用递归基构造变分量子电路，高效逼近目标量子态

In this work, a scalable algorithm for the approximate quantum state preparation problem is proposed, facing a challenge of fundamental importance in many topic areas of quantum computing. The algorithm uses a variational quantum circuit based on the Standard Recursive Block Basis (SRBB), a hierarchical construction for the matrix algebra of the $SU(2^n)$ group, which is capable of linking the variational parameters with the topology of the Lie group. Compared to the full algebra, using only diagonal components reduces the number of CNOTs by an exponential factor, as well as the circuit depth, in full agreement with the relaxation principle, inherent to the approximation methodology, of minimizing resources while achieving high accuracy. The desired quantum state is then approximated by a scalable quantum neural network, which is designed upon the diagonal SRBB sub-algebra. This approach provides a new scheme for approximate quantum state preparation in a variational framework and a specific use case for the SRBB hierarchy. The performance of the algorithm is assessed with different loss functions, like fidelity, trace distance, and Frobenius norm, in relation to two optimizers: Adam and Nelder-Mead. The results highlight the potential of SRBB in close connection with the geometry of unitary groups, achieving high accuracy up to 4 qubits in simulation, but also its current limitations with an increasing number of qubits. Additionally, the approximate SRBB-based QSP algorithm has been tested on real quantum devices to assess its performance with a small number of qubits.