Learning and certification of local time-dependent quantum dynamics and noise

TL;DR

This work extends Hamiltonian learning to time-dependent local quantum dynamics, proving that time-dependent generators on bounded-degree graphs can be learned efficiently under minimal experimental access. The authors develop a protocol that combines Lieb-Robinson locality, process shadow tomography, and semidefinite programming to construct stable, diagonally dominant linear systems for the time-dependent coefficients, with a sample cost scaling as $S= ilde{O}(rac{1}{\epsilon^2}\text{poly}(m)\log(n\delta^{-1}))$ and polynomial overhead in the function-class dimension $m$. By introducing Markov-stable function systems (MSFS) and robust interpolation, the method yields stable estimates of the time-dependent coefficients $h_{\alpha}(t)$ and $\ell_{j,P}(t)$ on $[0,T]$, enabling practical verification of time-dependent processes and drift diagnostics in quantum devices. The framework supports extrapolation to larger times with controlled overhead and opens several directions, including structure learning, extension to more general Lindbladians, and experimental demonstrations. Overall, the paper delivers a scalable, rigorous tool for certifying dynamic quantum protocols and characterizing time-varying noise in near-term devices.

Abstract

Hamiltonian learning protocols are essential tools to benchmark quantum computers and simulators. Yet rigorous methods for time-dependent Hamiltonians and Lindbladians remain scarce despite their wide use. We close this gap by learning the time-dependent evolution of a locally interacting $n$-qubit system on a graph of effective dimension $D$ using only preparation of product Pauli eigenstates, evolution under the time-dependent generator for given times, and measurements in product Pauli bases. We assume the time-dependent parameters are well approximated by functions in a known space of dimension $m$ admitting stable interpolation, e.g. by polynomials. Our protocol outputs functions approximating these coefficients to accuracy $ε$ on an interval with success probability $1-δ$, requiring only $O\big(ε^{-2}poly(m)\log(nδ^{-1})\big)$ samples and $poly(n,m)$ pre/postprocessing. Importantly, the scaling in $m$ is polynomial, whereas naive extensions of previous methods scale exponentially. The method estimates time derivatives of observable expectations via interpolation, yielding well-conditioned linear systems for the generator's coefficients. The main difficulty in the time-dependent setting is to evaluate these coefficients at finite times while preserving a controlled link between derivatives and dynamical parameters. Our innovation is to combine Lieb-Robinson bounds, process shadows, and semidefinite programs to recover the coefficients efficiently at constant times. Along the way, we extend state-of-the-art Lieb-Robinson bounds on general graphs to time-dependent, dissipative dynamics, a contribution of independent interest. These results provide a scalable tool to verify state-preparation procedures (e.g. adiabatic protocols) and characterize time-dependent noise in quantum devices.

Figures (2)

• Figure 1: Overview of the three phases of our protocol. (P1) we compute the necessary degree of the functions to achieve a desired precision with high success probability. Given that, we draw random times to perform a stable interpolation. For each time, we compute the necessary number of samples and draw random Paulis used for quantum data acquisition. (P2) process shadow tomography: we run the quantum circuits above for the prescribed times and random initial Pauli rotations and bases, recording the measurement outputs. (P3) we form SDPs to get stable linear systems; stably interpolate derivatives, solve for values at chosen times, then re-interpolate to recover the generator’s time-dependent functions.
• Figure 2: Workflow of the learning protocol. Preprocess to pick stable Pauli pairs and interpolation nodes; collect process-shadow data; then perform SDP-based inversion, coefficient estimation, derivative computation, and interpolation to recover parameter functions.

Theorems & Definitions (81)

• Definition 2.1: Markov-stable function system
• Remark 2.1
• Remark 2.2
• Theorem 2.1
• Proposition 3.1
• Lemma 5.1
• proof
• Corollary 5.1
• proof
• Proposition 5.1