Benchmarking Quantum Simulations of the Lipkin-Meshkov-Glick Model Using Large Tensor Networks

Technical Summary: Benchmarking Quantum Simulations of the Lipkin-Meshkov-Glick Model Using Large Tensor Networks

Problem Statement
As quantum computing matures, there is a critical need to benchmark the real-world problem-solving performance of Noisy Intermediate-Scale Quantum (NISQ) devices against competitive classical methods. While hardware characterizations (e.g., gate fidelity, coherence time) are standard, they do not fully capture performance on applications of interest. The Lipkin-Meshkov-Glick (LMG) model serves as an ideal testbed for this comparison: it describes a many-body system with rich physical phenomena, such as quantum phase transitions, yet remains classically tractable enough to generate high-precision reference data at scales beyond the reach of exact diagonalization. Previous quantum simulations of the LMG model have been confined to modest system sizes due to hardware constraints. This work aims to establish a large-scale classical benchmark to evaluate the accuracy and scalability of popular NISQ algorithms, specifically the Variational Quantum Eigensolver (VQE) and Sample-Based Quantum Diagonalization (SQD).

Methodology
The study employs a hybrid approach utilizing high-performance classical computing and NISQ hardware:

1. Classical Benchmark (DMRG): The authors utilized the Density Matrix Renormalization Group (DMRG) algorithm, a tensor network method, to compute ground-state energies of the LMG Hamiltonian. The simulations were executed on the Perlmutter supercomputer at NERSC. The LMG Hamiltonian was defined with parameters $\epsilon=1$, $W=0$, and varying interaction strengths $V$. The system sizes ranged from $N=2$ to $N=1400$ particles, with profiling runs extending to $N=1550$. Additionally, a phase diagram was generated for $N \in [2, 100]$ and $V \in [0, 1]$ in 0.1 increments.
2. Quantum Algorithms:
   • VQE: The Variational Quantum Eigensolver was implemented on an IBM Eagle quantum processor. To address the exponential growth of the Hilbert space, a Hamiltonian compression scheme was employed, exploiting the model's symmetries (specifically $W=0$) to map the system into a collective spin basis. This reduced the effective system size, allowing for the use of smaller ansatze composed of $R_y$ and CNOT gates.
   • SQD: The Sample-Based Quantum Diagonalization algorithm was also run on the IBM processor. This hybrid approach used an ensemble of Dicke state circuits to sample bitstrings reflecting the system's symmetry (fixed Hamming weight parity). These samples were used to project the full Hamiltonian into a reduced subspace, which was then diagonalized classically. Error mitigation was applied via symmetry-based post-selection and bit-flip correction to ensure the sampled subspace faithfully represented the symmetry-restricted space.

Key Contributions
The paper makes two primary contributions to the field of quantum benchmarking:

1. Large-Scale DMRG Dataset: The authors generated one of the largest datasets of LMG ground-state energies in literature, providing accurate reference values for systems up to 1400 particles. This includes a detailed phase diagram across interaction strengths.
2. Comparative Benchmarking: The study provides a rigorous comparison of VQE and SQD against the DMRG baseline on an IBM quantum processor, establishing performance baselines and identifying the specific scaling limitations of each method in the NISQ era.

Results

• DMRG Performance: The DMRG results demonstrated high reliability, achieving over 99.99% accuracy compared to exact numerical solutions for small systems ($N \le 19$). The energy curves for large $N$ showed smooth, monotonic scaling, confirming the convergence of the tensor network representation. The computational bottleneck was identified as the $O(N^2)$ Hamiltonian construction due to all-to-all interactions, rather than tensor contraction.
• VQE Performance: VQE achieved results within 1% error for small systems ($N=6$). However, accuracy degraded rapidly as system size increased. Beyond $N \approx 15$, errors exceeded the 1% threshold, reaching approximately 17% at $N=23$. The errors grew steadily, reflecting the difficulty of classical optimization in the compressed Hilbert space.
• SQD Performance: SQD demonstrated superior accuracy to VQE within its accessible regime. For $N \le 18$, SQD achieved relative errors below 1% (and even below $10^{-12}\%$ for $N \in [3, 15]$), effectively matching the exact solution. However, performance was strictly limited by the shot budget (106 shots). Once the number of basis states in the symmetry sector ($2^{N-1}$) exceeded the shot limit (around $N \approx 20$), the error rose sharply, reaching ~95% at $N=50$.

Significance and Claims
The paper claims that these results highlight the distinct trade-offs between accuracy, circuit depth, and noise resilience in the NISQ era.

• Subspace-Based Approaches: The authors suggest that subspace-based approaches like SQD may strike the best balance between accuracy and resource constraints, provided the symmetry sector can be fully covered by the available shot budget.
• Benchmarking Necessity: The work asserts that for quantum computers to demonstrate a genuine advantage, they must surpass the capabilities of classical tensor network methods (like DMRG) on the same problem Hamiltonians. The generated datasets serve as a high-precision reference for this purpose.
• Limitations: The study modestly concludes that while SQD shows promise, its scalability is currently gated by shot budgets and circuit depth. Similarly, VQE struggles to scale beyond modest system sizes without significant improvements in ansatz design or initialization strategies. The authors do not claim quantum advantage for the current implementations but rather use the data to define the boundaries of current NISQ utility.