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

TL;DR

This paper introduces new algorithms that significantly reduce C-NOT gate counts for quantum state preparation and block encoding, enhancing efficiency for quantum algorithms in scientific computing.

Contribution

It presents improved C-NOT count algorithms for state preparation and block encoding, including a novel diagonal matrix migration technique, surpassing previous bounds and optimizing for low-rank matrices.

Findings

Reduced C-NOT count for state preparation from (23/24)2^n to (11/12)2^n.

Achieved C-NOT count leading term (11/48)4^n for block encoding, surpassing the lower bound.

Optimized C-NOT count to (K+(11/12))2^n for rank-K matrices.

Abstract

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 from Plesch-Brukner algorithm, proposed in 2011, from $(23/24)2^n$ to $(11/12)2^n$. 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 $(11/48)4^n$. This result even exceeds the lower bound of $(1/4)4^n$ for $n$-qubit unitary synthesis. Further optimization is performed for low-rank matrices, which frequently arise in practical applications. Specifically, we achieve the C-NOT count leading term $(K+(11/12))2^n$ for a rank-$K$ matrix. 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.