Sparse identification of quantum Hamiltonian dynamics via quantum circuit learning

TL;DR

SIQHDy is introduced as a SINDy-inspired quantum circuit learning method to identify quantum Hamiltonian dynamics from time-series measurements, representing unitary steps as $U(\Delta t)=e^{-i H \Delta t}$ and aiming to recover both the structure and parameters of $H$ from data. It models the short-time evolution as a product of basis quantum circuits $u(\boldsymbol{\theta}\Delta t)$ and learns the sparse parameter vector $\boldsymbol{\theta}$ by fitting observed input-output pairs, using a loss that combines data fidelity with a sparsity-inducing threshold. The method is validated on single-, three-, and five-spin Hamiltonians, including a transverse-field Ising model, demonstrating accurate reconstructions and robustness to measurement noise; an extension handles limited measurement access to enable two-spin identification and five-spin network-structure reconstruction. This work presents a data-driven, interpretable quantum system identification framework powered by hybrid quantum-classical computation, with potential for experimental validation and application to open quantum systems in future work.

Abstract

Sparse identification of nonlinear dynamics (SINDy) is a data-driven framework for estimating classical nonlinear dynamical systems from time-series data. In this approach, system dynamics is represented as a linear combination of a predefined set of basis functions, and the corresponding coefficients are sparsely estimated from observed time-series data. In this study, we propose sparse identification of quantum Hamiltonian dynamics (SIQHDy), a SINDy-inspired quantum circuit learning framework for estimating quantum Hamiltonian dynamics from time-series data of quantum measurement outcomes. In SIQHDy, the unitary evolution of a quantum Hamiltonian system is expressed as a product of basis quantum circuits, and the corresponding circuit parameters are estimated through sparsity-promoting optimization. We numerically demonstrate that SIQHDy accurately reconstructs the dynamics of single-, three-, and five-spin systems, and exhibits robustness to measurement noise in the three-spin case. Furthermore, we propose an extension of SIQHDy for scenarios with limited accessible observables and evaluate its performance in identifying two-spin systems and in network-structure identification for five-spin systems.

Figures (12)

• Figure 1: (a) Schematic diagram of SIQHDy. (b) Correspondence between SINDy and SIQHDy.
• Figure 2: A schematic diagram of the SIQHDy algorithm.
• Figure 3: Basis quantum circuits for the single-qubit case (a) and the two-qubit case (b). (a) Three single-qubit rotation gates about $X$, $Y$, and $Z$ axes. (b) Fifteen rotation gates, consisting of $3 \times 2$ rotation gates acting on single qubits and $3 \times 3$ rotation gates acting on two qubits. Here, $q_1$ and $q_2$ denote the first and second qubits, respectively.
• Figure 4: Results of the single-spin case. (a-c) Comparison between the true dynamics (red solid lines) of the original quantum system and the dynamics predicted by SIQHDy (black dots) for the expectation values of observables $X$ (a), $Y$ (b), and $Z$ (c). The dashed vertical line indicates $t=1$, up to which the data are used for learning. For $t \leq 1$ single-step predictions are plotted, whereas for $t > 1$ the learned quantum circuit is recursively applied starting from $t = 1$, and the resulting predicted dynamics are plotted. The SIQHDy results are plotted every $10$ sampling points.
• Figure 5: Results of the three-spin case. (a-d) Comparison between the true dynamics (red solid lines) of the original quantum system and the dynamics predicted by SIQHDy (black dots) for the expectation values of observables $IXY$ (a), $XYI$ (b), $YXZ$ (c), and $ZIZ$ (d). The dashed vertical line indicates $t=1$, up to which the data are used for learning. For $t \leq 1$ single-step predictions are plotted, whereas for $t > 1$ the learned quantum circuit is recursively applied starting from $t = 1$, and the resulting predicted dynamics are plotted. The SIQHDy results are plotted every $10$ sampling points.