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Jülich Researchers Find Oscillating Limits in an Exact Ising Model
Researchers present an exactly solvable Ising model as a benchmark for quantum and classical optimization, finding recurring system-size fluctuations in quantum annealing and QAOA. Tests on a D-Wave processor and simulations also show how t
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An exact answer, but an uneven route
The Generalized Mean-Field Ising model gives researchers its exact ground state in closed form; quantum annealing and QAOA still show recurring, system-size-dependent fluctuations in reaching it. In a paper submitted to arXiv on October 28, Madita Willsch, Gopal Chandra Santra and six co-authors at Forschungszentrum Jülich present the model as a controlled benchmark for quantum and classical optimization.
The model is densely connected, with local fields and pairwise couplings that vary with the spins’ positions. Despite that complexity, its ground state has a simple structure: a single boundary separates down spins from up spins. The optimal number of up spins comes from a formula in the system size and the ratio of the field and coupling strengths; rounding that value down or up gives the ground state, with both choices optimal in certain cases.
“The GMFI family thus exhibits an indefinitely recurring oscillatory difficulty, providing a natural setting for studying unconventional finite-size effects and the limitations of standard scaling arguments in quantum optimization.”
Madita Willsch and co-authors, authors of the paper
A second gap complicates annealing forecasts
The authors analyze quantum annealing’s changing energy spectrum and distinguish two gaps between low-energy states. Their calculations indicate that the first gap is consistent with polynomial closing, but the second gap oscillates as system size changes. For the model’s chosen parameters, the pattern tracks the fractional part of the exact ground-state formula and has an asymptotic quasiperiod of about 4.7 system sizes.
That second gap can become arbitrarily small at infinitely many system sizes, the authors conclude, so a conventional scaling law cannot describe it. The result complicates a common shortcut in annealing analysis: assuming the first gap alone will eventually set the runtime. It does not show that quantum annealing fails on optimization problems generally; it identifies a recurring difficulty in this particular, deliberately structured family.
D-Wave tests show both model and hardware effects
The team tested instances on a D-Wave Advantage2_system2.1 processor at Forschungszentrum Jülich, focusing on sizes up to 30 spins so they could compare hardware results with numerical simulations. The measured success rates display the same size-dependent pattern, though the standard annealing schedule shows discrepancies at some sizes. The paper’s simulations reproduce the fast-annealing results more closely when they account for minor embedding and errors associated with qubit susceptibility.
Embedding matters because the model’s near-all-to-all connections do not match the processor’s sparse hardware graph: the researchers say instances with five or more spins require minor embedding. They also discard samples containing broken chains rather than correcting them in post-processing, and repeat runs to collect roughly 5,000 to 6,000 valid samples. The hardware data therefore captures the algorithm’s behavior alongside the practical costs of mapping this interaction pattern onto the device.
QAOA and classical methods sharpen the comparison
For QAOA, the researchers use matrix product state simulations, not a hardware experiment, to study instances with 5 to 35 spins at circuit depths from 15 to 20. The optimized states have energy approximation ratios above 0.9, yet ground-state fidelity still oscillates with size and its overall envelope declines as systems grow. That distinction matters: a strong average energy does not guarantee a high probability of measuring the exact optimum.
A quantum-inspired classical greedy method provides a counterpoint. In the study, it reaches at least 90% ground-state fidelity with roughly 35 steps in most tested cases, and the authors report that their chosen rotation angle works up to at least 250 spins. That result applies to this model, whose hidden low-dimensional structure helps explain why the method succeeds; the paper explicitly cautions that the approach need not find ground states for general Ising problems. The benchmark’s value lies in keeping the optimum known while exposing how differently algorithms approach it.