Probing many-body localization crossover in quasiperiodic Floquet circuits on a quantum processor

Using up to 144 qubits on an IBM Quantum processor, this study experimentally probes the many-body localization crossover in quasiperiodic Floquet Ising systems by implementing deep circuits up to 5000 cycles, revealing smooth ergodic-to-localized transitions and logarithmic entanglement growth in both one- and two-dimensional regimes.

The Gist

The Big Idea: When Chaos Stops

Imagine you have a cup of hot coffee and a cold spoon. If you leave them alone, the heat spreads until everything is the same temperature. In physics, this is called thermalization. It's the universe's way of saying, "Let's mix everything up until it's all the same."

But what if you had a magical cup of coffee where the heat refused to spread? The spoon stays cold, the coffee stays hot, and the memory of where the heat started never fades. This strange phenomenon is called Many-Body Localization (MBL).

In this paper, scientists used a super-advanced quantum computer to prove that this "frozen" state of matter actually exists, even in complex, messy systems, and they did it by watching how long it takes for the system to "forget" its past.

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The Analogy: The "Party" vs. The "Library"

To understand what the scientists did, let's use two analogies:

1. The Ergodic Party (Thermalization): Imagine a huge dance party. Everyone is moving, bumping into each other, and mixing. If you drop a red balloon in the corner, within minutes, the red color is everywhere because people are carrying it around. The system has "thermalized." It has forgotten where the balloon started.
2. The Library (Localization): Now, imagine a library where everyone is glued to their specific chair. They can talk to the person next to them, but they can't move. If you drop a red balloon in the corner, it stays in the corner. The system remembers exactly where it started. It is "localized."

The Problem: For a long time, scientists could only simulate small "libraries" on regular computers. As soon as the library got too big (too many people), the math became impossible to solve. They needed a new way to test this.

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The Experiment: The Quantum "Time Machine"

The researchers used an IBM Quantum Processor (specifically the "Heron" chip) with up to 144 qubits (quantum bits). Think of these qubits as the "people" in our library or party.

They set up a specific game called a Floquet Circuit.

• The Game: Imagine a drumbeat that repeats over and over. Every time the beat hits, the "people" (qubits) perform a specific dance move.
• The Twist: They added a "Quasiperiodic Potential." Think of this as a bumpy floor. In some spots, the floor is smooth (easy to dance); in others, it's very bumpy (hard to move).
  • Smooth Floor (Weak Disorder): People dance freely, mix, and the system becomes a "Party" (Thermalization).
  • Bumpy Floor (Strong Disorder): People get stuck in their spots. They can't move far. The system becomes a "Library" (Localization).

The Challenge: Usually, quantum computers are very noisy. It's like trying to watch a movie in a room with a flickering light and loud construction noise. The "movie" (the quantum state) gets corrupted quickly, and the experiment fails after just a few seconds (or cycles).

The Solution: The team used special "Fractional Gates."

• Analogy: Imagine you want to turn a dial 45 degrees.
  • Old Way: You have to click a button that turns it 90 degrees, then another that turns it back 45. This takes two steps and introduces more chances for error.
  • New Way (Fractional Gates): The dial has a new setting that lets you turn it exactly 45 degrees in one smooth motion.
• Result: This made the circuit much shorter and cleaner. It allowed them to run the "movie" for 5,000 cycles (a very long time in quantum terms) without the signal getting lost in the noise.

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What They Found

They ran the experiment on two types of layouts:

1. A 1D Line: Like a single row of chairs.
2. A 2D Grid: Like a heavy-hexagonal honeycomb pattern (a more complex web).

The Results:

1. The Crossover: When the "floor" was smooth, the system mixed quickly (Party). When the "floor" was bumpy, the system stayed stuck (Library).
2. Long-Term Memory: Even after 5,000 cycles, the "Library" systems still remembered where they started. The "Party" systems had completely forgotten.
3. The 2D Surprise: They found that this "stuck" behavior happens even in the complex 2D honeycomb grid. This is huge because classical computers can't simulate this size of 2D system to prove it.
4. Slow Spreading: They measured something called "Quantum Fisher Information" (a fancy way of measuring how much the qubits are entangled or "holding hands"). In the "Library" state, this entanglement grew very slowly—like a snail crawling—over thousands of cycles. This is the smoking gun for Many-Body Localization.

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Why This Matters

1. Beating Classical Computers: This is a classic example of a quantum computer doing something a regular supercomputer cannot do. They simulated a system so complex that it would take a classical computer longer than the age of the universe to calculate.
2. New Physics: It proves that even in messy, 2D systems, matter can get "stuck" and refuse to thermalize. This helps us understand how quantum information can be stored without being destroyed by heat.
3. Better Hardware: By using these new "Fractional Gates," they showed that quantum computers are becoming stable enough to run deep, long experiments. This is a major step toward building useful quantum computers for real-world problems.

In a Nutshell

The scientists built a digital "frozen lake" using a quantum computer. They showed that if you make the ice bumpy enough, the skaters (quantum particles) get stuck and never mix. They watched this happen for a very long time, proving that this "frozen" state is real and stable, even in complex 2D shapes. This is a big win for understanding how quantum matter behaves when it refuses to settle down.

Technical Summary

1. Problem Statement

Many-body localization (MBL) is a phenomenon where interacting quantum systems fail to thermalize, retaining memory of initial conditions and exhibiting slow entanglement growth. While MBL is well-understood in one-dimensional (1D) systems via classical simulations (e.g., tensor networks), several critical challenges remain:

• Dimensionality: It is unclear if MBL persists in higher dimensions (2D) due to the exponential growth of entanglement, which makes classical tensor-network simulations (like PEPS) computationally intractable.
• Time Scales: Probing the crossover between ergodic (thermalizing) and MBL regimes requires simulating long-time dynamics (thousands of cycles). Classical methods often fail to converge at these timescales, and previous quantum experiments were limited by circuit depth and noise.
• Noise Sensitivity: Current noisy intermediate-scale quantum (NISQ) devices suffer from error accumulation, making it difficult to distinguish genuine physical localization from noise-induced stagnation.

2. Methodology

The authors utilized the IBM Quantum Heron processor (ibm_kobe) to simulate quasiperiodic (QP) Floquet Ising systems.

• Models Studied:
  • 1D Model: A kicked Ising chain with 129 qubits. The Floquet unitary operator includes global $X$-rotations, nearest-neighbor $ZZ$ couplings, and a spatially modulated longitudinal field $h_z$ (incommensurate cosine potential).
  • 2D Model: A kicked Ising system on a heavy-hexagonal lattice with 144 qubits. The lattice features three coupling layers (Red, Green, Blue) corresponding to different lattice directions, with QP potentials applied along these paths.
• Key Hardware Innovation: Fractional Gates
  • Instead of decomposing interactions into standard Clifford gates (CZ, SX, etc.), the team utilized hardware-native fractional gates ($R_{ZZ}(\theta)$ and $R_X(\theta)$) available on the Heron processor.
  • These gates allow continuous-angle rotations, significantly reducing circuit depth compared to transpiled circuits. This reduction is crucial for suppressing error accumulation over long evolution times.
• Experimental Protocol:
  • Initial State: All qubits initialized in the ground state ($|0\rangle^{\otimes N}$), minimizing excited-state relaxation errors.
  • Evolution: Deep Floquet circuits executed up to 5,000 cycles.
  • Measurements:
    • Autocorrelation Function ($A(t)$): Measures the persistence of initial magnetization to detect thermalization failure.
    • Quantum Fisher Information (QFI): Used as a proxy for entanglement entropy to track the spreading of quantum information.
  • Validation: Results were cross-validated against time-dependent Density Matrix Renormalization Group (tDMRG) simulations (for 1D) and exact state-vector simulations (for smaller systems).

3. Key Contributions

• Deep Circuit Execution: Successfully executed Floquet circuits up to 5,000 cycles on a 144-qubit system, a regime previously inaccessible to both classical simulations and prior quantum experiments.
• Fractional Gate Advantage: Demonstrated that native fractional gates are essential for observing MBL signatures. Transpiled circuits (using standard Clifford gates) showed rapid decay of correlations even in the strong-disorder regime due to accumulated gate errors.
• 2D Localization Evidence: Provided the first experimental evidence of localization signatures in a two-dimensional quasiperiodic system on a quantum processor, a regime where classical tensor networks fail.
• Long-Time Dynamics: Observed logarithmic growth of entanglement (via QFI) over thousands of cycles, a hallmark of the MBL regime.

4. Key Results

• Ergodic-MBL Crossover:
  • Weak Disorder ($W \lesssim 3.0$): The system rapidly thermalizes; autocorrelation decays to zero, and QFI approaches the Haar-random (thermal) value.
  • Strong Disorder ($W \gtrsim 3.5$ in 1D, $W \gtrsim 4.0$ in 2D): The system exhibits persistent autocorrelations and slow entanglement growth.
• Autocorrelation Behavior:
  • In the MBL regime, $A(t)$ remains finite even at $t=5000$.
  • For $W < 4.0$, the long-time autocorrelation follows a power-law dependence $A(t \gg 1) \sim W^\alpha$ (exponent $\approx 5.4$ in 1D, $\approx 3.9$ in 2D).
  • In the limit $W \to \infty$, $A(t) \to 1$ (bond-decoupling limit).
• Entanglement Spreading (QFI):
  • In the MBL regime, the QFI exhibits logarithmic growth ($F_Q \sim \ln t$), consistent with theoretical predictions for MBL.
  • In the ergodic regime, QFI grows rapidly and saturates at the thermal value ($F_Q \approx 4/N$).
• Dimensionality Effects:
  • The threshold for the MBL regime ($W^*$) shifts to higher values in 2D ($W^* \sim 4$) compared to 1D ($W^* \sim 3.5$), suggesting that higher connectivity makes the system more prone to thermalization, requiring stronger disorder to localize.
• Noise Analysis:
  • The observed dynamics could not be fully explained by simple depolarizing noise models. The authors suggest that biased coherent errors in fractional gate modes play a significant role, highlighting the need for advanced error mitigation.

5. Significance

• Beyond Classical Simulation: This work demonstrates that programmable quantum processors can access physical regimes (large system sizes, long times, 2D geometries) that are strictly beyond the reach of classical supercomputers.
• Validation of MBL in 2D: It provides strong experimental support for the existence of localization-like behavior in 2D driven systems, addressing a long-standing theoretical debate regarding the stability of MBL in higher dimensions.
• Hardware Efficiency: The study establishes fractional gates as a critical tool for NISQ-era quantum simulation, enabling deep circuits without the prohibitive overhead of gate decomposition.
• Future Directions: The results pave the way for investigating complex non-equilibrium phenomena, such as quantum avalanches and the stability of time crystals, using quantum hardware as a primary experimental platform.