Operator Learning for efficient Quantum Computation

To address this, we propose a complete full-stack variational framework to transform arbitrary operators to compact quantum circuits. Using a hierarchical optimization strategy, we do not compute Riemann derivatives, cf. [7], but compute the derivatives by backpropagation [9]. The resulting variational circuits can be tailored to the connectivity and long-range interactions of the target hardware and are learned through a cost function that efficiently optimizes unitary operators, block-encodings, or operators restricted to specific symmetry sectors. The proposed operator learning protocol is demonstrated for both quantum mechanical and engineering applications. For practical reasons we use a noise-free quantum computer emulation.

We benchmark our framework by learning propagators arising in native quantum problems such as quantum simulation and quantum chemistry, where both cases improved resource scaling in comparison to standard Suzuki--Trotter expansions. 

Importantly, we demonstrate the ability of the operator programming to implement the central second-order approximation of the Laplacian relevant for solving partial differential equations on quantum hardware, improving state-of-the-art error measures. The final case learns a dense non-unitary operator arising in computational fluid dynamics when analyzing the inviscid flow around an airfoil modeled by a panel method [10]. The complexity and the accuracy of the framework are evaluated based on the relative error and the computational effort.

The presented methodology accomplishes, that any non-unitary operator can generally be treated by using a single ancilla, maintaining optimal success probabilities in variational quantum algorithms applications. Due to its general applicability, the approach can be easily adapted to implement other non-unitary dynamics and to optimize existing gate sequences. For example, applications in scientific computing could address the realization of higher-order approximation schemes, or most importantly, the treatment of boundary conditions on non-uniform discretizations. 

Collectively, the present framework lays the foundation for a quantum computer programming protocol and opens the door for solving many problems beyond prototypical engineering and quantum applications.