Superdiffusion resilience in Heisenberg Chains with 2D interactions on a quantum processor

This study utilizes quantum hardware to demonstrate that while 2D interactions generally break superdiffusive spin transport in Heisenberg chains, $SU(2)$-preserving interactions exhibit the highest resilience, a finding validated by both theoretical scattering analysis and accurate quantum simulations.

The Gist

Imagine a long line of people holding hands, passing a secret message down the chain. In a perfectly organized line (what physicists call an "integrable" system), this message doesn't just walk slowly; it zooms along in a very specific, unusual way called superdiffusion. It's faster than a normal walk but slower than a sprint. This is a known phenomenon in certain one-dimensional magnetic materials.

However, real life is messy. Real materials aren't perfect lines; they have extra connections, like people in the line reaching out to grab hands with neighbors in a second, parallel line. These extra connections are 2D interactions. The big question this paper asks is: How much can we mess up the line with these extra connections before the "super-fast" message passing breaks down and turns into a normal, slow walk (diffusion) or a chaotic sprint (ballistic motion)?

Here is how the researchers tackled this, using a quantum computer as their laboratory:

1. The Setup: Building a "Heavy-Hex" Lattice

The researchers didn't just simulate a straight line. They built a digital model that looks like a ladder or a grid (specifically, a "heavy-hex" shape) that fits perfectly onto IBM's quantum computers.

• The Base: They started with the perfect, super-fast 1D line.
• The Twist: They slowly added "rungs" to the ladder (the 2D connections) to see what happens.
• The Test: They watched how a "spin" (a tiny magnetic arrow) at one end of the line moved and correlated with itself over time.

2. The Experiment: Different Types of "Handshakes"

The researchers realized that not all extra connections are the same. They tested different "flavors" of these 2D interactions:

• The "Symmetry-Preserving" Handshake: Some connections respect the rules of the original line (specifically, they keep the $SU(2)$ symmetry). Think of this as a handshake that follows the exact same etiquette as the people in the line.
• The "Symmetry-Breaking" Handshake: Other connections ignore the rules. They are like people grabbing hands in a way that confuses the original flow.

3. The Discovery: Resilience Varies

The results were fascinating. When they turned up the strength of these extra connections:

• The Breakdown: In almost all cases, the "super-fast" message passing eventually broke down. The message slowed down to a normal walk or sped up into a chaotic sprint.
• The Resilient One: However, the symmetry-preserving connection was a superhero. It could withstand much stronger "messiness" before the super-fast behavior broke down. It was the most resilient.
• The Weak Links: The connections that broke the rules (symmetry-breaking) caused the super-fast behavior to collapse much faster.

4. The "Why": Scattering Coefficients

To understand why one type was tougher than the other, the researchers looked at how the "message" (the spin) scattered when it hit these extra connections.

• The Weak Link: When the message hit a "symmetry-breaking" connection, it often got reflected back or couldn't cross over to the other side of the ladder effectively. It was like hitting a wall.
• The Resilient Link: The "symmetry-preserving" connection allowed the message to flow through and cross over to the other side of the ladder more easily. Because the message could keep moving and spreading out, the system stayed in its "super-fast" state for longer.

5. The Hardware Test: Real Quantum Computers

The researchers didn't just run this on a supercomputer; they ran it on actual IBM quantum processors (specifically the Heron chips).

• The Challenge: Quantum computers are currently "noisy." They make mistakes easily, especially when the calculation gets long and complex.
• The Result: Despite the noise, the real quantum hardware successfully reproduced the pattern they saw in the perfect simulations. It correctly identified that the symmetry-preserving connection was the most resilient. This proves that current quantum computers are already good enough to study these complex, non-equilibrium physics problems.

Summary

In simple terms, this paper shows that if you want to keep a special, fast-moving energy flow alive in a 2D magnetic material, you need to be very careful about how you connect the atoms. If you connect them in a way that respects the underlying rules of the system, the fast flow survives longer. If you connect them randomly, the flow breaks down quickly. The researchers proved this using a quantum computer, showing that these machines can act as powerful microscopes for understanding how real-world materials behave when they aren't perfect.

Technical Summary

Technical Summary: Superdiffusion Resilience in Heisenberg Chains with 2D Interactions on a Quantum Processor

Problem Statement
The integrable one-dimensional (1D) Heisenberg model is a cornerstone of non-equilibrium quantum many-body physics, known for exhibiting superdiffusive spin transport scaling as $t^{-2/3}$, a behavior associated with the Kardar-Parisi-Zhang (KPZ) universality class. While this phenomenon has been theoretically characterized and experimentally observed in materials like KCuF$_3$, real-world materials often contain imperfections and integrability-breaking interactions. A critical open question is how these perturbations, specifically the introduction of two-dimensional (2D) couplings, affect the stability of superdiffusion. Understanding the onset of "superdiffusion breakdown"—the transition from superdiffusive to diffusive or ballistic regimes—is essential for connecting idealized theoretical models to realistic materials. Current quantum hardware offers a platform to simulate these non-equilibrium phenomena, but the specific resilience of superdiffusion against various types of 2D interactions requires systematic investigation.

Methodology
The authors propose a digital quantum simulation of a 2D model that generalizes the 1D Floquet Heisenberg chain. The model is designed to be natively compatible with the heavy-hex lattice topology found in IBM Quantum processors.

• Model Construction: The system consists of a 1D chain of spins (black bonds in Fig. 1) governed by a Floquet Hamiltonian known to exhibit superdiffusion. This is extended into a 2D geometry by introducing "rung" interactions (vertical bonds) between parallel chains. The 2D interactions are parameterized by a vector $\vec{J}_\perp = (\lambda_x, \lambda_y, \lambda_z) \times J_\perp$, where $\lambda$ defines the interaction type (e.g., XX, YY, ZZ, or combinations) and $J_\perp$ is the interaction strength relative to the 1D coupling $J$.
• Time Evolution: The system evolves via a first-order Trotterization of the Hamiltonian, applying sequential unitary gates for red, green, and blue bond types. The time step $\tau$ is set to 1 to balance the need for simulating continuous-time physics with the circuit depth constraints of current hardware.
• Observables: The primary observable is the infinite-temperature spin-spin autocorrelation function, $C_{pp}^{ii}(t)$, measured at an edge probe qubit. The scaling exponent of this correlation function determines the transport regime: $-2/3$ indicates superdiffusion, $-1/2$ indicates diffusion, and $-1$ indicates ballistic transport.
• Simulation Platforms:
  1. Noiseless Simulations: Performed on a 28-qubit system (and a 45-qubit system for scattering analysis) to establish baseline behavior and calculate scattering coefficients (reflection and transmission) to explain resilience mechanisms.
  2. Hardware Experiments: Executed on IBM's Heron Quantum Processing Units (QPUs) using a 20-qubit system. The study employs random state preparation, dynamical decoupling for error mitigation, and averages over multiple runs to approximate infinite-temperature correlations.

Key Contributions and Results
The study comprehensively analyzes how different 2D interaction types influence the breakdown of superdiffusion:

1. Interaction-Type Dependent Breakdown: The nature of the 2D interaction dictates the direction of the breakdown.

   • Diffusive Breakdown: Interactions that preserve total spin conservation ($S_z$) and parity, such as $\vec{\lambda} = (0,0,1)$ (ZZ) and $\vec{\lambda} = (1,1,0)$ (XX+YY), drive the system toward diffusive scaling ($t^{-1/2}$).
   • Ballistic Breakdown: Interactions that break total spin conservation but preserve parity, such as $\vec{\lambda} = (1,0,0)$ (XX) and $\vec{\lambda} = (1,0,1)$, drive the system toward ballistic scaling ($t^{-1}$).
   • Highest Resilience: The interaction type preserving the full $SU(2)$ symmetry, $\vec{\lambda} = (1,1,1)$ (Heisenberg/XXX), exhibits the highest resilience. It maintains superdiffusive scaling even at relatively high 2D interaction strengths ($J_\perp \sim 4J$), requiring significantly stronger perturbations to deviate from the $t^{-2/3}$ scaling compared to other types.

2. Mechanism of Resilience: Through scattering coefficient analysis in 45-qubit simulations, the authors link resilience to cross-chain transmission.

   • The less resilient $\vec{\lambda} = (0,0,1)$ (ZZ) interaction suppresses transmission across the rung ($T_{cross} \approx 0$), enhancing reflection and accelerating the breakdown.
   • The more resilient $\vec{\lambda} = (1,1,0)$ (XX+YY) allows for cross-chain transmission, reducing reflection and sustaining the transport behavior longer.

3. Hardware Validation: The relative resilience of different interaction types was successfully captured on IBM Heron QPUs. Despite the noise inherent in deep circuits required for Trotterization and random state preparation, the hardware results showed remarkable agreement with noiseless simulations regarding the relative ordering of resilience. The experiments confirmed that $SU(2)$-preserving interactions are the most robust against superdiffusion breakdown.

Significance and Claims
The paper claims that this work establishes the utility of current quantum hardware in simulating complex non-equilibrium quantum many-body phenomena, specifically the breakdown of anomalous transport. By demonstrating that specific symmetries (like $SU(2)$) confer resilience against integrability-breaking terms, the study provides a pathway for identifying how superdiffusive spin transport might be sustained in realistic two-dimensional lattices and quasi-one-dimensional materials. The authors emphasize that their model is natively implementable on existing heavy-hex hardware, validating the capability of near-term devices to probe emergent hydrodynamic behaviors that are difficult to access via classical simulation or direct material experimentation. The work serves as a bridge between theoretical integrable models and the imperfect, interacting reality of physical materials.