Researchers at Friedrich-Alexander-Universität Erlangen-Nürnberg have used quantum annealing hardware to determine the ground state of complex Ising models. The team applied a unit-cell-based optimization scheme, performing finite optimizations on each unit cell using commercial quantum annealing hardware, to problems featuring algebraically decaying long-range interactions. To demonstrate the capabilities of the approach, researchers chose to evaluate the ground state of the Kagomé lattice motivated by research into artificial spin ice metamaterials. The researchers write that the approach provides a timely connection between available quantum hardware and relevant systems.
Long-Range Ising Model Formulation for Quantum Simulation
The determination of ground states for Ising models featuring algebraically decaying, competing long-range interactions is now possible using superconducting qubit quantum annealing devices, a technique that allows exploration in the thermodynamic limit. Researchers chose three problems with relevance to both quantum simulation and materials science to demonstrate the capabilities of the approach, showcasing its versatility across multiple research areas. One key application involved calculating devil’s staircases of magnetization plateaux within the long-range Ising model, specifically on a triangular lattice subjected to a longitudinal field.
This calculation was motivated by advancements in atomic and molecular quantum simulators, where precise control over interactions is paramount. These metamaterials, engineered to mimic magnetic frustration, require a deep understanding of ground state properties for effective design and implementation.
The UCBOS approach allows for the investigation of these complex lattice structures without truncation of long-range interactions, a significant advantage over traditional methods. The researchers also extended their method to study models incorporating additional few-nearest-neighbor interactions, relevant to frustrated Ising compounds potentially exhibiting long-range interactions. These compounds, like Er₂Be₂GeO₇, are examples of systems where the method can be applied. The optimization process itself relies on performing finite optimizations on each unit cell using existing quantum technology, ultimately revealing the optimal state of matter for the investigated long-range interacting model in the thermodynamic limit.
The team employed the D-Wave Advantage system for their quantum annealing computations, using commercially available technology to tackle these complex problems. The approach, according to the authors, provides a useful and realistic application of existing quantum annealing technology, applicable across many research areas in which lattice problems with resummable long-range interactions are relevant.
The ability to accurately model these interactions is important for understanding the behavior of complex materials and developing new quantum technologies. The work represents a timely connection between available quantum hardware and relevant systems with long-range interactions and applications across various experimental platforms in quantum optics and condensed matter physics. Note, for the application of the UCBOS in this work, the decay of the long-range interaction must be larger than the spatial dimension of the system.
Unit-Cell-Based Optimization Scheme for Lattice Problems
The ability to map complex, long-range interactions onto finite-connectivity quantum annealers has been significantly advanced through a unit-cell-based optimization scheme, allowing researchers to use quantum annealing hardware to determine the ground state of Ising models with algebraically decaying competing long-range interactions in the thermodynamic limit. This approach circumvents limitations by effectively translating all-to-all connected lattice problems into forms suitable for devices like the commercially available D-Wave Advantage system.
The scheme’s core innovation lies in dividing the larger problem into smaller, manageable unit cells, each optimized independently before being reassembled to approximate the ground state of the entire system. This methodology was demonstrated across three distinct models, each chosen to demonstrate the capabilities of the approach and relevant to ongoing research in quantum simulation and materials science.
Further expanding the scheme’s utility, the researchers also investigated models incorporating few-nearest-neighbor interactions alongside long-range connections, mirroring the complexities found in frustrated Ising compounds such as Er₂Be₂GeO₇. The team found that the results obtained with the UCBOS and quantum annealing are consistent with those obtained through classical optimization. This is particularly significant given the current state of quantum computing, where commercially available devices are still limited in scale and coherence.
Increased commercial availability of quantum annealing time will also facilitate the study of larger unit cells, further enhancing the accuracy and scope of the method. Future applications could extend to modeling more complex systems, such as long-range density-density interacting Fermi- or Bose-Hubbard models, relevant to quantum dot arrays, Moiré materials, or ultracold gases in optical traps. This work demonstrates a pathway for using existing quantum technology to address fundamental challenges in condensed matter physics and quantum simulation.
D-Wave Advantage System Implementation of Quantum Annealing
The D-Wave Advantage system served as a hardware accelerator to determine the ground state of Ising models, replacing classical binary optimization methods previously used with a unit-cell-based optimization scheme. Researchers used the quantum device to tackle optimization problems inherent in modeling magnetic materials and quantum simulators, moving beyond purely classical approaches for these calculations.
While classical stochastic searches proved sufficient for smaller unit cells, the D-Wave system offered a pathway to explore larger, more intricate configurations. These staircases, representing distinct ordered states of matter, were computed using unit cells up to 36 spins, demonstrating the quantum annealer’s ability to reproduce classical findings within the tested parameters.
Examining the Kagomé lattice, a two-dimensional structure with a unique arrangement of interconnected triangles, researchers determined magnetic configurations using the D-Wave system, visualizing spin alignments with blue and red circles representing spin-up and spin-down states. These configurations, calculated without an external field, provide insight into the behavior of artificial spin ice metamaterials, engineered materials designed to mimic magnetic properties. The visualization of these states highlights the potential for quantum annealing to aid in the design and understanding of novel magnetic materials.
The connectivity of the current D-Wave Advantage system presents limitations; embeddings with chain lengths exceeding seven are discouraged, influencing the maximum viable unit cell size. Consequently, the largest unit cell successfully embedded contained 64 sites.
Determining the optimal unit cell size where quantum annealing reliably finds the ground state remains a key consideration for future applications of this approach. Beyond its role as an optimizer, the code developed also functions as a classical repetition error-correcting code, adding another layer of functionality to the system. Both the D-Wave and other quantum annealing architectures could serve as optimizers for the unit-cell-based optimization scheme, each with its own strengths and weaknesses.
The team utilized the Ocean software development kit provided by D-Wave to formulate the optimization problems and communicate with the Advantage system, employing the DWaveCliqueSampler class for all-to-all connected problems. Each optimization problem was addressed with an unspecified number of annealing runs, each lasting 200 microseconds, utilizing the standard annealing protocol. This parameter selection reflects a balance between computational cost and the desire to achieve reliable results.
The work received funding from the Jülich Supercomputing Centre, providing access to the D-Wave Advantage System JUPSI through the Jülich UNified Infrastructure for Quantum computing (JUNIQ). The researchers stated in their published work, detailing a specific aspect of their optimization process.
Ground State Determination via Resummed Couplings
The approach relies on effectively “resummed” couplings, a mathematical technique represented by the equation 1/2∑(i ≠ j)J/(|r_i-r_j|^α)σ_i^zσ_j^z = = 0)^KJ^α(i,j)σ_i^zσ_j^z, which defines how interactions are calculated across the unit cell. This method facilitates the analysis of systems relevant to multiple quantum simulation platforms and material science, as demonstrated by investigations into three specific problems. Researchers chose to calculate the devil’s staircase of magnetization plateaux for the long-range Ising model in a longitudinal field as one of these exemplary problems, motivated by the development of atomic and molecular quantum simulators.
The annealing protocol itself is carefully tuned to maintain the system in its ground state throughout the computational process, ensuring the final state represents a global minimum of the encoded optimization problem. Observations on the Kagomé lattice revealed a striking pattern in the ground state of the dipolar long-range Ising model. The Kagomé lattice itself was defined using a triangular lattice with three sites per elementary unit cell, utilizing primitive translation vectors t_1^( KGM) = (2,0)^T and t_2^( KGM) = (1, sqrt(3))^T.
This specific lattice structure, with its corner-sharing triangles, presents a unique challenge for determining ground states due to its inherent magnetic frustration. The researchers also investigated the impact of van-der-Waals interactions (α = 6), finding that both truncated and untruncated studies yielded consistent predictions regarding the ground state. This consistency reinforces the robustness of the unit-cell-based optimization scheme in handling different interaction types.
The team applied their method to an anisotropic Shastry-Sutherland lattice, modeling the Er_2Be_2GeO_7 compound as an example of a system where the method can be applied, and incorporating realistic Ising coupling values for J_1, J_1^′, J_2, and J_2^′ alongside interatomic distances d_(J_1), d_(J_1^′), d_(J_2), and d_(J_2^′). As the researchers note, underscoring the mathematical foundation of their approach.
Application to Magnetization Plateaus and Spin Ice Models
This calculation, performed using commercial quantum annealing hardware, extends beyond theoretical modeling to address systems increasingly realized in analogue quantum simulation platforms based on atomic or molecular quantum optics and condensed matter systems. Parameters within the Ising model, including the coupling amplitude J, longitudinal field h, and decay of long-range interaction α, were all incorporated into the calculations, with the requirement that α exceeds the system’s spatial dimension for the application of the UCBOS in this work.
This work demonstrates a robust method for tackling lattice problems with algebraically decaying, resummable long-range interactions, and is particularly significant given the increasing relevance of these lattice models in diverse areas of quantum simulation and materials science, bridging the gap between theoretical predictions and experimental realization.
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