Reducing C-NOT Counts for State Preparation and Block Encoding via Diagonal Matrix Migration

Quantum state preparation and block encoding are versatile and practical input models for quantum algorithms in scientific computing. The circuit complexity of state preparation and block encoding frequently dominates the end-to-end gate complexity of quantum algorithms. We give algorithms with lower C-NOT counts for both the state preparation and block encoding. For a general $n$-qubit state, we improve the C-NOT count of the Plesch-Brukner algorithm (2011) from $\frac{23}{24}2^n$ to $\frac{11}{12}2^n$. For block encoding, our single-ancilla protocol for $2^{n-1}\times 2^{n-1}$ matrices uses the spectral norm as subnormalization and achieves a C-NOT count leading term $\frac{11}{48}4^n$. Further optimization is performed for low-rank matrices, which frequently arise in practical applications. Specifically, we achieve the C-NOT count leading term $({\lceil\log_{2}K\rceil}+\frac{11}{12})2^n$ for a rank-$K$ matrix. This is the first quantum algorithm that encodes matrices using the optimal normalization factor while also allowing the C-NOT count to be adjusted according to the matrix rank. Our approach builds upon the recursive Block-ZXZ decomposition from Krol et al. and introduces a diagonal matrix migration technique based on the commutativity of the diagonal matrix and the uniformly controlled rotation about the $z$-axis to minimize the use of C-NOT gates.