Algebraic Reduction to Improve an Optimally Bounded Quantum State Preparation Algorithm

TL;DR

The paper tackles the resource cost of quantum state preparation (QSP) under ancilla-assisted settings by building on SUN's asymptotically optimal framework. It introduces OSUN, a two-part decomposition that assigns a single Λ-type diagonal constructor to each UCG for the real-part preparation and a final Λ for the complex phases, coupled with a diagonal phase unitary; this yields depth-prefactor improvements without altering the overall complexity class $\mathcal{O}(n^2+\frac{2^n}{m})$. The authors provide explicit constant-bound analyses and demonstrate the approach with a PennyLane implementation, benchmarking against SUN and Mottonen's standard, across dense, sparse, real, and complex target states up to 10 qubits. The results show notable depth reductions starting from moderate numbers of qubits, while gate counts remain comparable to the reference, highlighting practical gains for ancilla-based QSP in the first parametric range and suggesting directions for hardware-aware extensions.

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 $Λ$ 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 Möttönen 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.

Figures (13)

• Figure 1: QSP traditional framework.
• Figure 2: UCG in terms of diagonal operators $F_n(R_z)$.
• Figure 3: UCG in terms of $\Lambda$-type operators.
• Figure 4: $\Lambda_n$ quantum circuit divided into 5 sub-unitaries (stages).
• Figure 5: Quantum circuit for $\Lambda_4$. Figure taken from sun2021asymptotically.