Hamiltonian learning via quantum Zeno effect

TL;DR

The paper introduces a scalable Hamiltonian learning protocol for geometrically local quantum systems that leverages the quantum Zeno effect via unitary kicks to dynamically reshape the system’s interactions into independent patches. Each patch is characterized in parallel using quantum process tomography, allowing recovery of local Hamiltonian coefficients from simple product-state preparations and local measurements. The authors provide analytical performance guarantees linking the reconstruction error to Zeno fidelity and QPT sampling, and demonstrate the approach numerically up to 128-qubit chains and experimentally on IBM hardware learning a 109-qubit Hamiltonian. The method exhibits favorable scaling, relies on hardware-friendly operations (virtual Z gates and short coherent evolutions), and holds promise as a practical benchmarking tool for large-scale quantum devices.

Abstract

Determining the Hamiltonian of a quantum system is essential for understanding its dynamics and validating its behavior. Hamiltonian learning provides a data-driven approach to reconstruct the generator of the dynamics from measurements on the evolved system. Among its applications, it is particularly important for benchmarking and characterizing quantum hardware, such as quantum computers and simulators. However, as these devices grow in size and complexity, this task becomes increasingly challenging. To address this, we propose a scalable and experimentally friendly Hamiltonian learning protocol for Hamiltonian operators made of local interactions. It leverages the quantum Zeno effect as a reshaping tool to localize the system's dynamics and then applies quantum process tomography to learn the coefficients of a local subset of the Hamiltonian acting on selected qubits. Unlike existing approaches, our method does not require complex state preparations and uses experimentally accessible, coherence-preserving operations. We derive theoretical performance guarantees and demonstrate the feasibility of our protocol both with numerical simulations and through an experimental implementation on IBM's superconducting quantum hardware, successfully learning the coefficients of a 109-qubit Hamiltonian.

Figures (6)

• Figure 1: Benchmarking digital quantum simulators via Hamiltonian learning. Top: a digital quantum simulator implements the target evolution. Bottom: our protocol uses Zeno-based reshaping to isolate local patches, enabling parallel quantum process tomography and accurate reconstruction of the simulated Hamiltonian.
• Figure 2: Sketch of the protocol. (a) Three reshaping configurations are required to learn an $N$-qubit Hamiltonian with $k=2$ and a linear geometry. Red circles represent the frozen qubits, while the green blurred regions represent the learned interactions. (b) Circuit-like scheme of the protocol for the first reshaping configuration. The effect of alternative application of unitary evolution and unitary kicks can be approximated by the evolution under the Zeno Hamiltonian, where different pairs of target qubits evolve independently. By applying this procedure to a set of initial states and performing a selected set of measurements at the end, the learning of the interactions in each target subsystem can be achieved. Note that no back-kick is shown in the scheme, since in our setting ($Z$ kicks with even $r$) the accumulated kick factor is the identity.
• Figure 3: Numerical results (a). Euclidean norm of the difference between reconstructed and true Hamiltonian coefficients as a function of the total number of copies used in the tomography for system sizes up to $N=128$ . The shaded areas indicate one standard deviation over 10 different Hamiltonians. We fix the evolution time to $T=0.01$ and number of kicks to $r=10$. In the bottom left insert panel, we show the average absolute error for the coefficients at a fixed number of copies $\approx10^{9}$ (b). Absolute errors (1) and comparison of true versus reconstructed Hamiltonian coefficients (2) for $N=9$ and $\approx10^{10}$ total copies.
• Figure 4: Experimental results. The experiment was performed with a total evolution time of $T=1$, $r=10$ kick repetition for the Zeno reshaping and $900$ shots for each setting ($36\times9\times3$) on the ibm_brisbane device. (a). Selection of qubits used to perform the 109-qubit experiment. (b). Relative differences between real and reconstructed coefficients. (c). Reconstructed Hamiltonian coefficients: one-body (orange) and two-body (purple). Error bars denote one standard deviation obtained via Monte Carlo resampling Feigelson_Babu_2012 ($10^4$ instances).
• Figure 5: Euclidean norm of the difference between reconstructed and true Hamiltonian coefficients as a function of the number of kicks, for system sizes $N=3,6,9$ and fixed total evolution time $T=0.01$. The reconstruction is performed under ideal conditions with exact tomography and noiseless measurements, so the plotted error reflects only the contribution from imperfect Zeno confinement.

Theorems & Definitions (2)

• Theorem 1
• Corollary 1.1