Probing the Ground State of the Antiferromagnetic Heisenberg Model on the Kagome Lattice using Geometrically Informed Variational Quantum Eigensolver

Technical Summary: Probing the Ground State of the Antiferromagnetic Heisenberg Model on the Kagome Lattice using Geometrically Informed Variational Quantum Eigensolver

Problem Statement
The paper addresses the challenge of finding the ground state of the antiferromagnetic Heisenberg model on a kagome lattice, a system known for geometric frustration and potential quantum spin liquid phases. Determining the ground state of such $k$-local Hamiltonians is a QMA-hard problem, rendering classical simulation intractable for large systems due to the exponential growth of the Hilbert space. While Variational Quantum Eigensolvers (VQE) offer a hybrid quantum-classical approach suitable for Noisy Intermediate-Scale Quantum (NISQ) devices, they face significant hurdles: the need for deep circuits, susceptibility to noise (coherent and incoherent errors), barren plateaus where gradients vanish, and the difficulty of designing ansätze that are both expressive and hardware-efficient. Specifically, the authors aim to characterize the ground state properties, such as spin-spin correlations and static structure factors, on real quantum hardware without relying on heavy problem-specific reductions or extensive error mitigation during the optimization loop.

Methodology
The authors propose a geometrically informed VQE approach tailored for the kagome lattice, focusing on two fundamental geometries: a single triangle and a kagome star (six corner-sharing triangles).

1. Geometrically Informed Ansatz:
   The core innovation is a custom, shallow, hardware-efficient ansatz. Unlike the Hamiltonian Variational Ansatz (HVA), which can be too deep for NISQ devices, this ansatz is constructed using a pattern of parametrized single-qubit rotations ($R_y$) and entangling gates restricted to native hardware connectivity (avoiding SWAP gates).

   • Euclidean Parameter Space: A critical feature of this design is that the Fubini-Study metric tensor is analytically proven to be diagonal and constant. This ensures the parameter space is naturally Euclidean, singularity-free, and free of over-parameterization. Consequently, the quantum natural gradient is equivalent to the standard gradient, simplifying the optimization landscape.
   • Hardware Compatibility: The circuit depth is minimized (one entangling layer), and the number of parameters is reduced by alternating the placement of $R_y$ gates.

2. Optimization Strategy (AQNGD/AGD):
   The authors employ an Adaptive Gradient Descent (AGD) optimizer. Because the ansatz yields a diagonal and constant Fubini-Study metric, the Adaptive Quantum Natural Gradient Descent (AQNGD) reduces to AGD. The learning rate is dynamically adapted using the Armijo backtracking rule. This allows for large initial steps to accelerate convergence while automatically reducing the step size as the optimizer approaches the minimum, avoiding stagnation in narrow geometries. This is compared against the Simultaneous Perturbation Stochastic Approximation (SPSA).

3. Error Mitigation (Post-Processing):
   Error mitigation is applied after the VQE optimization to refine the final energy estimates and structural properties, rather than during the iterative training.

   • Readout Error Mitigation (REM): A partitioned response matrix approach is used to reverse SPAM (State Preparation and Measurement) errors, scaling more efficiently than full matrix inversion for sparse measurements.
   • Zero-Noise Extrapolation (ZNE): The authors apply ZNE using quadratic extrapolation (and Bayesian polynomial regression) to extrapolate results to the zero-noise limit. They note that while ZNE does not strictly obey the Rayleigh-Ritz variational principle, it is used here to improve accuracy.

Key Results
The study was conducted on real quantum hardware (IBM Oslo, IBM Kyoto, IBM Torino, and IBM Algiers) and validated via state-vector simulations.

• Convergence and Optimization:

  • On a 3-qubit triangle system, the AGD-optimized VQE converged smoothly to the ground state energy ($E \approx -3$) with high fidelity (99.89% on simulation) using only 19 iterations. The final parameters closely matched theoretical exact values.
  • On a 12-qubit kagome star, AGD demonstrated superior convergence compared to SPSA, requiring fewer iterations to reach the ground state energy ($E \approx -18$). The final fidelity on simulation was 96.37%.
  • The step-size adaptation via the Armijo rule was shown to effectively navigate the cost landscape, starting with large steps and refining as convergence approached.

• Noise Resilience of Structural Properties:

  • A significant finding is that spin-spin correlations and the static structure factor ($S(q)$) are resilient to noise. Even without error mitigation, the experimental $S(q)$ maps retained the qualitative signature of the ground state (e.g., the suppression at zero momentum indicating no ferromagnetic order).
  • The Pearson correlation coefficient between the noisy experimental $S(q)$ and the exact theoretical $S(q)$ remained extremely high (>99.9%).
  • Simulation experiments revealed a "step-like" dependence of the static structure factor on noise levels: the correlation structure remains robust up to a critical noise threshold, after which it rapidly degrades, resembling a phase transition. This contrasts with energy, which decays monotonically with noise.

• Error Mitigation Efficacy:

  • Applying REM and ZNE post-optimization significantly improved energy estimates. For the kagome star, raw energy was $\approx -14.9$, while REM alone improved it to $\approx -17.3$, and ZNE (quadratic) brought it closer to the exact value of $-18$.
  • The authors observed that combining REM and ZNE sometimes led to overshooting the ground energy (violating the variational bound), a behavior consistent with ZNE's non-variational nature. ZNE alone often provided the best accuracy among the tested configurations.

Significance and Claims
The paper claims that a geometrically informed ansatz design, which ensures a naturally Euclidean parameter space, enables efficient training of VQE on current NISQ hardware without the need for complex, problem-specific reductions.

• Robustness: The work demonstrates that meaningful physical properties, specifically spin-spin correlations and the static structure factor, can be extracted from noisy quantum hardware with sufficient accuracy to characterize the ground state, even without error mitigation during the optimization loop.
• Optimization Efficiency: The use of AGD with adaptive step sizes proves more efficient than SPSA for this specific hardware-efficient ansatz, achieving faster convergence in terms of iterations.
• Scalability: By avoiding deep circuits and heavy error mitigation during training, the approach offers a scalable path for probing frustrated quantum magnets. The authors suggest this framework can be extended to larger kagome lattices and other frustrated quantum systems.
• Modesty: The authors acknowledge that while their method successfully characterizes small fundamental cells (triangle and star), the general ground state of the infinite kagome lattice remains an open question in the literature. They do not claim to have solved the infinite lattice problem but rather demonstrated a viable methodology for probing these systems on current devices. They also note that while ZNE improves energy estimates, it does not preserve the variational principle, and the combination of REM and ZNE requires careful interpretation.