Detecting errors in trapped-ion quantum processors beyond the Markovian regime using reduced Choi-matrix tomography.

Understanding the performance of quantum processors and their components is an essential prerequisite for achieving scalable fault-tolerant (FT) quantum error correction (QEC). The majority of QEC codes and FT thresholds are designed for incoherent Markovian errors. If these errors are sufficiently small and satisfy the threshold theorem, reliable quantum computation can be ensured. However, real quantum processors also exhibit non-Markovian errors and coherent errors.

Thus, in order to optimize QEC codes to function close to FT thresholds, it is important to detect the presence of coherent and non-Markovian errors. This allows for the application of subsequent error mitigation techniques and debugging of circuits affected by errors that are not relevant, and to make the noise more ``Pauli'' for FT-QEC thresholds.

To accomplish this without scaling up to a holistic QEC circuit characterization protocol, we focus on differentiating between the various types of noise present in trapped-ion native gates [1] by employing a recently developed benchmarking technique based on reduced Choi-matrix (RCM) tomography [2,3]. We leverage the Choi-Jamiolkowski isomorphism, which, through a quantum channel-state duality enables us to examine the characteristics of an unknown quantum process affecting the target unitary. This examination is achieved by looking at the properties of the Choi density matrix. If non-Markovian and coherent errors are present, error mitigation techniques are applied until these errors are no longer detectable by RCM tomography, and subsequent active QEC techniques are then employed to deal with stochastic Markovian processes.

[1] Andrea Rodriguez-Blanco, Farid Shahandeh, and Alejandro Bermudez. Witnessing entanglement in trapped-ion quantum error correction under realistic noise, 2023.

[2] Bharath Hebbe Madhusudhana. Benchmarking multi-qubit gates – I: Metrological aspects, 2023.

[3] Bharath Hebbe Madhusudhana. Benchmarking multi-qubit gates – II: Computational aspects, 2023.