Algebraic Reduction to Improve an Optimally Bounded Quantum State Preparation Algorithm

TL;DR

This paper introduces an algebraic decomposition method that simplifies and reduces the circuit complexity of an existing optimal quantum state preparation algorithm, especially when ancillary qubits are available, improving efficiency.

Contribution

A novel algebraic approach to quantum state preparation that separates real and complex parts, reducing circuit depth and gate count compared to prior methods.

Findings

Reduced circuit depth, gates, and CNOTs with ancillary qubits.

Effective implementation tested on dense and sparse states.

Performance comparison with Möttönen et al.'s algorithm highlights advantages.

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.