⚛️ quantum physics

Unveiling clean two-dimensional discrete time crystals on a digital quantum computer

Using IBM's 133-qubit Heron processor, researchers experimentally demonstrated the existence of a robust, clean two-dimensional discrete time crystal and an incommensurately modulated variant in a driven kicked Ising model, validating these non-equilibrium phenomena through comparison with classical simulations and highlighting the potential of gate-based quantum computers for studying complex many-body dynamics.

Original authors: Kazuya Shinjo, Kazuhiro Seki, Tomonori Shirakawa, Rong-Yang Sun, Seiji Yunoki

Published 2026-03-13

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CC BY 4.0

Original authors: Kazuya Shinjo, Kazuhiro Seki, Tomonori Shirakawa, Rong-Yang Sun, Seiji Yunoki

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you have a giant, complex clockwork machine made of 133 tiny, spinning gears (these are the qubits on a quantum computer). Normally, if you keep shaking this machine up and down (applying a periodic drive), all the gears would eventually get so hot and chaotic that they stop moving in any organized pattern. They would reach "thermal equilibrium," which is basically a state of total, boring chaos where nothing interesting happens.

However, this paper discovers a magical trick: for a surprisingly long time, the gears don't just spin randomly. Instead, they lock into a rhythm where they only flip their direction every second time you shake the machine, even though you are shaking it every time. It's like a dancer who only turns around on the second beat of the music, ignoring the first. This stubborn, rhythmic defiance is called a Discrete Time Crystal (DTC).

Here is the breakdown of their discovery in everyday terms:

1. The Problem: The "Hot Mess" of Time

In the world of quantum physics, if you keep pushing a system, it usually absorbs energy until it melts into a hot, disordered soup. Scientists used to think the only way to stop this melting and keep the "time crystal" rhythm alive was to make the machine messy and full of "junk" (disorder) so the gears couldn't talk to each other. This is like trying to keep a dance party going by locking everyone in separate rooms so they can't get distracted.

2. The Breakthrough: A Clean, Organized Dance

The researchers, using IBM's powerful Heron quantum computer (which has 133 qubits arranged in a special "heavy-hexagon" shape, like a honeycomb), proved you don't need the "junk" or the "locked rooms."

They created a clean system (no disorder, no mess). They started with the gears in a neat, organized pattern (a "product state") and shook them.

The Result: The gears kept their rhythm for over 100 shakes (time steps).
The Analogy: Imagine a stadium full of people doing "The Wave." Usually, if you keep clapping, the wave gets messy and stops. But here, the crowd kept doing a perfect wave that only moved every other clap, and they did it for a long time without anyone getting confused or the wave collapsing.

3. The "Prethermal" State: The Long Pause

Why didn't they melt immediately? The paper explains they are in a prethermal state.

The Metaphor: Think of a spinning top. When you first spin it, it wobbles wildly but stays upright for a long time before finally falling over. That "wobbly but upright" phase is the prethermal state. The system is technically heading toward chaos (melting), but it's taking a very long detour, allowing the time crystal rhythm to exist for a while.

4. The Surprise: The "Incommensurate" Rhythm

When the researchers added a second type of push (a longitudinal field), something even stranger happened.

The Analogy: Imagine the dancers are doing their "every-second-beat" turn. Suddenly, a new, slightly different beat is added to the music. The dancers don't just stop; they start doing a complex, slow-motion "breathing" motion on top of their turns.
The Science: This created an IM-DTC (Incommensurately Modulated Time Crystal). The rhythm wasn't just a simple "flip-flop" anymore; it was a flip-flop with a slow, wavy modulation that didn't perfectly line up with the main beat. It's like a metronome that ticks, but the volume of the tick slowly swells and fades in a pattern that never quite repeats the same way.

5. Why This Matters: The Quantum vs. Classical Battle

This is a huge deal for two reasons:

New Physics: It proves that you can have these exotic, time-bending states in a clean, 2D system without needing disorder. It opens the door to new materials and states of matter.
The Computer Power: The researchers compared their quantum computer results with the best supercomputers in the world (using "tensor networks").
- The Analogy: Imagine trying to predict the path of a billion leaves in a hurricane. Classical supercomputers are like trying to calculate every leaf's path with a calculator; they get overwhelmed and give up after a while because the math gets too messy (entanglement grows too fast).
- The Quantum Win: The quantum computer is the hurricane. It doesn't calculate the leaves; it becomes the leaves. It simulated 100 steps of this complex dance, while the classical supercomputers started to struggle and lose accuracy after about 50 steps.

Summary

The team used a 133-qubit quantum computer to show that a clean, organized group of quantum particles can resist the natural urge to become chaotic. They found a way to make them dance to a rhythm that breaks the rules of time, and they even discovered a new, complex "wavy" version of this dance. Most importantly, they proved that quantum computers are now powerful enough to simulate these complex dances better than our best classical supercomputers can.

1. Problem Statement

Periodically driven (Floquet) quantum systems typically suffer from rapid heating, evolving into an infinite-temperature thermal state due to continuous energy absorption. While Discrete Time Crystals (DTCs) represent a non-equilibrium phase of matter characterized by subharmonic responses (breaking discrete time-translation symmetry), sustaining them is challenging.

The Challenge: In one-dimensional (1D) systems, DTCs usually require Many-Body Localization (MBL) (induced by disorder) to prevent thermalization. However, MBL is unstable in higher dimensions.
The Gap: While "prethermal" DTCs (long-lived meta-stable states before thermalization) have been theorized for clean (disorder-free) systems in 2D, experimental realization on digital quantum computers has been limited. Previous studies were restricted to 1D or small system sizes where classical simulations could easily keep up. There was a lack of systematic examination of clean DTCs in genuinely 2D geometries with system sizes large enough to challenge classical tensor-network methods.

2. Methodology

The authors utilized the IBM Quantum Heron processor (ibm torino), a superconducting quantum computer with 133 qubits arranged on a heavy-hexagonal lattice.

Model: They simulated a Kicked Ising Model governed by a time-dependent Hamiltonian with period $T$ $T$ . The system involves:
- Transverse field ( $h_x$ )
- Longitudinal field ( $h_z$ )
- Exchange interaction ( $J$ )
- The system is driven by a Floquet operator $\hat{U}_F$ composed of single-qubit rotation gates ( $R_X, R_Z$ ) and two-qubit entangling gates ( $R_{ZZ}$ ).
Initial State: Product states with a stripe pattern of $|0\rangle$ and $|1\rangle$ (domain walls).
Simulation Depth: The system was evolved for up to 100 Floquet time steps.
Error Mitigation: To address noise inherent in NISQ devices, the authors employed a global depolarizing noise model. They estimated the noise parameter $f$ using the trivial case ( $\theta_x = \pi$ ) where the ideal signal is known to be 1, and applied a correction factor to raw data for other parameters:
$\langle \hat{Z}_{avg}(t) \rangle \approx \frac{\langle \hat{Z}_{avg}(t) \rangle_0}{|\langle \hat{Z}_{avg}(t) \rangle_0|_{\theta_x=\pi}}$
Validation: Results were cross-validated against:
1. State-vector simulations (exact diagonalization) for a 28-qubit subsystem.
2. Two-Dimensional Tensor Network State (2dTNS) simulations for the full 133-qubit system (up to bond dimension $\chi=400$ ).

3. Key Contributions

Realization of Clean 2D DTCs: The first demonstration of a clean (disorder-free) DTC in a two-dimensional geometry on a digital quantum computer, without relying on MBL or high-frequency prethermalization limits.
Discovery of IM-DTC: Identification of a new phase called Incommensurately Modulated DTC (IM-DTC). When a longitudinal field ( $h_z$ ) is introduced, the magnetization exhibits additional amplitude modulations with frequencies incommensurate with the driving period.
Benchmarking Quantum vs. Classical: The study pushes the boundary of classical simulation. While 2dTNS agreed with quantum results for $\sim$ 50 steps, the quantum computer successfully simulated up to 100 steps in a parameter regime where classical tensor networks began to fail due to rapid entanglement growth.
Hardware Validation: Demonstrated the reliability of the IBM Heron processor for long-time dynamics (100 steps) on a 133-qubit system, significantly outperforming previous experiments on similar hardware (e.g., IBM Eagle) which were limited to ~20 steps.

4. Key Results

Stability of Clean DTC:
- For transverse field parameters $\theta_x \in [0.8\pi, \pi]$ , the system exhibited robust period-doubling oscillations (subharmonic response at $T/2$ ) persisting for at least 100 time steps.
- This stability was confirmed to be a prethermal phenomenon, as quantum scrambling (measured via Out-of-Time-Order Correlators, OTOCs) was suppressed in this regime, unlike in 1D systems where thermalization occurred rapidly.
- The heavy-hexagonal lattice geometry was found to suppress scrambling more effectively than a square lattice with the same parameters.
Emergence of IM-DTC:
- Introducing a longitudinal field ( $\theta_z > 0$ ) broke the Ising symmetry and induced side peaks in the Fourier spectrum of the magnetization.
- These side peaks corresponded to frequencies incommensurate with the driving frequency, creating a modulation envelope.
- The envelope frequency $\omega_{env}$ was found to be proportional to the perturbation parameter $\epsilon$ (deviation of $\theta_x$ from $\pi$ ), resembling Bloch oscillations.
Comparison with Classical Simulations:
- Agreement: For $\theta_x = 0.9\pi$ , quantum results matched 2dTNS simulations (with low bond dimensions) up to 100 steps.
- Divergence: Near the crossover to thermalization ( $\theta_x \approx 0.8\pi$ ), classical 2dTNS simulations showed slow convergence and discrepancies with state-vector results for 28 qubits at long times. The quantum computer provided data in this "hard" regime where classical methods struggled, highlighting the utility of quantum hardware for exploring entanglement-rich dynamics.

5. Significance

Fundamental Physics: The work provides experimental evidence that clean, disorder-free 2D systems can host stable prethermal DTCs, validating theoretical predictions that such phases do not strictly require MBL. It also characterizes the transition from standard DTC to IM-DTC.
Quantum Advantage: This study serves as a strong benchmark for Quantum Advantage in simulating non-equilibrium many-body dynamics. It demonstrates that current gate-based quantum computers can access dynamical regimes (long-time evolution in 2D) that are computationally intractable for state-of-the-art classical tensor-network methods due to entanglement growth.
Hardware Progress: It showcases the maturity of the IBM Heron processor, capable of maintaining coherence and signal integrity over 100 time steps on a 133-qubit lattice, a significant leap from previous generations.
Future Directions: The findings suggest that digital quantum computers are essential tools for exploring the "prethermal" window of quantum matter, bridging the gap between short-time coherent dynamics and long-time thermal equilibrium.