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Exploring variational quantum eigensolver ansatzes for the long-range XY model

This paper investigates various VQE ansatzes for the long-range XY model, demonstrating that while full-entanglement CRX and TQR gates can accurately capture ground state energies, only the full-entanglement TQR ansatz achieves high-fidelity wavefunction representation, and that restricted-entanglement structures can also yield acceptable solutions.

Original authors: Jia-Bin You, Dax Enshan Koh, Jian Feng Kong, Wen-Jun Ding, Ching Eng Png, Lin Wu

Published 2026-09-01

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CC BY 4.0

Original authors: Jia-Bin You, Dax Enshan Koh, Jian Feng Kong, Wen-Jun Ding, Ching Eng Png, Lin Wu

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the microscopic world of quantum physics, particles do not behave like the solid objects we see around us. Instead, they exist in a haze of probabilities, and when they interact, they can become inextricably linked in a phenomenon known as entanglement. This link means that the state of one particle instantly influences another, no matter how far apart they are. Understanding these connections is essential for describing how materials behave, from superconductors that carry electricity without resistance to magnets that hold their shape. However, calculating the exact behavior of even a small group of these interacting particles is a task so complex that it overwhelms the most powerful classical supercomputers. To solve this, scientists are turning to a new kind of machine: the quantum computer. These devices use the very laws of quantum mechanics to simulate other quantum systems, offering a potential shortcut to understanding nature's most intricate puzzles.

The challenge, however, is that the quantum computers available today are still in their infancy. They are small, prone to errors, and cannot run the long, complex calculations required for perfect accuracy. To work around these limitations, researchers use a hybrid method called the variational quantum eigensolver. This approach acts like a guided search. A quantum computer prepares a specific state of particles based on a set of adjustable settings, while a classical computer acts as a coach, tweaking those settings to find the configuration with the lowest possible energy. This lowest energy state is crucial because it represents the system's ground state, its most stable and natural form. The success of this method depends entirely on the "ansatz," or the blueprint used to build the quantum state. If the blueprint is too simple, it misses the complex connections between particles; if it is too complex, the noisy hardware breaks down before the answer is found.

In a recent study, researchers set out to test different blueprints for simulating a specific theoretical model known as the long-range XY model. This model describes a chain of particles where each one interacts with every other particle in the chain, not just its immediate neighbors. The team tested several different designs for the quantum circuit, varying both the types of gates used to manipulate the particles and the pattern of connections between them. They compared circuits that used standard logic gates, which are fixed and rigid, against circuits that used more flexible, adjustable gates. They also tested two distinct connection patterns: one where every particle was linked to every other particle, and another where particles were only linked to their nearest neighbors.

The results revealed that the choice of blueprint matters immensely. The researchers found that circuits using simple, fixed logic gates failed to capture the true nature of the system, no matter how many times they were repeated or how the gates were arranged. These designs simply lacked the flexibility to describe the deep connections required by the model. In contrast, circuits built with adjustable rotation gates performed much better. Specifically, a design that allowed every particle to interact with every other particle using these flexible gates was able to accurately predict the system's energy. Even more impressively, one specific type of adjustable gate, known as a two-qubit rotation, was the only design capable of reproducing the actual state of the particles with near-perfect accuracy.

Perhaps the most surprising discovery was that total connectivity was not strictly necessary to get a good answer. The team found that they could achieve acceptable results by restricting the connections to only those particles that were a fixed distance apart, rather than linking everyone to everyone. This "restricted-entanglement" approach is significant because it reduces the complexity of the circuit. In the noisy environment of current quantum hardware, fewer connections mean fewer opportunities for errors to creep in, making the simulation faster and more reliable. The researchers also observed that by packing more adjustable settings into each layer of the circuit, they could achieve high accuracy with a shallower, simpler circuit structure. This trade-off is vital for today's machines, which struggle to maintain stability as circuits grow deeper.

To understand why some designs worked and others failed, the team looked at a measure called entanglement entropy, which quantifies how deeply the particles are linked. They found that the most successful designs were the only ones capable of generating the maximum possible amount of entanglement for the system. This confirmed that the ability to create strong, widespread connections between particles was the key to solving the problem. The study concludes that while the most powerful designs require complex, fully connected circuits, there is a practical middle ground. By using flexible gates and limiting connections to a specific range, scientists can find accurate solutions to difficult quantum problems without overwhelming the fragile hardware of current quantum computers. This work provides a clear roadmap for how to build better quantum simulations in the near future, balancing the need for accuracy with the reality of hardware limitations.

Technical Summary: Exploring Variational Quantum Eigensolver Ansatzes for the Long-Range XY Model

Problem Statement
Finding the ground state energy and wavefunction of quantum many-body systems is a fundamental challenge in quantum physics and chemistry. While fault-tolerant quantum computers promise efficient solutions, current hardware is limited to Noisy Intermediate-Scale Quantum (NISQ) devices. These devices lack full coherence and error correction, necessitating algorithms that can operate effectively with shallow circuits and limited qubit counts. The Variational Quantum Eigensolver (VQE) is a leading hybrid classical-quantum algorithm designed for this regime. This paper investigates the efficacy of various VQE ansatzes in solving the long-range XY model, a system relevant to Bardeen-Cooper-Schrieffer (BCS) superconductivity and condensed matter physics. The core challenge addressed is identifying which ansatz structures and gate sets can sufficiently capture the entanglement and correlations of the ground state without incurring prohibitive circuit depths or noise-induced errors.

Methodology
The authors apply the VQE algorithm to the long-range XY model with $N$ spins, defined by the Hamiltonian:
$H = -J \sum_{i<j} (X_i X_j + Y_i Y_j) - h \sum_{i} Z_i$
where $J$ is the coupling strength, $h$ is the Zeeman field, and $X, Y, Z$ are Pauli matrices.

The study systematically compares different ansatz architectures constructed from a mean-field layer (single-qubit rotations) and an entangler layer. The authors vary two primary dimensions:

Building Gates: Three types of two-qubit gates are tested:
- CNOT (Controlled-NOT)
- CRX (Parametric Controlled-Rotation)
- TQR (Parametric Two-Qubit Rotation, specifically $R_{xx}$ and $R_{yy}$ gates)
Entanglement Structures: Two connectivity patterns are evaluated for each gate type:
- Full Entanglement: Gates connect all possible pairs of qubits.
- Linear Entanglement: Gates connect only nearest-neighbor qubits.

Additionally, the authors explore "restricted-entanglement" ansatzes where gates are applied only between qubits at a fixed distance $r$ , and investigate the impact of gate ordering and multi-layer setups (increasing the number of entangler repetitions, $L$ ). The simulations are performed using Qiskit's QASM and Statevector simulators. Performance is evaluated based on the ground state energy per site ( $E_{GS}/N$ ) and the fidelity ( $F = |\langle \psi_{VQE} | \psi_{exact} \rangle|^2$ ) against the exact analytical solution. The expressive power of the ansatzes is characterized using entanglement entropy.

Key Contributions and Results

Limitations of Mean-Field and CNOT Ansatzes:
- The mean-field ansatz (product state) yields results consistent with analytical mean-field theory but deviates significantly from the exact solution, particularly near the phase transition ( $|h/J|=1$ ), with low fidelity ( $F < 0.5$ ).
- Adding CNOT entanglers (both full and linear) improves the energy slightly in the ferromagnetic phase but fails to capture the ground state accurately in the paramagnetic phase. The fidelity remains low ( $<0.9$ ), indicating insufficient expressive power.
Superiority of Parametric Gates (CRX and TQR):
- Full-Entanglement CRX: This ansatz significantly improves energy accuracy, approaching the exact solution, though fidelity remains imperfect near critical points.
- Full-Entanglement TQR: This ansatz demonstrates the highest performance. It accurately reproduces the ground state energy across both ferromagnetic and paramagnetic phases. Crucially, it achieves a fidelity close to one (except near critical points), successfully representing the ground state wavefunction.
Linear vs. Restricted Entanglement:
- Linear entanglement (nearest-neighbor only) is generally insufficient for the full CRX and TQR ansatzes to describe the ground state energy and wavefunction accurately, particularly in the ferromagnetic phase.
- However, the authors find that "restricted-entanglement" ansatzes, where entangling gates are applied only between qubits within a fixed distance $r$ (e.g., $r=2$ ), are sufficient to yield acceptable solutions. This approach reduces the overall running time of the optimization compared to full entanglement.
Gate Order and Multi-Layer Effects:
- The performance of CNOT and CRX ansatzes is sensitive to the specific ordering of gates in the circuit.
- In contrast, the full TQR ansatz is robust; the gate order has negligible effect on the results (except near phase boundaries).
- Increasing the number of layers ( $L$ ) improves stability and accuracy for CRX and TQR ansatzes, but the single-layer full TQR ansatz already performs well.
Expressive Power via Entanglement Entropy:
- The study uses von Neumann entanglement entropy to quantify the "expressive power" of the ansatzes.
- Full CRX and TQR ansatzes are the only ones capable of reaching the maximum possible entanglement entropy ( $N/2$ ) for an $N$ -qubit chain, which is necessary to represent the ground state of the long-range XY model.
- The full TQR ansatz converges to the maximally entangled state faster (in terms of optimization iterations) than the full CRX ansatz.

Significance and Claims
The paper claims that for the long-range XY model, the full-entanglement TQR ansatz is the most effective VQE architecture among those tested. It uniquely combines high fidelity in representing the wavefunction with the ability to reduce circuit depth by increasing the number of variational parameters per layer rather than stacking more layers. This is particularly significant for NISQ devices, where noise scales with circuit depth.

Furthermore, the authors demonstrate that restricted-entanglement ansatzes (using gates between qubits at a fixed distance) can provide acceptable solutions with reduced computational overhead, offering a practical trade-off between simulation time and accuracy. The work highlights that the choice of gate set (specifically using parametric two-qubit rotations like $R_{xx}$ and $R_{yy}$ ) and the entanglement structure are critical determinants of VQE success, with the TQR ansatz offering the highest expressive power for this specific many-body problem.

Technical Summary: Exploring Variational Quantum Eigensolver Ansatzes for the Long-Range XY Model

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