Researchers V. M. Tkachuk and M. I. Samar of Ivan Franko National University of Lviv and SoftServe have successfully simulated a 36-spin Kagome antiferromagnet using the ibm_kingston quantum processor. Their work introduces a hybrid quantum-classical method employing a single ancilla qubit to measure the partition function at imaginary inverse temperatures, enabling reconstruction of energy-level degeneracies and thermodynamic properties.

Despite “substantial circuit depth and hardware noise,” the reconstructed spectra retained dominant features, and calculated heat capacities mirrored exact results, demonstrating the method’s robustness. This approach offers a new path to understanding the complex thermodynamics of frustrated spin systems.

Hybrid Quantum-Classical Method for Spin System Thermodynamics

The researchers demonstrate the method for frustrated Ising antiferromagnets on periodic Kagome graphs with up to 36 spins, using the ibm_kingston quantum processor. The method relies on encoding the normalized partition function into expectation values measured from a single ancilla qubit, bypassing the need for explicit summation over all 2^N spin configurations.

This approach is particularly advantageous for systems where couplings are bounded integer multiples of a common energy scale, limiting the growth of candidate energy levels relative to the exponentially increasing number of spin configurations. By matching sampling times to the energy grid, the team minimized finite-window broadening and spectral leakage, although statistical and hardware errors remained a factor.

The researchers note that “errors at different energy levels may partially compensate,” contributing to the robustness of the reconstructed data. The ability to accurately determine thermodynamic properties hinges on reconstructing the density of states, a process now facilitated by this hybrid quantum-classical approach. The team demonstrated the method’s efficacy on a five-spin benchmark before scaling up to periodic Kagome Ising clusters of varying sizes. This resilience is due to the method’s ability to use weighted spectral contributions, allowing errors at different energy levels to partially offset each other.

However, the study acknowledges that low-temperature properties remain particularly sensitive to inaccuracies in determining low-energy degeneracies. The technique extends beyond simple Ising models, offering a potential route to analyze quantum many-body systems with spectra lying on a known discrete grid.

In these cases, preparing the system in a maximally mixed state allows the ancilla protocol to measure the normalized trace of the evolution operator. The work demonstrates a promising path toward using quantum computation to tackle challenging problems in condensed matter physics.

Inverse Discrete Fourier Transform Reconstructs Energy Degeneracies

For energy spectra defined on a known, equally spaced grid, an inverse discrete Fourier transform can reconstruct energy level degeneracies without introducing spectral leakage, even in the absence of noise. This reconstruction relies on the discrete Fourier orthogonality relation, enabling recovery of degeneracies from a limited number of time points, a technique distinct from continuous Fourier transform approximations.

The method’s efficiency stems from using the known energy grid, eliminating the need for prolonged evolution times to achieve high energy resolution. Measurements of the ancilla qubit yield complex signals that, when subjected to the inverse discrete Fourier transform, reveal the spectral weights and, consequently, the degeneracies of the energy levels.

Initial benchmarking involved a five-vertex graph, confirming the protocol’s functionality before scaling up to the more complex system. Analysis of ancilla expectation values, specifically and, provided the data for reconstructing the energy spectrum. “The discrete Fourier transform yields unconstrained estimates of the degeneracies,” the study notes, adding that statistical and hardware noise can introduce minor deviations, but these are mitigated through post-processing.

The ability to achieve accurate results with circuits of this complexity suggests a pathway toward tackling more challenging quantum many-body problems. This approach differs from other quantum spectral estimation methods that rely on real-time measurements of the time-evolved Hamiltonian, as it specifically exploits the discrete structure of the energy spectrum.

Kagome Antiferromagnets Studied with 36-Spin Quantum Processor

The ibm_kingston quantum processor successfully simulated a frustrated Ising antiferromagnet arranged on a periodic Kagome graph. This simulation, detailed in recent work, used a novel approach to reconstruct energy-level degeneracies and thermodynamic properties, moving beyond traditional computational methods like Wang-Landau sampling and polynomial expansions. The team optimized using commuting interactions and Eulerian ordering to maximize efficiency.

This technique differs from previous quantum approaches that focused on additive approximations of Ising partition functions, instead directly addressing the reconstruction of key spectral features. Analysis of measurement precision revealed a bound of $σ_{w_k}$ ≤ 1/ $\sqrt{Q M}$ for the linear estimator of spectral weights, where Q represents the number of sampling times and M the number of shots per ancilla component.

The reconstructed spectral weights were then used to calculate heat capacity across a range of temperatures, allowing for direct comparison with exact finite-cluster results obtained through classical enumeration of spin configurations. Experiments with a six-spin Kagome cluster, sampled with 2048 shots per measurement setting on the ibm_kingston processor, showed that the reconstructed spectra captured dominant spectral contributions and the broad structure of the heat capacity.

Optimized Quantum Circuits via Commuting Interactions & Euler Ordering

This achievement represents a significant reduction in logical two-qubit gate counts, decreasing from 4L to 2L + 2, where L is the number of bonds in the system. The method’s efficiency stems from exploiting the commutativity of Ising terms, allowing for an ordered arrangement of bond interactions along an Euler circuit and cancellation of adjacent parity gates. The resulting circuits, however, still present challenges due to their depth and susceptibility to hardware limitations. Despite these complexities, the researchers applied readout-error mitigation techniques.

Heat Capacity Analysis Validates Reconstructed Spectra

Analysis of heat capacity data confirms the accuracy of reconstructed energy spectra from a novel quantum simulation technique. Specifically, the reconstructed heat capacities mirrored the main temperature dependence observed in the exact calculations, validating the approach’s ability to capture essential thermodynamic behavior. Analysis of both measurement precision and the impact of weak spectral weights further refined the reconstruction process. The team’s work demonstrates a method for extracting thermodynamic information even with imperfect data, as the reconstructed degeneracies were used to calculate the real-temperature partition function and heat capacity.

Comparisons between experimental datasets and exact results showed that post-processing and readout mitigation techniques improved the accuracy of the reconstructed spectra. For a 24-spin cluster, simulations closely reproduced the dominant spectral weights, while hardware results exhibited larger deviations, highlighting the influence of device noise.

Despite this limitation, the reconstructed spectrum retained its dominant features, and the calculated heat capacity followed the overall temperature dependence of the exact result. The researchers acknowledge that both limited sampling precision and hardware noise contribute to deviations in the reconstructed heat capacity, but emphasize that the method’s resilience is noteworthy.

Measurement Precision Limits Weak Spectral Weight Resolution

Reconstructing the thermodynamic properties of complex magnetic materials demands increasingly precise measurements as system size grows, and recent work highlights the limits imposed by statistical uncertainty and hardware limitations. The study quantified the impact of finite sampling, establishing a shot-noise bound of approximately 3. 1 x 10^-3 for 24-spin clusters and 2. These values become critical when considering the diminishing ground-state weight, dropping from 1. 1 x 10^-2 to roughly 1. 0 x 10^-3 as the cluster size increases, demonstrating that noise increasingly obscures the signal of subtle energy level differences.

👉 More information
🗞 Quantum computation of energy level degeneracy and thermodynamical properties of spin systems
✍️ V. M. Tkachuk and M. I. Samar
🧠 ArXiv: https://arxiv.org/abs/2610.35660

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