Ground state energy and phase transitions of Long-range XXZ using VQE

This paper proposes a Variational Quantum Eigensolver (VQE) approach with a constrained ansatz circuit to detect infinite-order phase transitions in the long-range XXZ chain by analyzing sensitivity in ground state energy errors across different phases, while also validating the method against exact diagonalization results.

The Gist

Imagine you are trying to find the lowest point in a vast, foggy landscape. In the world of physics, this "lowest point" is called the ground state energy, and it tells us how a system of particles (like atoms in a magnet) wants to settle down when it's completely calm.

Usually, finding this lowest point for complex systems is like trying to solve a puzzle that is too big for any human brain or even the world's most powerful supercomputers. This is where the authors of this paper bring in a new tool: a Variational Quantum Eigensolver (VQE). Think of VQE as a smart, hybrid robot that uses a quantum computer to make a "guess" at the lowest point and a classical computer to refine that guess until it's as close as possible to the truth.

The Challenge: Two Types of "Borders"

The researchers were studying a specific model called the Long-Range XXZ chain. Imagine a line of tiny magnets (spins) that can talk to each other. Usually, magnets only talk to their immediate neighbors, but in this model, they can shout across the whole line (long-range interaction).

The team wanted to find the "borders" where the behavior of these magnets changes completely. These are called phase transitions. They found two types of borders:

1. The "Cliff" (First-Order Transition): This is like walking off a steep cliff. The energy changes suddenly and sharply. It's easy to spot because the ground just drops.
2. The "Slope" (Infinite-Order Transition): This is much trickier. It's like walking up a very gentle, smooth hill. There is no sudden drop or cliff; the change happens so gradually that standard tools can't see the border at all. Usually, scientists need a special "global map" (a complex order parameter) to find this, which is hard to calculate.

The Secret Weapon: The "Bad Guess" Strategy

Here is the clever part of the paper. Usually, scientists use VQE just to get the exact number for the lowest energy. But the authors realized something interesting: The quality of the guess depends on where you are.

They designed their quantum robot (the "ansatz circuit") with a specific rule: It must keep the total spin (magnetization) constant.

• In the "Right" Neighborhood: If the robot is in a phase where the magnets naturally want to be in that specific constant-spin state, the robot makes a great guess. The error (the difference between the robot's guess and the true answer) is tiny.
• In the "Wrong" Neighborhood: If the robot is in a phase where the magnets don't want to be in that state, the robot struggles. It tries to force the magnets into the wrong shape, and the error gets huge.

The "Compass" Analogy

To find the invisible borders, the authors didn't just look at the size of the error. They looked at the direction of the error.

Imagine you are walking through a forest, and you drop a compass at every step.

• In one part of the forest (Phase A), the compass needles point in random directions, spinning wildly.
• In the other part (Phase B), the compass needles all point neatly in the same direction.

The authors used a technique called Directional Coherence to measure this. They calculated the "error" at thousands of points and looked at the direction of the change.

• When the compass needles were chaotic, they knew they were in one phase.
• When the needles suddenly aligned, they knew they had crossed a border.

This allowed them to spot both the easy "cliff" border and the hidden "slope" border just by watching how the robot's mistakes behaved. They didn't need a new, complex map; they just needed to watch how the robot stumbled.

The Results

• For the easy border (First-Order): They saw the error jump up suddenly, like a cliff.
• For the hard border (Infinite-Order): They saw the "compass needles" (the error gradients) go from chaotic to organized. This revealed the border that standard methods missed.
• Accuracy: By making the robot's "brain" slightly deeper (adding more layers to the circuit), they could also calculate the exact energy of the system with very high accuracy (within 3-7% error), even for difficult cases.

The Bottom Line

The paper claims that you don't always need a perfect calculation to find where a system changes. Sometimes, studying how your calculation fails (the error) can actually tell you more about the system's structure than the calculation itself. They successfully used this "error analysis" method to map out both simple and complex phase transitions in a long-range magnetic chain, proving that quantum algorithms can be used not just to solve problems, but to discover the hidden rules of how matter behaves.

Technical Summary

Technical Summary: Ground State Energy and Phase Transitions of the Long-Range XXZ Chain Using Variational Quantum Eigensolver

Problem Statement
The paper addresses the challenge of identifying phase transitions in quantum many-body systems, specifically the Long-Range XXZ (LRXXZ) chain. While Variational Quantum Eigensolvers (VQE) are widely established for finding ground state energies of Hamiltonians lacking analytical solutions, their utility in detecting phase boundaries—particularly infinite-order phase transitions—remains underexplored. Conventional methods for detecting phase transitions rely on singularities in the derivatives of ground state energy or two-point correlation functions. These techniques are effective for finite-order transitions but often fail for infinite-order transitions, which typically require globally ranged order parameters or complex methods like Quantum Monte Carlo. The authors aim to demonstrate that VQE, traditionally used solely for energy minimization, can be adapted to identify both first-order and infinite-order phase transition boundaries by analyzing the behavior of the energy error relative to the phase of the system.

Methodology
The study employs the VQE algorithm on a 1D LRXXZ chain defined by the Hamiltonian:
$$H = -J \sum_{i \neq j} \frac{S^x_i S^x_j + S^y_i S^y_j + \Delta S^z_i S^z_j}{|i - j|^\alpha}$$
where $J$ is the coupling constant, $\Delta$ is the anisotropy, and $\alpha$ defines the interaction range.

1. Ansatz Circuit Design: The authors designed a specific variational ansatz inspired by the Hamiltonian Variational Ansatz (HVA) but modified for efficiency.

   • Initialization: The system starts in a Néel state ($|0101\dots\rangle$), prepared by applying alternating Pauli-X gates to the $|000\dots\rangle$ state.
   • Constraint: The circuit is constructed to maintain a constant net spin (zero magnetization in the z-direction) throughout the optimization.
   • Structure: The Hamiltonian is decomposed into even and odd bond terms. Unlike standard HVA where commuting terms share parameters, the authors parameterize every single gate individually. This allows for a low circuit depth ($p=2$) while maintaining a total of $4(N-1)$ parameters.
   • Gate Modification: Initial implementations using standard $R_Z$ gates were found ineffective; these were replaced with Controlled-$R_Z$ (CRZ) gates to improve ground state convergence.

2. Phase Transition Detection Strategy:

   • The core hypothesis is that the accuracy of the VQE ground state energy depends on the phase in which the system resides relative to the initial ansatz state.
   • The authors calculate the energy difference (error), $E_d = E_{exact} - E_{VQE}$, between the exact diagonalization (ED) result and the VQE result.
   • Directional Coherence: To map phase boundaries, the authors analyze the gradient vector of this energy difference ($E_d$) across the parameter space ($\Delta, \alpha$). They compute the normalized gradient components and determine the "directional coherence," defined as the magnitude of the mean sine and cosine of the gradient angles over a moving window.
   • Logic: In phases where the ansatz is well-suited (e.g., the AFM phase given the Néel initialization), the error is low and the gradient vectors exhibit a systematic pattern. In phases where the ansatz fails (e.g., FM or disordered PM phases), the error is higher, and the gradient vectors become random. The boundary between these behaviors marks the phase transition.

Key Results
The study was performed on a system with $N=12$ sites and open boundary conditions, comparing results for $J=1$ (depth 1 and 2) and $J=-1$ (depth 2).

1. First-Order Transition (PM to FM):

   • For the transition from Paramagnetic (PM) to Ferromagnetic (FM) at $\Delta = 1$, the method successfully identified the boundary.
   • Because the initial Néel state has zero net magnetization, the VQE ansatz cannot access the FM ground state. Consequently, the error $E_d$ increases abruptly as the system enters the FM phase.
   • The directional coherence plot showed a sharp change in the orientation of gradient vectors at $\Delta = 1$, clearly delineating the transition, consistent with mean-field theory predictions.

2. Infinite-Order Transition (PM to AFM):

   • The transition from the PM phase to the Antiferromagnetic (AFM) phase is an infinite-order transition, which is notoriously difficult to detect using ground state energy derivatives alone.
   • The authors observed that while the error $E_d$ increased gradually during this transition, the directional coherence of the gradient vectors provided a clear distinction.
   • In the AFM region, gradient vectors were randomly aligned; in the PM region, they were coherent. A threshold of directional coherence (0.7) successfully separated these regions.
   • The identified critical point at $\Delta = -3$ was found to be $\alpha_c \approx 1.75$, which aligns closely with previous results obtained via geometric entanglement ($\alpha \approx 1.78$).
   • Crucially, the authors note that this transition was not identifiable in the standard ground state energy gradient or energy gap plots, validating the necessity of their error-based approach.

3. Ground State Energy Accuracy:

   • Using a depth-2 circuit, the authors achieved high accuracy in ground state energy calculations.
   • For $J < 0$, the maximum relative error was approximately 3%.
   • For $J > 0$, the maximum relative error was 7%. The authors attribute the higher error for $J > 0$ to the difficulty of the optimizer reaching the ground state without additional modifications (such as applying $R_y$ gates to expand the Hilbert space), though the method remained effective.

Significance and Claims
The paper claims to be the first to utilize VQE specifically to identify the boundary of an infinite-order phase transition. The primary contribution is a technique that leverages the failure of the ansatz in certain phases to map phase boundaries, rather than relying solely on the success of energy minimization.

• Novelty: The method demonstrates that ground state energy data, when analyzed through the lens of error gradients and directional coherence, can probe infinite-order transitions that are conventionally inaccessible to standard energy-based diagnostics.
• Efficiency: The proposed ansatz achieves accurate results with low circuit depth ($p=2$) and a manageable number of parameters, making it suitable for near-term quantum devices.
• Generalizability: The authors suggest that this approach is not limited to the LRXXZ model but offers a new paradigm for identifying phase transitions in other systems where ground state symmetries change across boundaries. The technique relies on designing an ansatz that is sensitive to the specific phase being evaluated.

The authors maintain a modest tone, acknowledging that noise effects were not considered and that the directional coherence metric is a tool for analyzing error behavior rather than a fundamental physical law. They emphasize that the success of the method relies on the specific design of the ansatz and the initial state selection.