Quantum Circuits Learn System Behaviour from Data Alone

Researchers are developing new methods to understand the underlying rules governing quantum systems, and a new study introduces a technique for identifying the dynamics of Hamiltonian systems using quantum circuit learning. Yusei Tateyama and Yuzuru Kato, both from the Department of Complex and Intelligent Systems at Future University Hakodate, Hokkaido, Japan, present a framework called sparse identification of Hamiltonian dynamics (SIQHDy) which builds upon existing sparse identification of nonlinear dynamics (SINDy) methods. This innovative approach represents quantum evolution as combinations of fundamental circuits, allowing for the estimation of system parameters from observed data. Demonstrating accurate reconstruction of dynamics in systems ranging from single to five spins, and showing resilience to measurement noise, this work represents a significant step towards characterising complex quantum behaviours and offers a pathway to understanding systems even with limited observational access.

Scientists are developing techniques to unlock the full potential of quantum computers, moving beyond theoretical possibilities towards practical application. A crucial challenge lies in accurately characterising how these complex systems evolve over time. New work offers a promising method for deciphering quantum behaviour from limited observational data, potentially accelerating progress in quantum simulation and control.

Scientists have developed a new method for accurately reconstructing the dynamics of quantum systems directly from measurement data. This work introduces sparse identification of quantum Hamiltonian dynamics (SIQHDy), a framework inspired by techniques used to model classical systems but adapted for the unique challenges of quantum mechanics. Unlike existing approaches that often require prior knowledge of a system’s Hamiltonian, its description of energy and evolution, SIQHDy simultaneously identifies both the structure and parameters of the Hamiltonian from time-series data obtained through quantum measurement.

The core innovation lies in representing the quantum system’s evolution as a combination of fundamental quantum circuits, with their parameters refined through a sparsity-promoting optimisation process. Numerical demonstrations confirm that SIQHDy successfully reconstructs the dynamics of systems ranging in complexity from single quantum spins to networks of five interacting spins, including the commonly studied transverse-field Ising model.

Crucially, the research also demonstrates the method’s resilience to measurement noise, a significant practical hurdle in quantum experiments. By incorporating sparsity into the model, SIQHDy not only improves accuracy but also yields simpler, more interpretable representations of the underlying quantum dynamics. This advancement addresses a critical gap in quantum system identification, particularly for solid-state systems where the Hamiltonian structure is often unknown.

The framework leverages quantum circuit learning, a hybrid quantum-classical approach, to optimise circuit parameters and extract the governing dynamics. An extension of SIQHDy is also proposed to handle scenarios where only limited measurements are accessible, successfully identifying two-spin systems and reconstructing network structures within five-spin systems. This work paves the way for more efficient and accurate modelling of complex quantum systems, potentially accelerating progress in areas like quantum simulation and materials science.

Quantum circuit construction via sparsity-promoting optimisation reveals Hamiltonian dynamics

Sparse identification of quantum Hamiltonian dynamics (SIQHDy) builds upon the established framework of sparse identification of nonlinear dynamics (SINDy) by adapting it for quantum systems. Rather than directly estimating a Hamiltonian, SIQHDy represents the unitary evolution of a quantum system as a product of carefully chosen basis quantum circuits.

These circuits, acting as fundamental building blocks, are combined with parameters that are then optimised to best match observed time-series measurement data. This approach allows for simultaneous identification of both the structure and parameters defining the underlying quantum Hamiltonian dynamics. The core innovation lies in translating the SINDy concept of basis functions into the language of quantum circuits.

A predefined library of circuits is constructed, each representing a potential contribution to the overall Hamiltonian. The research then employs sparsity-promoting optimisation, a technique that encourages many of these circuit parameters to be zero, effectively identifying the dominant terms governing the system’s evolution. This optimisation is achieved through quantum circuit learning, leveraging the capabilities of quantum computers to refine the circuit parameters.

To validate the methodology, the study numerically simulated the dynamics of single-, three-, and five-spin systems, including the transverse-field Ising model. These simulations served as a controlled environment to assess SIQHDy’s ability to accurately reconstruct the underlying dynamics from measurement data. Furthermore, the robustness of the framework to measurement noise was specifically investigated using the three-spin system, demonstrating the benefits of incorporating sparsity to achieve simpler, more accurate models. An extension of SIQHDy was also developed to address scenarios where only limited observables are accessible, and its performance was evaluated on two- and five-spin systems, including the reconstruction of network structures.

Precise Hamiltonian reconstruction via sparse identification for single and multi-qubit systems

Applying the sparse identification of Hamiltonian dynamics (SIQHDy) framework to a single-spin system governed by the Hamiltonian H = 3/2Y, the optimised circuit parameters were determined as θopt = [0, 1.503, 0] · (2∆t), achieving a cost function value of 7.058 × 10−7. This exceptionally low value confirms the accuracy of the parameter estimation process. The framework’s ability to precisely reconstruct the system’s dynamics is demonstrated. Moving to a three-spin system defined by the Hamiltonian H = 3/2(XXI + ZZI) + IXX + IZZ, the study utilised a total of 36 basis quantum circuits, encompassing both single- and two-qubit rotations.

Data collection involved sampling expectation values of 4N orthogonal operators, where N represents the number of qubits, at discrete time steps of ∆t = 0.01, spanning from k = 0 to 1000. The initial 101 data points were employed for learning, while the remaining data served for validation of the learned circuit. Parameter optimisation, performed using the COBYLA method, minimised the cost function across all input-output pairs, ensuring accurate convergence.

In the case of the single-spin system, the estimated parameter θx1 corresponded to a value of a · (2∆t) when the Hamiltonian was given by H = aX, indicating a direct relationship between the estimated parameter and the Hamiltonian’s coefficient. The framework’s ability to accurately predict dynamics beyond the learning interval, achieved through recursive application of the learned circuit starting at t = 1, further validates its effectiveness. These results demonstrate the potential of SIQHDy for accurately reconstructing and predicting the behaviour of quantum systems.

Reconstructing quantum Hamiltonians via sparse identification of dynamical structures

The persistent challenge of reverse-engineering quantum systems, determining the underlying rules governing their behaviour, has long been hampered by the sheer complexity of quantum mechanics. While we can readily simulate small quantum systems, accurately reconstructing the Hamiltonian, the mathematical description of the system’s energy, from observed data has remained elusive.

This work offers a significant step forward by adapting techniques from classical dynamical systems, specifically, sparse identification methods, to the quantum realm. It isn’t merely about finding a Hamiltonian that fits the data, but one that does so efficiently, leveraging the inherent simplicity often hidden within complex quantum interactions. This approach, termed SIQHDy, moves beyond simply estimating parameters; it aims to identify the fundamental structure of the quantum dynamics itself.

The ability to accurately reconstruct these dynamics, even in the presence of noise, is crucial not only for verifying the performance of quantum devices but also for potentially discovering new, uncharacterised quantum systems. Real-world applications extend to optimising quantum control protocols and improving the accuracy of quantum simulations. However, scaling this method to larger, more realistic quantum devices remains a considerable hurdle. The limited accessibility of observables, the properties we can actually measure, poses a significant challenge.