Exploring Hilbert-Space Fragmentation on a Superconducting Processor

Using a 24-qubit superconducting processor, this study experimentally demonstrates Hilbert-space fragmentation in Stark systems by observing initial-state dependent non-equilibrium dynamics that persist and intensify with system size, providing strong evidence for a weak breakdown of ergodicity distinct from disordered systems.

The Gist

Imagine you have a giant, complex ballroom filled with dancers (the quantum particles). In a normal party, if you drop a single dancer into the crowd, they will eventually mix with everyone else, forget where they started, and the whole room will reach a state of "thermal equilibrium"—a chaotic, well-mixed dance where no one remembers their original spot. This is how most quantum systems behave, a concept known as thermalization.

However, this paper explores a very strange, "frozen" party where the dancers get stuck in specific patterns and refuse to mix. The researchers used a superconducting quantum computer (a high-tech dance floor made of 24 tiny quantum bits, or "qubits") to prove that this happens in a specific type of environment called a Stark system.

Here is the breakdown of their discovery using simple analogies:

1. The Setup: The Tilted Dance Floor

Usually, quantum particles move around freely. But in this experiment, the researchers created a "Stark potential," which is like tilting the entire dance floor.

• Imagine the floor is a giant ramp.
• Because of this tilt, it's very hard for the dancers to move "uphill." They get stuck in their local spots.
• This is similar to how water flows downhill but doesn't spontaneously flow up a hill.

2. The Mystery: The "Domain Wall" Secret

The researchers prepared the dancers in two different starting patterns. Crucially, both patterns had the same total energy and the same number of dancers. By all normal physics rules, they should behave exactly the same way.

But they didn't. The difference was in the arrangement of the dancers, specifically the number of "boundaries" or domain walls between groups of dancers.

• Pattern A (Few Boundaries): Imagine two large groups of dancers standing together, with only one or two lines separating them.
• Pattern B (Many Boundaries): Imagine the dancers are arranged in a checkerboard pattern, creating many small lines separating different groups.

The Surprise: Even though both patterns had the same energy, the "Few Boundaries" group got stuck and refused to mix. The "Many Boundaries" group, however, managed to dance around and mix with the crowd.

3. The Concept: Hilbert-Space Fragmentation

This is the core discovery, called Hilbert-Space Fragmentation.

• The Metaphor: Imagine the ballroom isn't one big open space. Instead, the tilted floor has created invisible, magical walls that split the room into thousands of tiny, disconnected rooms (fragments).
• The Trap: If you start in a "Few Boundaries" room, the magic walls prevent you from ever leaving that specific small room. You are trapped in a tiny corner of the ballroom, unable to reach the rest of the party.
• The Freedom: If you start in a "Many Boundaries" room, you happen to be in a much larger room that connects to the rest of the ballroom, allowing you to mix freely.

This means the future of the system depends entirely on how you started, not just how much energy you have. This breaks the usual rule of physics (called the Eigenstate Thermalization Hypothesis) which says that if two things have the same energy, they should end up the same.

4. The Experiment: The Quantum Ladder

To prove this, the team used a "ladder-type" processor (a quantum chip shaped like a ladder with 24 rungs).

• They acted like conductors, precisely tuning the frequency of each qubit to create that "tilted ramp" effect.
• They prepared the "Few Boundary" and "Many Boundary" starting states.
• They watched what happened over time.

The Result:

• The "Many Boundary" state quickly scrambled and mixed (thermalized).
• The "Few Boundary" state stayed frozen in its original pattern for a very long time. It was as if the dancers were holding hands in a rigid formation that the tilted floor couldn't break.

5. Why This Matters

This isn't just about a weird party trick. It helps us understand weak ergodicity breaking.

• Ergodicity is the idea that a system will eventually visit every possible state.
• Breaking Ergodicity means the system gets stuck.
• The "Weak" Break: In this case, the system doesn't get stuck because of random messiness (disorder), but because of the structure of the rules themselves. It's like a maze where the walls are built into the design, not just randomly placed.

The Takeaway

The researchers showed that in a quantum world with a "tilted" potential, the shape of your starting arrangement determines your destiny. If you start with a specific pattern (low domain walls), you are trapped in a tiny, isolated corner of the universe (Hilbert space) and can never explore the rest.

This is a major step forward in understanding how quantum computers might store information without it getting scrambled, and it reveals a hidden, complex landscape in quantum physics where the "rules of the road" depend entirely on where you start your journey.

Technical Summary

1. Problem Statement

The paper addresses the phenomenon of ergodicity breaking in isolated interacting quantum systems. While the Eigenstate Thermalization Hypothesis (ETH) predicts that generic closed systems thermalize, exceptions exist, such as Many-Body Localization (MBL) and Hilbert-Space Fragmentation (HSF).

• The Specific Challenge: HSF occurs when the Hilbert space fragments into exponentially many disconnected Krylov subspaces due to conservation laws (e.g., charge and dipole moment). This leads to dynamics that are strongly dependent on initial conditions.
• The Gap: While HSF has been studied theoretically in dipole-conserving systems, its experimental realization in Stark systems (systems under a linear potential) remains debated. Previous experiments in Stark systems (e.g., tilted Fermi-Hubbard models) showed initial-state dependence, but the initial states used often differed in total energy or quantum numbers (charge/dipole), making it difficult to isolate HSF from other effects like mobility edges or energy-dependent thermalization.
• Goal: To provide unambiguous experimental evidence of HSF in Stark systems by comparing the dynamics of initial states that share the exact same energy, charge, and dipole moment, but differ in their domain wall numbers.

2. Methodology

The authors utilized a ladder-type superconducting quantum processor to simulate the dynamics of interacting spinless fermions (mapped from a ladder-XX model).

• Hardware: A 30-qubit transmon processor arranged in a two-legged ladder structure. For the experiments, they utilized up to 24 qubits ($12 \times 2$).
  • Coupling: Nearest-neighbor coupling $\bar{J}_{NN}/2\pi \approx 7$ MHz; next-nearest-neighbor coupling $\bar{J}_{NNN} \lesssim \bar{J}_{NN}/6$.
  • Control: Precise control of qubit frequencies via Z-bias lines to engineer site-dependent potentials.
• Hamiltonian Engineering:
  • The effective Hamiltonian includes hopping terms and an on-site potential $W_{jm}$.
  • Stark Potential: $W_{jm} \propto -j\gamma$ (linear gradient).
  • Disordered Potential: $W_{jm}$ drawn from a uniform distribution $[-W, W]$ (for comparison).
• Initial State Preparation:
  • The team prepared product states with identical global quantum numbers (Total U(1) charge $Q$ and dipole moment $P$) and energy $E$.
  • Variable: The domain wall number ($n_{DW}$), which counts the number of interfaces between domains of excitations.
  • Key Comparison: They compared states with low $n_{DW}$ (e.g., $n_{DW}=2$) against states with high $n_{DW}$ (e.g., $n_{DW}=L-2$ or $n_{DW}=10$).
• Measurement Techniques:
  • Imbalance ($I$): A local observable measuring the memory of the initial state distribution.
  • Participation Entropy (PE): A global measure defined as $S_{PE} = -\sum p_i \log p_i$, where $p_i$ is the probability of finding the system in spin configuration $i$. PE quantifies the volume of the Hilbert space explored by the state.
  • Local PE: For larger systems ($L=12$), where full readout is challenging, they measured PE on subsystems of length $l$ and extrapolated to estimate the total PE.
  • Post-Selection: Data was post-selected within the $Q=L$ sector to correct for leakage due to qubit relaxation ($T_1$).

3. Key Contributions

1. First Direct Experimental Evidence of HSF in Stark Systems: The study isolates HSF as the mechanism for ergodicity breaking in linear potentials by rigorously controlling initial state quantum numbers.
2. Differentiation from Weak Disorder: The paper establishes a clear distinction between Stark systems (weak linear potential) and weakly disordered systems. It demonstrates that while both may show slow relaxation for specific initial states, the underlying mechanisms differ (HSF vs. finite-time effects in ETH).
3. Scalable Measurement Protocol: The authors propose and validate a method to estimate the upper bound of Participation Entropy for large systems ($L \ge 12$) by measuring local participation entropy on subsystems and extrapolating, overcoming the readout limitations of current quantum processors.
4. Numerical Validation: Extensive numerical simulations (using Exact Diagonalization for small systems and Matrix Product States/TDVP for larger systems up to $L=20$) corroborate the experimental findings and extend the analysis to longer timescales.

4. Key Results

• Initial-State Dependent Dynamics:
  • In Stark systems ($\gamma = 1, 2, 4$), states with low domain wall numbers ($n_{DW}=2$) exhibited slow relaxation and maintained a high imbalance over long times, whereas states with high domain wall numbers ($n_{DW} \approx L$) thermalized (imbalance $\to 0$) much faster.
  • This distinction persisted and became more pronounced as system size increased ($L=8 \to 12$), a hallmark of HSF.
• Participation Entropy (PE) Analysis:
  • Stark Systems: The PE for low-$n_{DW}$ states oscillated at a significantly lower value compared to high-$n_{DW}$ states, indicating the state is confined to a small, disconnected fragment of the Hilbert space.
  • Disordered Systems: In contrast, for weakly disordered systems ($W=3, 4$), the PE for low-$n_{DW}$ states eventually grew to match the high-$n_{DW}$ states (approaching the ETH prediction), indicating that the initial slow relaxation was a finite-time effect, not true fragmentation.
• Scaling Behavior:
  • Numerical simulations up to $L=20$ confirmed that in Stark systems, the dimension of the Krylov subspace for low-$n_{DW}$ states scales sub-extensively (vanishing in the thermodynamic limit), while high-$n_{DW}$ states explore a volume scaling with the total Hilbert space.
  • In disordered systems, both initial states eventually explored the full Hilbert space.
• Local PE Extrapolation: The linear growth of local PE with subsystem size $l$ allowed for a reliable estimation of the total PE, confirming the fragmentation signature even for $L=12$.

5. Significance

• Fundamental Physics: The work resolves the debate regarding the relationship between Stark localization and HSF. It confirms that Stark systems exhibit "weak" ergodicity breaking due to HSF, characterized by a small set of non-thermal eigenstates coexisting with a thermal spectrum, distinct from the "strong" ETH behavior found in weakly disordered systems.
• Mobility Edge Landscape: The results suggest a complex mobility edge landscape in Stark systems where the nature of thermalization depends not just on energy, but also on the specific structure of the initial state (domain wall number).
• Quantum Simulation: This study demonstrates the capability of superconducting processors to simulate complex non-equilibrium many-body phenomena that are intractable for classical computers, particularly in the regime of large system sizes and long timescales.
• Methodological Advance: The technique of using local observables to infer global Hilbert space fragmentation provides a scalable pathway for studying HSF in future quantum devices with even larger qubit counts.

In summary, the paper provides a rigorous experimental demonstration that Hilbert-Space Fragmentation is the mechanism driving non-ergodic dynamics in Stark systems, fundamentally distinguishing them from disordered systems and enriching the understanding of quantum thermalization.