Efficient Complex-Valued State Preparation on Bucket Brigade QRAM

TL;DR

This paper improves quantum state preparation efficiency by precomputing angles and extending to complex matrices, reducing quantum processing to retrievals and controlled rotations with minimal arithmetic.

Contribution

It introduces a method to precompute and store rotation angles classically, extending state preparation to complex matrices within the same architecture-aware framework.

Findings

Reduces quantum state preparation to BBQRAM retrievals and controlled rotations.

Stores precomputed fixed-point angles in QRAM, eliminating the $U_{2CR}$ subroutine.

Extends the method to complex-valued matrices with minimal additional quantum complexity.

Abstract

Efficient quantum state preparation is a critical component in quantum algorithms that process large classical data, and it is fundamental to realizing quantum advantage in domains such as machine learning, quantum linear algebra, and quantum finance. Building on the framework of~\cite{berti2025efficient}, which integrates Bucket Brigade QRAM (BBQRAM) with a segment tree to achieve amplitude encoding in polylogarithmic query time, we present two improvements within the same architecture-aware framework. First, we remove the $U_{2\mathrm{CR}}$ subroutine by classically precomputing the rotation angles determined by the segment tree and storing these angles directly in the BBQRAM cells. The tradeoff is that the classically loaded QRAM stores precomputed fixed-point angles rather than raw subtree weights. Second, we extend the construction to complex-valued matrices $A \in \mathbb{C}^{M \times N}$ by storing a leaf phase alongside each precomputed rotation angle and using a two-step magnitude-then-phase procedure; the real signed case is naturally subsumed as a one-bit phase specialization. At unchanged $\mathcal{O}(\log_2^2(MN))$ BBQRAM query complexity, the QPU procedure reduces to BBQRAM retrievals and controlled-rotation cascades, with $\mathcal{O}(MN)$ memory cells per matrix and no reversible arithmetic on the QPU.