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

In this work, a scalable quantum neural network is introduced as a means to approximate any unitary evolution through the Standard Recursive Block Basis (SRBB) and, subsequently, redesigned with a number of CNOTs asymptotically reduced by an exponential contribution. 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 original SRBB-based scalability scheme, already known in the literature only from a theoretical point of view, is reformulated for efficient algorithm implementation and complexity management. Remarkably, 2-qubit operators emerge as a special case outside the original scaling scheme. Furthermore, an algorithm is proposed to reduce the number of CNOTs, thus deriving a new implementable scaling scheme that requires only one layer of approximation. 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 CNOT-reduced SRBB-based synthesis algorithm is also tested on real hardware and compared with other valid approximation and decomposition methods available in the literature.