Stability of thermal equilibrium in long-range quantum systems

This paper demonstrates that the stability of local observables in long-range quantum systems at thermal equilibrium is guaranteed by correlation decay and Lieb-Robinson bounds at high temperatures, with numerical evidence suggesting this robustness extends to broader interaction regimes, thereby supporting the reliability of analog quantum simulators.

The Gist

Imagine you are trying to bake a perfect cake (the thermal equilibrium) in a very large, complex kitchen (a quantum system). The recipe involves ingredients that interact with each other. In some kitchens, ingredients only talk to their immediate neighbors (like a whisper passed down a line). But in the kitchens this paper studies, ingredients can shout across the room to talk to people on the other side (these are long-range interactions).

The big worry for scientists is: What happens if you make a tiny mistake? Maybe you accidentally add a pinch too much salt, or the oven temperature fluctuates slightly. In a normal kitchen, a small mistake might ruin one cookie, but the rest of the cake stays fine. However, in a kitchen where everyone is shouting to everyone else, a small mistake in one corner could theoretically ripple through the whole room and ruin the entire cake.

This paper asks: Is the cake actually safe? If we have these "shouting" long-range interactions, does a small error in the recipe destroy the final result, or does the system stay stable?

The Main Discovery: The "Whisper Network" Saves the Day

The authors prove that yes, the cake is safe. Even with long-range shouting, the system remains stable. A small error in the Hamiltonian (the recipe) only causes a small, local change in what you measure.

They found that this stability relies on two key "superpowers" of the system:

1. The "Distance Decay" Rule (Correlation Decay): Even though ingredients can shout across the room, the importance of their conversation drops off quickly as they get farther apart. If two ingredients are very far away, they barely care about each other. The paper proves that if this "shouting gets quieter with distance" rule holds, the system is robust.
2. The "Speed Limit" (Lieb-Robinson Bound): In quantum physics, information can't travel infinitely fast. Even with long-range interactions, there is a speed limit on how fast a disturbance can spread. Think of it like a rumor: even if everyone has a phone, it still takes time for the rumor to reach the person on the other side of the country. This speed limit prevents a small error from instantly destroying the whole system.

The Analogy of the "Local Perturbation"

Imagine you are in a crowded stadium (the quantum system).

• The Error: Someone in the back row drops a soda can.
• The Fear: Will this cause a panic that ripples through the entire stadium, making everyone jump?
• The Reality (According to this paper): Because of the "Distance Decay" and "Speed Limit," the people in the front row (the local observables) barely notice the soda can. They might hear a faint noise, but their view of the game (the measurement) remains almost exactly the same.

The authors proved mathematically that local observations are "indistinguishable" from the global chaos. You can measure a small part of the system, and it will look just as if the error never happened, provided the system isn't too cold (high temperature) or the interactions aren't too strong.

The "Digital vs. Analog" Connection

The paper also touches on why this matters for Quantum Simulation.

• Digital Quantum Computers are like trying to build a castle with individual Lego bricks. You have to be perfect; if one brick is wrong, the whole thing might fall. This is hard and error-prone.
• Analog Quantum Simulators (the focus of this paper) are like molding clay. You shape the whole thing at once. If you make a tiny mistake in the shape, the whole clay blob just shifts slightly, but it doesn't collapse.

The authors show that these "clay" simulators are incredibly robust. Even if the hardware isn't perfect (which it never is), the physics of the system naturally filters out the noise. This means we can trust these machines to give us accurate answers about complex materials, even without the expensive "error correction" needed for digital computers.

The Numerical "Stress Test"

The authors didn't just do the math; they also ran computer simulations (like a video game test) to see what happens when they push the rules to the limit.

• They tested systems where the "shouting" was extremely loud (very strong long-range interactions).
• The Result: The system was even more stable than their math predicted! The errors didn't spread as wildly as feared. This suggests that real-world quantum simulators might be even more reliable than the strict mathematical proofs guarantee.

Summary in One Sentence

This paper proves that even in quantum systems where everything talks to everything else, small mistakes don't ruin the whole picture because the influence of those mistakes fades away with distance and time, making these powerful new quantum simulators much more reliable than we thought.

Technical Summary

1. Problem Statement

Experimental realizations of quantum simulators (e.g., Rydberg atom arrays, trapped ions) are prone to "coherent errors," modeled as perturbations to the system Hamiltonian ($H \to H + \epsilon V$). While the stability of dynamics and ground states in short-range systems is well-understood, the stability of thermal equilibrium (Gibbs states) in long-range interacting systems remains an open challenge.

In long-range systems, interactions decay as a power law $d^{-\alpha}$. The rapid propagation of perturbations in these systems makes standard stability proofs for finite-range interactions inapplicable. The authors aim to:

1. Analytically prove the stability of local expectation values against global perturbations in long-range systems.
2. Establish the relationship between this stability, the decay of correlations, and the Lieb-Robinson bound.
3. Investigate the regime where analytical bounds fail (strong long-range interactions, $\alpha \leq D$) using numerical methods.

2. Methodology

A. Analytical Framework

The authors develop a perturbative analysis of matrix exponentials for the Gibbs state $\rho_\beta = e^{-\beta H}/\text{tr}(e^{-\beta H})$. The proof relies on two fundamental pillars:

1. Decay of Correlations: The assumption that the covariance between local observables $O_A$ and $O_B$ decays polynomially with distance:
   $$\text{Cov}_{\rho_\beta}(O_A, O_B) \leq K \|O_A\| \|O_B\| \frac{|A||B|e^{(|A|+|B|)/k}}{(1+d(A,B))^\alpha}$$
2. Lieb-Robinson (LR) Bounds: Bounds on the speed of information propagation. The authors utilize different forms of LR bounds depending on the interaction exponent $\alpha$:
   • Weak LR Bound: Valid for $\alpha > D$ (requires high temperature $\beta < \beta^*$).
   • Strong LR Bounds: Valid for $\alpha > 2D$ (allows arbitrary temperatures).

Key Technical Steps:

• Local Perturbations Perturb Locally (LPPL): The authors first prove that a local perturbation $V_B$ affects local observables $O_A$ only if $A$ and $B$ are close. The error decays with the distance $d(A,B)$.
• Global to Local Reduction: They demonstrate that a global error (sum of local terms) can be decomposed into a telescopic sum of local perturbations. By applying the LPPL property iteratively and utilizing the convolution properties of power-law decays, they bound the total error.
• Matrix Exponential Analysis: They combine a careful analysis of the derivative of the matrix exponential (using Quantum Belief Propagation) with the LR bound to handle the non-local nature of the perturbations.

B. Numerical Approach

To test regimes beyond the analytical assumptions (specifically $\alpha \leq D$), the authors perform numerical simulations using Tensor Networks (specifically Matrix Product Operators - MPOs).

• Model: A long-range transverse-field Ising model (LR-TFI) in 1D ($D=1$).
• Perturbation: A global magnetic field error ($V = \sum \sigma_z^i$).
• Algorithm: A TEBD-like algorithm with swap gates to simulate power-law interactions, evolving the system in imaginary time to reach the Gibbs state.

3. Key Contributions

1. Proof of Stability for Long-Range Systems: The authors prove that local expectation values are stable against global perturbations in arbitrary dimensions, provided the system satisfies the decay of correlations and a Lieb-Robinson bound.

   • Result: The difference in expectation values scales as $O(\epsilon)$ and is independent of the system size $N$, depending only on local quantities and the perturbation strength $\epsilon$.
   • Regimes:
     • For $D < \alpha \leq 2D$: Stability holds at high temperatures ($\beta < \beta^*$).
     • For $\alpha > 2D$: Stability holds at arbitrary temperatures.

2. Equivalence of Clustering Notions: The paper establishes a "circle of implications" between three concepts in long-range systems:

   • Decay of Correlations.
   • Local Perturbations Perturb Locally (LPPL).
   • Local Indistinguishability (the ability to approximate global Gibbs states using local Hamiltonians).
   • Note: Unlike short-range systems where these are exponentially equivalent, in long-range systems, the polynomial overheads in the proofs result in a degradation of the decay exponent (e.g., $\alpha \to \alpha - 2D$) when cycling through these properties.

3. Numerical Evidence for Strong Long-Range Regimes: The simulations reveal that stability persists even in the strong long-range regime ($\alpha \leq D$), where analytical proofs currently fail. The error scales linearly with $\epsilon$ and stabilizes as system size increases, suggesting the analytical bounds are conservative.

4. Results

• Theorem III.1: Formalizes the stability bound. For a local observable $O_A$:
  $$|\text{tr}[O_A \rho_\beta(H)] - \text{tr}[O_A \rho_\beta(H+\epsilon V)]| \leq C(\beta, \alpha) \cdot \epsilon \cdot \|O_A\| \cdot e^{|A|/k}$$
  The constant $C$ depends polynomially on $\beta$ and super-exponentially on $\alpha$, but crucially, the bound does not grow with the system size $N$.
• Numerical Findings:
  • The difference in expectation values between perturbed and unperturbed states remains bounded as $N \to \infty$.
  • The error grows linearly with the perturbation magnitude $\epsilon$, confirming the optimality of the bound with respect to $\epsilon$.
  • Stability holds for $\alpha \leq 1$ (strong long-range), a regime not covered by the analytical proofs.
  • Correlations decay faster than the worst-case analytical bounds predict, likely due to the system being in a disordered phase at the simulated temperatures.

5. Significance

• Robustness of Analog Simulation: The results provide rigorous theoretical guarantees that analog quantum simulators for long-range models are robust against coherent errors. This supports the feasibility of using these platforms to study thermal equilibrium and many-body physics without requiring full fault tolerance.
• Computational Advantage: The findings suggest that computing physical quantities (like local observables) in long-range thermal states is significantly easier than performing arbitrary quantum computations, as the system's local properties are insensitive to global noise.
• Theoretical Bridge: The work bridges the gap between the well-understood stability of short-range systems and the complex physics of long-range interactions, identifying the specific conditions (decay of correlations + LR bounds) required for stability.
• Future Directions: The numerical evidence suggests that stability may hold even more broadly than proven, motivating further research into the stability of long-range bosonic, fermionic, and open quantum systems.

In summary, the paper demonstrates that despite the rapid propagation of information in long-range systems, thermal equilibrium remains locally stable against global perturbations, provided correlations decay sufficiently fast. This validates the reliability of current and future analog quantum simulators for studying long-range interacting matter.