Exact Quantum State Preparation with the Standard Recursive Block Basis

Variational approaches to Quantum State Preparation (QSP) allow for low-depth and structured circuit ansatz, at the cost of non-convex optimization landscapes and barren plateaus that severely limit scalability. In contrast, exact synthesis methods typically involve exponential growth in the number of independent parameters, leading to a rapid increase in circuit depth. In this work, it is shown that the algebraic properties of the Standard Recursive Block Basis (SRBB) can partially overcome this delicate trade-off. By exploiting the algebraic structure of the SRBB decomposition and the CNOT-optimized circuit for its diagonal component, an exact parameter-solving procedure for the variational SRBB-based QSP problem is defined by analytically inverting the foundational parametric Lie map. As a consequence, a variationally trained circuit is converted to an exact one without increasing circuit depth or gate count. The resulting framework preserves the expressive power of the variational ansatz, thanks to the completeness of the SRBB, while eliminating optimization-induced limitations, shifting the computational problem entirely to a classical precomputation stage. The new exact QSP algorithm has been implemented via the PennyLane library and tested in HPC simulations (up to 12 qubits) to demonstrate the scalability of the parametric map resulting from the recursive structure of the diagonal component of SRBB. In this new exact framework, circuit design (and hence rotation parameters) is twinned with the Unitary Group Hermitian generators and their properties, opening up new optimization possibilities.