Resource-efficient Variational Block-Encoding

Abstract: Block-encoding operators are one of the essential components in quantum algorithms based on Quantum Signal Processing. Their gate complexity largely determines the overall gate complexity of the full algorithm. Using variational methods, we aim to compile block-encoding unitaries with near-optimal resource requirements for a large range of input matrices. We find that the number of variational parameters in the parameterized quantum circuit approaches the number of free parameters in the input matrices, depending on whether they are real, complex and/or hermitian. Additionally, symmetries present in the input matrix can be incorporated into the ansatz circuit, reducing the parameter count further and making optimization possible for up to n=8 qubits. While determining variational block-encodings ceases to be computationally feasible for large system sizes, the constructed operators can be used as components of larger block-encodings via a linear combination of block-encodings.