Experimental characterization of coherent and non-Markovian errors using tangent space decomposition

Abstract

Accurate characterization of coherent and non-Markovian errors remains a central challenge in quantum information processing, as conventional benchmarking techniques typically rely on Markovian and time-independent noise assumptions. In practice, however, quantum devices exhibit both systematic coherent miscalibrations and temporally correlated fluctuations, which complicate error diagnosis and mitigation. Here, we apply a technique based on tangent-space decomposition to characterize such error in single-qubit quantum gates implemented on a trapped ion platform. Small imperfections in a quantum operation are treated as perturbations of the target quantum map, represented as tangent vectors in the space of quantum channels. This formulations enables a natural decomposition of the deviation into three components corresponding to coherent, Markovian and non-Markovian processes.The relative weights of these components provide a quantitative measure of the contribution from each type of error mechanism, directly from a single tomographic snapshot. We experimentally validate this method on a single-qubit gates implemented on a trapped $^{40}$Ca$^+$ ion, where control is achieved through laser-driven optical transitions. By analyzing experimentally reconstructed process matrices, expressed in the Pauli Transfer Matrix and Choi representations, we identify and quantify non-Markovian effects arising from controlled injection of slow fluctuations in the experimental environment. We also characterize deterministic coherent miscalibrations using the same technique. This approach provides a physically transparent and experimentally accessible tool for diagnosing complex error sources in quantum control systems.

Figures (4)

• Figure 1: Tangent space decomposition:a. The Choi matrix is a $4\times 4$, Hermitian, positive semi-definite matrix that characterized the quantum process completely. It can be measured using a process tomography. b. The space of Choi matrices form a manifold and the errors in quantum control are a perturbation on the surface of the manifold. The difference between the target Choi matrix and the measured Choi matrix can be decomposed into three components corresponding to three distinct sources of error. c. Experimental sequence for process tomography of a target unitary gate $U$ on a single qubit. The gates $A$ and $B$ are varied across $12$ combinations (see text) to recover the Choi matrix completely.
• Figure 2: Energy level scheme of the Ca$^+$ ion showing the main transitions used for the experiment.
• Figure 3: Process tomography for a single qubit rotation with induced coherent errors. (a) Contributions of different types of error $\epsilon^2$ to the total gate infidelity $r$ when a $\pi$ rotation about the $x$ axis, $U= R_x(\pi/2)$, is miscalibrated by 10% under-rotation, i.e., the miscalibrated $\pi$-pulse time is $0.9$ of the true $\pi$-pulse time. A non-Markovian contribution arising from background fluctuations is also found to be present. (b) Rotation-angle miscalibration inferred from the tangent-space decomposition: the angle estimates are extracted via he axis-angle representation (Eq. \ref{['ax-angle']}). (c) Measured rotation-angle error (y-axis) as a function of the introduced $\pi$-pulse time miscalibration for the rotation about the $x$-axis. Solid lines show experimental data, dashed lines show numerical simulations (see Appendix \ref{['app:numerical_sim']}). Error bars for the histograms in (a) and (b) are indicated by the vertical black lines. (d) Measured total errors. While the non-Markovian errors are constant, the coherent errors show the expected parabolic structure (see main text).
• Figure 4: (a) Non-Markovian noise is emulated by introducing controlled fluctuations in the Rabi frequency via laser power modulation, which is implemented by modulating the RF signal fed on a Acousto-Optic Modulator (AOM). (b) Contributions of the different errors to the total gate infidelity as a function of different RF drive amplitudes $V_{pp}$. Measured non-Markovian errors and contribution to the infidelity monotonically increases with $V_{pp}$. (c) Measured total error ($y$-axis) versus injected $V_{pp}$ RF amplitude ($x$-axis). Solid lines show experimental data; dashed lines show numerical simulations (see Appendix \ref{['app:numerical_sim']}). Blue (gray) dashed lines corresponds to normal (top hat) error distribution.