A Scalable Quantum Neural Network for Approximate SRBB-Based Unitary Synthesis

In this work, scalable quantum neural networks are introduced to approximate unitary evolutions through the Standard Recursive Block Basis (SRBB) and, subsequently, redesigned with a reduced number of CNOTs. This algebraic approach to the problem of unitary synthesis exploits Lie algebras and their topological features to obtain scalable parameterizations of unitary operators. First, the recursive algorithm that builds the SRBB is presented, framed in the original scalability scheme already known to the literature only from a theoretical point of view. Unexpectedly, 2-qubit systems emerge as a special case outside this scheme. Furthermore, an algorithm to reduce the number of CNOTs is proposed, thus deriving a new implementable scaling scheme that requires one single layer of approximation. From the mathematical algorithm, the scalable CNOT-reduced quantum neural network is implemented and its performance is assessed with a variety of different unitary matrices, both sparse and dense, up to 6 qubits via the PennyLane library. The effectiveness of the approximation is measured with different metrics in relation to two optimizers: a gradient-based method and the Nelder-Mead method. The approximate SRBB-based synthesis algorithm with CNOT-reduction is also tested on real hardware and compared with other valid approximation and decomposition methods available in the literature.