Data-driven learning of non-Markovian quantum dynamics

TL;DR

This work develops a data-driven learning protocol for characterizing quantum gates that builds off previous work on learning the Nakajima-Mori-Zwanzig (NMZ) formulation of open system dynamics from time series data, which allows detailed reconstruction of quantum evolution, including non-Markovian dynamics.

Abstract

Fault-tolerant quantum computing requires extremely precise knowledge and control of qubit dynamics during the application of a gate. We develop a data-driven learning protocol for characterizing quantum gates that builds off previous work on learning the Nakajima-Mori-Zwanzig (NMZ) formulation of open system dynamics from time series data, which allows detailed reconstruction of quantum evolution, including non-Markovian dynamics. We demonstrate this learning technique on three different systems: a simulation of a qubit whose dynamics are purely Markovian, a simulation of a driven qubit coupled to stochastic noise produced by an Ornstein-Uhlenbeck process, and trapped-ion experimental data of a driven qubit whose noise environment is not characterized ahead of time. Our technique is able to learn the generators of time evolution, or the NMZ operators, in all three cases and can learn the timescale in which the qubit dynamics can no longer be accurately described by a purely Markovian model. Our technique complements existing quantum gate characterization methods such as gate set tomography by explicitly capturing non-Markovianity in the gate generator, thus allowing for more thorough diagnosis of noise sources.

Figures (4)

• Figure 1: Demonstration of the NMZ learning algorithm on a qubit undergoing Markovian dynamics (under \ref{['eq:ME_general']} with parameters defined in the text). (a) shows how an NMZ model is able to accurately predict the dynamics of the simulated qubit. The blue points are the simulated qubit Bloch vector data, and the red lines are predictions from the trained NMZ model. (b) shows predictive performance ($\overline{\text{RMSE}}$, dashed magenta line) and the averaged norms of the learned NMZ operators ($\overline{\bm{\hat{\Omega}}_\Delta^{(l)}}$, cyan markers) as a function of memory kernel length. (c) shows the average error in the predicted Markov matrix, $\overline{||\hat{\bm{\Omega}}_{0.1}^{(0)} - \bm{\Omega}_{0.1}^{(0)}||_2}$, as a function of the length and number of time series used for training.
• Figure 2: NMZ operator learning for simulated driven qubit with OU noise. (a) The $\overline{\text{RMSE}}$ of the learned NMZ model predictions (dashed magenta line) and the averaged L2 norms of the learned NMZ operators (cyan markers) as a function of memory kernel length for $\gamma = 0.5$ (strongly non-Markovian) and $\gamma = 10.0$ (weakly non-Markovian). (b) shows an example of predictions from the learned model when applied to an unseen initial state. The top set of plots shows predictions for the $\gamma = 0.5$ data when the memory kernel length is truncated at $h^\star = 5.0$$(arb.)$. The bottom set of plots shows predictions for the $\gamma = 10.0$ data where the memory kernel length is truncated at $h^\star = 0.2$$(arb.)$. The blue points are the simulated data, and the red lines corresponds to the predictions made by the learned NMZ models.
• Figure 3: A continuous Rabi drive is applied to a single qubit causing it to rotate about its x-axis for 5000 $\mu$s. (a)$\overline{\text{RMSE}}$ of the learned NMZ model predictions (dashed magenta line) and averaged L2 norms of the learned NMZ operators (cyan markers) as a function of memory kernel length. The orange line is a fit of $\overline{\text{RMSE}}$ to an exponential function, which is $\overline{\text{RMSE}} = 0.35e^{-8.73 (2.5 \mu s)l}+0.34$, where $l$ is the memory kernel length index. The blue line is a fit of the average NMZ operator norms to an exponential function, which is $\overline{|| \hat{\bm{\Omega}}_{2.5 \mu s}^{(l)} ||} = 1.67e^{-0.68 (2.5 \mu s) l}+0.01$. Guided by the decay of the operator norms, we truncate the memory kernel at $h^\star = 22.5 \mu$s. (b) Predictions from one of the learned NMZ models for an unseen initial state. The blue datapoints are the measured Pauli expectations, and the red line corresponds to the predictions made by the learned NMZ model.
• Figure 4: A single qubit undergoes a Pauli-X rotation of angle $\pi$ on the Bloch sphere, or equivalently a Pauli-X gate. (a)$\overline{\text{RMSE}}$ (dashed magenta line) and average NMZ operator norms (cyan markers) for the learned NMZ models as a function of memory kernel length. (b) Example of predictions of the Bloch vector evolution during the gate execution for one of the learned models given the initial state for one of the time series. The blue points are the measured Pauli expectations, and the red lines are the predictions.