Quantum Annealer Resolves Finite-Temperature Criticality in Two-Dimensional Ising Model

Understanding how systems change state – known as criticality – is fundamental to physics, yet investigating these transitions at realistic temperatures presents a major computational hurdle. Gianluca Teza at the Max Planck Institute for the Physics of Complex Systems, Francesco Campaioli from RMIT University and the Universitá degli Studi di Padova, and colleagues demonstrate a significant advance in overcoming this challenge by harnessing the power of quantum annealing. The team successfully uses a quantum annealer to precisely map and analyse the behaviour of a magnetic material undergoing a phase transition, capturing the full range of critical behaviour at finite temperatures. This breakthrough offers a new pathway to study complex systems – from magnetism to potentially even biological systems – and promises to unlock deeper insights into equilibrium and non-equilibrium phenomena.

Methods for studying critical phenomena face persistent challenges, including limitations with critical slowing down and entanglement growth. Quantum computing and simulation platforms are emerging as promising alternatives, potentially overcoming these limitations. While digital quantum processors and analog platforms have shown promise, quantum annealers – analog devices designed to find low-energy configurations – are gaining attention for simulating the thermodynamics of complex systems.

Recent studies demonstrate their ability to capture features of both classical and quantum phase transitions, highlighting their potential for exploring equilibrium statistical mechanics and finite-temperature criticality. Accurately determining finite-temperature criticality with quantum annealers has proven difficult due to system response time, temperature fluctuations, and hardware constraints. However, a recent study demonstrates that careful programming can enable quantum annealers to precisely resolve finite-temperature criticality in complex many-body systems.

Researchers leveraged the two-dimensional ferromagnetic Ising model – a system with a known critical point – as a benchmark. They mitigated magnetic leakage, hardware asymmetries, and edge effects through a tailored embedding strategy that maps the simulated system to the annealer’s physical architecture. By varying the energy scale of the model and calibrating the quantum processing unit (QPU) temperature, they were able to embed lattices with over 2500 spins.

From local observables and two-point correlations, the critical temperature and critical exponents were extracted. The team demonstrated the ability to measure and control all relevant parameters of an exactly solvable model through two key measurements. First, they tracked magnetization during a non-equilibrium relaxation after a sudden change in conditions, demonstrating acceleration gained by the presence of a transverse field.

Second, they initiated the system in a metastable state and measured the probability of tunneling through an energy barrier, revealing the annealer’s ability to reach lower-energy states. This establishes a systematic and scalable method for studying finite-temperature criticality, applicable to a broad class of systems including frustrated magnets, spin glasses, and lattice gauge theories. The programmable framework can also be adapted to investigate non-equilibrium dynamics and quenched disorder, offering a path for exploring critical phenomena challenging for classical methods.

Trapped Magnetization and Relaxation in Transverse Fields Researchers investigated the behaviour of the system when initialized in a fully polarized state opposite to the direction favoured by the longitudinal field – a metastable configuration. They quenched the system to a transverse-field Ising Hamiltonian and measured the probability of reaching the correct symmetry-broken ground state as a function of the transverse field. The results demonstrate that at small transverse fields, the system remains trapped in the metastable state.

As the transverse field increases, the success probability rises sharply, signalling the onset of barrier crossing. This transition reflects the role of the transverse field in facilitating escape from metastability. Researchers determined that the system can tunnel through a barrier approximately 60 times larger than the thermal energy at the experimental temperature.

The values of the transverse field explored remain well below the critical field at which the ordered phase disappears, ensuring the observed transition reflects a meaningful crossing between locally stable configurations. ## Criticality Found with Quantum Annealers This work demonstrates that carefully calibrated quantum annealers can accurately determine finite-temperature criticality in complex many-body systems. Crucially, calibration of the QPU temperature and mitigation of freeze-out effects proved essential for reliably extracting critical temperature and exponents from a two-dimensional Ising ferromagnet.

Previously considered limitations – the native topology of the QPU and the freeze-out point – are bypassed by this operational line, enabling reliable analysis for finite-temperature criticality in both current and older architectures. The embedding strategy extends the approach to a broad class of systems and problems mappable to the Ising model, including quantum gravity models, non-equilibrium quantum thermodynamics, and quantum circuit compilation. Further research should address memory effects and systematic comparisons with classical algorithms to clarify the potential advantage of quantum sampling or optimization near criticality.