit gate errors in various standard data encoding tasks. Moreover, all algorithms are available as open source.