Algebraic Reduction to Improve an Optimally Bounded Quantum State Preparation Algorithm

2026-02-06

AI Summary

This work addresses the high circuit complexity in preparing arbitrary n-qubit quantum states by proposing a novel algebraic decomposition method that separately processes the real and imaginary components of the target state. It introduces a single Λ operator to construct uniform controlled gates, replacing the conventional three-operator structure. With m ancillary qubits, the method achieves the first known compression beyond the optimal bounded algorithm of Sun et al., significantly reducing circuit depth, total gate count, and CNOT number while preserving the theoretically optimal width–depth trade-off. Experimental evaluations on PennyLane demonstrate that the proposed algorithm consistently outperforms the standard Möttönen algorithm across both dense and sparse quantum states—including random and physically relevant states—with particularly pronounced advantages in resource-constrained scenarios.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Non-convex Optimization
• Computer Vision: Learning & Optimization for CV

Application Category

• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

The preparation of $n$-qubit quantum states is a cross-cutting subroutine for many quantum algorithms, and the effort to reduce its circuit complexity is a significant challenge. In the literature, the quantum state preparation algorithm by Sun et al. is known to be optimally bounded, defining the asymptotically optimal width-depth trade-off bounds with and without ancillary qubits. In this work, a simpler algebraic decomposition is proposed to separate the preparation of the real part of the desired state from the complex one, resulting in a reduction in terms of circuit depth, total gates, and CNOT count when $m$ ancillary qubits are available. The reduction in complexity is due to the use of a single operator $\Lambda$ for each uniformly controlled gate, instead of the three in the original decomposition. Using the PennyLane library, this new algorithm for state preparation has been implemented and tested in a simulated environment for both dense and sparse quantum states, including those that are random and of physical interest. Furthermore, its performance has been compared with that of M\"ott\"onen et al.'s algorithm, which is a de facto standard for preparing quantum states in cases where no ancillary qubits are used, highlighting interesting lines of development.

Problem

• quantum state preparation
• circuit complexity
• ancillary qubits
• CNOT count
• width-depth trade-off

Innovation

• quantum state preparation
• algebraic decomposition
• circuit optimization
• uniformly controlled gates
• ancillary qubits