Quantum Wasserstein Compilation: Unitary Compilation using the Quantum Earth Mover's Distance

Quantum circuit compilation (QCC) is crucial to any quantum algorithm execution. It can translate a circuit into hardware-specific gates, optimize circuit depth, and adapt to noise.Variational quantum circuit compilation (VQCC) optimizes the parameters of an ansatz according to the goal of reproducing a given unitary transformation. In this work, we present a VQCC-objective function called the quantum Wasserstein compilation (QWC) cost based on the quantum Wasserstein distance of order 1. We show that the QWC function upper bounds the average infidelity of two circuits. An estimation method based on measurements of local Pauli-observable is utilized in a generative adversarial network to learn a given quantum circuit.We compare the efficacy of QWC cost with other cost functions, such as the Loschmidt echo test (LET) and the Hilbert-Schmidt test (HST). Finally, our experiments demonstrate that QWC as a cost function can mitigate the barren plateaus for the particular case we consider.