Classical Criticality via Quantum Annealing

Quantum annealing provides a powerful platform for simulating magnetic materials and realizing statistical physics models, presenting a compelling alternative to classical Monte Carlo methods. We demonstrate that quantum annealers can accurately reproduce phase diagrams and simulate critical phenomena without suffering from the critical slowing down that often affects classical algorithms. To illustrate this, we study the piled-up dominoes model, which interpolates between the ferromagnetic 2D Ising model and Villain’s fully frustrated ``odd model’’. We map out its phase diagram and for the first time, employ finite-size scaling and Binder cumulants on a quantum annealer to study critical exponents for thermal phase transitions. Our method achieves systematic temperature control by tuning the energy scale of the Hamiltonian, eliminating the need to adjust the physical temperature of the quantum hardware. This work demonstrates how, through fine-tuning and calibration, a quantum annealer can be employed to apply sophisticated finite-size scaling techniques from statistical mechanics. Our results establish quantum annealers as robust statistical physics simulators, offering a novel pathway for studying phase transitions and critical behavior.

💡 Research Summary

The authors demonstrate that a commercial quantum annealer can be used as a high‑fidelity simulator of classical statistical‑mechanical phase transitions, overcoming the critical slowing down that plagues conventional Monte‑Carlo methods. Their testbed is the “Piled‑Up Domino” (PUD) model, a two‑dimensional Ising system whose couplings are tuned by a parameter s. When s = 0 the model reduces to the ferromagnetic 2D Ising lattice, while s = 1 yields Villain’s fully‑frustrated “odd” model; intermediate values interpolate between these limits, producing a tunable degree of frustration. The exact solution of the PUD model provides a known critical temperature T_c(s) and a phase diagram containing ferromagnetic, antiferromagnetic, and paramagnetic regions.

The key methodological insight is that the D‑Wave device samples from a Boltzmann distribution at a fixed, unknown physical temperature T_sampler. By scaling the programmed Hamiltonian with a factor J (i.e., H_input = J H), the effective sampling temperature becomes T_eff ∝ 1/J. Thus, without changing the hardware temperature, the authors sweep an effective temperature simply by varying J. They implement this on the Advantage2 prototype (2.6 GHz) for toroidal lattices of size L = 6, 8, 10, 12 (up to 144 spins), collecting 10 000 samples per (s, J⁻¹) point.

Order parameters—average magnetization ⟨|m|⟩ and staggered magnetization ⟨|m_AFM|⟩—are computed from the samples. The resulting s–J⁻¹ maps reproduce the three‑phase structure predicted by the exact solution, confirming that the annealer correctly captures the global phase diagram. To locate critical points precisely, the fourth‑order Binder cumulant U = 1 − ⟨m⁴⟩/(3⟨m²⟩²) is evaluated for each system size. The crossing of U‑curves for different L yields a critical energy scale J_c⁻¹(s). These crossing points align linearly with the exact T_c(s), validating the assumed relation T_eff ∝ 1/J.

Further, the authors extract the magnetic susceptibility χ and heat capacity C_V via χ/β = N(⟨m²⟩ − ⟨m⟩²) and C_V/β² = N(⟨E²⟩ − ⟨E⟩²). Plots of χ/β and C_V/β² versus J⁻¹ for various L show the characteristic growth of peak height and shift toward lower J⁻¹ (higher temperature) as L increases, mirroring finite‑size scaling behavior in Monte‑Carlo studies. Applying standard finite‑size scaling forms χ ∝ L^{γ/ν} f(t L^{1/ν}) and C_V ∝ L^{α/ν} g(t L^{1/ν}), where t = (T − T_c)/T_c, they obtain estimates of the critical exponents ν, γ, and α that are statistically consistent with the known 2D Ising values.

Importantly, the work shows that despite known non‑idealities of quantum annealers—biased ground‑state sampling, noise, residual quantum effects—systematic calibration (energy‑scale tuning, extensive sampling, periodic boundary conditions) yields data of sufficient quality to perform sophisticated statistical‑mechanics analyses such as Binder‑cumulant crossings and exponent extraction. The methodology eliminates the need for hardware temperature control, replacing it with a simple programmable scaling of the Hamiltonian, thereby simplifying experimental design.

In summary, the paper establishes three major points: (1) quantum annealers can emulate temperature sweeps by Hamiltonian scaling, (2) classical finite‑size scaling techniques, including Binder cumulant analysis, can be directly applied to annealer‑generated data to locate critical points, and (3) critical exponents can be extracted with accuracy comparable to traditional Monte‑Carlo simulations. This demonstrates that quantum annealing is a viable, potentially superior platform for studying phase transitions, especially in frustrated systems where classical algorithms suffer from critical slowing down. The results open a pathway for using analog quantum hardware as a general-purpose statistical‑physics simulator.