Zero noise extrapolation for a small diatomic molecule on a trapped-ion quantum computer

Current noisy intermediate-scale quantum (NISQ) ion-trap devices are subject to gate errors, particularly for two qubit gates. These errors significantly impact the accuracy of calculations if left unchecked. Zero noise extrapolation techniques such as Richardson extrapolation can reduce these errors without incurring a qubit overhead. We demonstrate and optimize this extrapolation on the quantum scientific computing open user testbed (QSCOUT) ion-trap device to calculate the ground state energy of the HeH+ molecule using the variational quantum eigensolver (VQE). This work uses two methods to scale the noise in our circuits: lengthening the duration of our laser control pulses and discretely inserting gates. Using simulations, we study how to best integrate Richardson extrapolation into an optimization procedure. We further study the effects of the extrapolation order on the accuracy and precision of the final energy estimate to maximize its effectiveness for a given sampling budget. Experimental extrapolation results are mixed; time-stretching the pulse duration does not increase the noise enough to extrapolate, but gate insertions can effectively scale the noise given a large enough sampling overhead. However, these extrapolated results are still sensitive to coherent errors. Further experiments should use more samples and layer additional error mitigation techniques such as Randomized Compiling on top of extrapolation in order to improve estimates.