Comparative Study of Quantum State Preparation Methods in Sparse Isometry Decomposition

Abstract

Initializing an isometry in a quantum circuit is a fundamental yet challenging task, espe- cially when aiming for efficient state preparation and resource optimization. In this work, we

present a comparative analysis of the Householder Decomposition for isometry implementation, leveraging three distinct state preparation methods starting from the original Pivot method and extending the study to two additional strategies: Merge and Low Rank. The study investigates how each method impacts the performance of the isometry decomposition, focusing on key metrics such as CNOT gate count and circuit depth. To provide a comprehensive evaluation, we examine variations in matrix size and sparsity levels, capturing the effects of structural complexity on resource requirements. Our results reveal that the Merge state preparation method generally outperforms the other two approaches, particularly in terms of scalability and gate efficiency. Building upon

these findings, we further compare the best-performing method, Merge, with the state-of- the-art isometry decomposition implementation available in Qiskit, a widely used quantum

computing framework. The analysis demonstrates that, for isometries involving up to 6 qubits, Qiskit’s implementation exhibits superior performance. However, beyond this threshold, our proposed decomposition method proves more effective, especially for highly sparse isometries or those characterized by a smaller number of columns. This work highlights the potential for optimizing isometry decompositions in scenarios where sparsity and structural constraints are critical factors. These findings contribute to advancing state preparation techniques and offer insights into improving the efficiency of quantum circuits for applications in quantum information processing.