Using Fourier Analysis and Maximum Likelihood Estimation to Identify and Model Non-Markovian Noise in Quantum Operations

The most well-studied error models for quantum operations or gates are Markovian, which assume that the error is “memoryless.” Generically, most error sources can lead to gate errors which violate this assumption. One such source includes periodic noise, which causes the amplitude of gate errors to fluctuate at a characteristic frequency, requiring a non-Markovian error model. Characterizing periodic errors, rather than just identifying them, requires data processing and error modeling beyond the standard procedures used to identify Markovian errors. Using an open-source Fourier transform implementation for qubit measurement data (pyGSTi), we were able to identify the presence of periodic error in quantum operations on trapped-ion qubits. This information allowed us to hypothesize the sources of periodic noise, and create a time-dependent error model, or waveform, whose parameters we optimized to fit the measured data via maximum likelihood estimation. This procedure allows us to quantitatively characterize and predict error sources in an experimental set up, as well as provide a basis for building more generalized non-Markovian error models.