Data-driven learning of non-Markovian quantum dynamics

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.

💡 Research Summary

The paper presents a data‑driven protocol for learning the generators of quantum gate dynamics without assuming a specific model of the environment. Building on recent work that casts the Nakajima‑Mori‑Zwanzig (NMZ) formulation of open‑system dynamics as a linear regression problem, the authors develop an efficient algorithm that extracts the Markov transition matrix M, the memory kernel K(t), and the Langevin noise term F(t) directly from time‑series measurements of observable expectations. By discretizing the NMZ integro‑differential equation, they obtain a linear relation g_{k+1}=∑{l=0}^{L}Ω(l)Δ g{k−l}+W_k, where Ω(0)Δ≈I+MΔ and Ω(l)Δ≈Δ²K(lΔ) for l>0. They argue that, in realistic quantum‑computing settings, the stochastic Langevin term is negligible, allowing the problem to be solved by ordinary least‑squares with O(N) computational cost.

The methodology is validated on three distinct cases. First, a purely Markovian qubit is simulated using a Lindblad master equation with realistic rates. Ten random pure initial states are evolved for a total time T=20 with a fine timestep δ=10⁻³, then down‑sampled to Δ=100δ for training. Using leave‑one‑out cross‑validation, the learned Ω(0) closely matches the true transition matrix, while the norms of Ω(l>0) are essentially zero, confirming that the algorithm correctly identifies the absence of memory. The root‑mean‑square error (RMSE) remains low and increases only when an artificial memory length is added, illustrating robustness.

Second, the authors simulate non‑Markovian dynamics by driving a qubit with a Hamiltonian H(t)=ω_zσ_z+η(t)σ_x where η(t) is an Ornstein‑Uhlenbeck (OU) stochastic process. Two regimes are explored: a strongly non‑Markovian case (γ=0.5, σ=0.1) and a weakly non‑Markovian case (γ=10, σ=1). Average dynamics are obtained via the diffusive hierarchical equations of motion (DHEOM). The NMZ learning algorithm recovers a memory kernel whose effective length h* matches the OU correlation time τ=1/γ: for γ=0.5 the optimal kernel length is about 5 (arbitrary units), while for γ=10 it shrinks to ≈0.2. The corresponding RMSE curves show a clear minimum at these lengths, demonstrating that the method can quantitatively extract the non‑Markovian timescale from noisy data.

Third, experimental data from a trapped‑ion platform are analyzed. The system consists of a driven single qubit whose noise environment is not characterized a priori. Applying the same learning pipeline yields a memory kernel with a characteristic decay on the order of a few microseconds, consistent with known technical noise sources (laser phase noise, magnetic field fluctuations). The learned Markov matrix also reproduces the observed Bloch‑vector trajectories with high fidelity.

Compared to gate‑set tomography (GST), which assumes time‑independent, Markovian generators and provides only a static process matrix, the NMZ approach delivers a time‑resolved generator and an explicit memory kernel. This enables direct diagnosis of non‑Markovian noise, informs the design of dynamical decoupling sequences, and can be incorporated into error‑mitigation or fault‑tolerant protocols that require knowledge of the underlying master equation. The authors also discuss connections to transfer‑tensor methods, STEADY, and recent convolution‑less master‑equation learning, emphasizing that their linear‑regression‑based scheme avoids the heavy data requirements of deep‑learning approaches while retaining physical interpretability.

In summary, the paper makes three key contributions: (1) a model‑free, data‑driven framework for learning NMZ operators from experimentally accessible observables; (2) a demonstration that the method accurately distinguishes Markovian from non‑Markovian dynamics and quantifies the memory timescale; and (3) an application to real trapped‑ion data showing practical feasibility. The work opens the door to scalable, physics‑informed characterization of quantum hardware, with immediate relevance to gate calibration, noise spectroscopy, and the development of more accurate error models for near‑term and fault‑tolerant quantum computers.