Algebraic Reduction to Improve an Optimally Bounded Quantum State Preparation Algorithm

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.

💡 Research Summary

The paper addresses the fundamental subroutine of quantum algorithms: the preparation of an arbitrary n‑qubit quantum state (QSP). While the Sun et al. algorithm (referred to as SUN) is known to achieve the asymptotically optimal width‑depth trade‑off, its implementation requires three diagonal phase operators (denoted Λ₁, Λ₂, Λ₃) for each uniformly controlled gate (UCG) together with several single‑qubit layers. This three‑Λ construction, although optimal in terms of asymptotic depth when ancillary qubits (m) are available, inflates the actual circuit depth and the number of CNOT gates, especially for large n.

The authors propose a new algebraic decomposition that separates the preparation of the real part of the target state from its complex phase part. Concretely, the overall unitary U that maps |0⟩ⁿ to the desired state |ψ⟩ is written as U = U_mod · D_ph, where U_mod prepares the real amplitudes and D_ph encodes the relative phases. By exploiting the identity R_y(γ) = S H · R_z(γ) · H S†, the R_y blocks in each UCG become diagonal in the computational basis, allowing each UCG to be implemented with a single Λ‑type operator (Λ₂) instead of three. The diagonal phase operator D_ph can also be realized by a single Λ operator after a global‑phase adjustment. Consequently, the entire QSP circuit collapses to a two‑stage structure: a ladder of UCGs (now each requiring only one Λ) followed by a single Λ for the phase correction.

Complexity analysis shows that for a given level k (1 ≤ k ≤ n) the depth contribution of a Λ operator is D_Λ(k,m) = O(log₂ m + 2ᵏ m). In the original SUN algorithm the total depth is roughly 3 ∑{k=1}^{n} D_Λ(k,m) + O(n), whereas the new algorithm (named OSUN) reduces the coefficient from 3 to 1, yielding depth O(∑{k=1}^{n} D_Λ(k,m) + O(n)). For the regime where ancillary qubits satisfy 2ⁿ ≤ m ≤ 2ⁿ n, this translates to a depth improvement from O(n·log m + 2ⁿ m) to O(n·log m + (2ⁿ m)/3). The CNOT count follows the same scaling because each Λ implementation is the dominant source of entangling gates. The authors also demonstrate that the new bound matches or improves the previously known optimal bound for the first parameter range identified by Sun et al.

Implementation is carried out using the PennyLane library. The Λ circuits are built from Gray‑code based prefix‑copy, suffix‑copy, and inverse‑copy stages, with Fourier‑space parallelisation of phase additions. Automatic differentiation in PennyLane is employed to optimise the parameters of the Λ operators. Extensive simulations were performed on the University of Parma’s HPC cluster for three categories of states: (i) dense random states, (ii) sparse states with ~1 % non‑zero amplitudes, and (iii) physically motivated states such as eigenvectors of molecular Hamiltonians. Results indicate average reductions of 15–30 % in circuit depth, 18–32 % in total gate count, and 20–35 % in CNOT count compared with the original SUN implementation. When benchmarked against the Möttönen et al. algorithm (which uses no ancillae), the new method with a modest number of ancilla qubits consistently outperforms in all metrics, often achieving more than a two‑fold improvement in depth and CNOT count.

The paper concludes with several avenues for future work: experimental validation on real quantum hardware, adaptive selection of the optimal number of ancilla qubits under hardware constraints, extension to multi‑state preparation and quantum channel simulation, and integration with quantum compiler pipelines to automatically invoke the single‑Λ decomposition when appropriate. Overall, the work delivers a practically more efficient realization of the theoretically optimal QSP, promising tangible speed‑ups for quantum machine learning, quantum chemistry, and linear‑system solvers that rely heavily on state preparation.