Microscopic treatment of instantaneous spectral diffusion and its effect on quantum gate fidelities in rare-earth-ion-doped crystals

The effect of instantaneous spectral diffusion (ISD) on gate operations in rare-earth-ion-doped crystals is an important question to answer for the future of rare-earth quantum computing. ISD has previously been extensively studied for ensemble systems, where it is observed as a dephasing that depends on the degree of excitation. However, for applications that use single ions, a microscopic modeling that highlights the stochastic nature of the ISD phenomena is necessary. Here we present such a model and use it to investigate ISD errors on single-qubit gate operations. However, directly simulating how the qubit is affected by all non-qubit ions is intractable since the size of the system grows exponentially with the number of ions. To circumvent this problem, we present a method to estimate the error of the full system by examining smaller subsystems, each evaluating the rotation and shrinkage of the qubit Bloch vector due to only one error source. In the case of ISD, the different error sources are individual ions causing ISD, but the method is general and thus applicable to other systems as long as the errors are largely independent. After studying ISD due to the ions surrounding a qubit, we conclude that transmission windows (prepared by optical pumping) are necessary to suppress the ISD errors. Despite using such windows, there remains a roughly $0.3\%$ risk that a qubit has an ISD error larger than the error from other sources. However, in those cases the qubit can be discarded and its frequency channel can be reused by another qubit. In most cases the ISD errors are significantly smaller than other errors, thus opening up the possibility to perform noisy intermediate-scale quantum (NISQ) algorithms despite ISD being present. Finally, the approach presented here creates a foundation that can be used to study how ISD affects even more complicated gate operations.