⚛️ quantum physics

Quantum many-body mixed phase space revealed by hybrid feedback control

This paper presents a hybrid quantum-classical feedback protocol implemented on a superconducting processor that autonomously discovers and stabilizes long-lived regular trajectories, thereby experimentally revealing a novel quantum many-body mixed phase space arising from nonlinear variational dynamics.

Original authors: Hang Dong, Jie Ren, Andrew Hallam, Han Wang, Zhengyi Cui, Yiren Zou, Junlin Wang, Hekang Li, Qiujiang Guo, Zhen Wang, Lei Ying, Zlatko Papic

Published 2026-07-17

📖 7 min read🧠 Deep dive

CC BY 4.0

Original authors: Hang Dong, Jie Ren, Andrew Hallam, Han Wang, Zhengyi Cui, Yiren Zou, Junlin Wang, Hekang Li, Qiujiang Guo, Zhen Wang, Lei Ying, Zlatko Papic

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 the universe as a giant, bustling dance floor. On one side, you have dancers moving in perfect, predictable circles, like planets orbiting a sun or a pendulum swinging back and forth. This is "order." On the other side, you have a mosh pit where everyone is bumping into each other, moving randomly, and eventually, the whole crowd just jumbles into a uniform, chaotic soup. This is "chaos." For a long time, scientists thought that if you had enough dancers (particles) interacting with each other, the whole group would inevitably turn into a chaotic mosh pit, losing all memory of how they started. This idea, known as thermalization, suggests that complex systems eventually forget their past and just heat up. But what if there's a secret corner of the dance floor where some dancers manage to keep their rhythm even in the middle of the chaos? Finding these "islands of order" inside a sea of chaos is a huge deal because it challenges our understanding of how the universe works and could help us build better quantum computers that don't fall apart from noise.

This paper is about a team of scientists who decided to hunt for these hidden islands of order in a quantum system—a system where particles behave like both tiny balls and waves. They didn't just look for them; they built a special "hybrid feedback" robot to find and hold onto them. Think of it like trying to balance a broom on your hand. If you just let go, it falls (chaos). But if you constantly watch it and make tiny, quick adjustments with your hand to keep it upright, you can stabilize it. The researchers used a superconducting quantum processor (a type of quantum computer made of tiny electrical circuits) to simulate a complex system of 24 interacting particles. They found that while most starting points led to chaos, there were specific starting positions that led to long-lasting, rhythmic motion. By using their hybrid robot—which alternates between letting the system evolve for a split second and then using a classical computer to "nudge" it back onto a stable path—they were able to discover and lock onto these stable rhythms. They didn't just find one; they mapped out a "mixed phase space," showing that regular, predictable motion and chaotic motion can coexist right next to each other in a quantum world, a phenomenon that had been theorized but never seen in such a complex system before.

The Dance of Order and Chaos

To understand what's happening here, we need to look at how things move. In the classical world, like a solar system, things are usually predictable. But if you add a little bit of chaos, like a third planet messing with the orbits, you get a "mixed phase space." Imagine a map where some areas are calm lakes (regular orbits) and others are raging whitewater rapids (chaos). In the quantum world, things are trickier. Usually, when you have a lot of particles interacting, they all get tangled up in a mess called "entanglement," and the system forgets its history, turning into a chaotic soup. This is called thermalization. However, scientists suspected that maybe, just maybe, some quantum systems could have their own version of those calm lakes, where particles keep dancing in a pattern without getting lost in the chaos.

The big question was: Can we find these calm lakes in a system with many interacting particles? And if we do, can we keep them there?

The Hybrid Feedback Robot

The researchers built a clever solution using a "hybrid" approach. They used a quantum processor (the dance floor) and a classical computer (the coach) working together. Here's how their "feedback loop" works:

The Quantum Step: They let the quantum system dance for a very short time (about 80 nanoseconds). During this time, the particles start to interact. If they are in a chaotic region, they start to get messy and tangled.
The Measurement: They stop the dance and take a snapshot. They don't look at the whole messy picture; instead, they check a specific "imbalance" score. Think of this like checking if the dancers on the left side of the floor are still mostly different from the dancers on the right. If the score is high, the system is still organized. If it's low, it's getting chaotic.
The Classical Nudge: The classical computer looks at that score and figures out how to tweak the starting position of the dancers to make the score higher next time. It's like the coach saying, "You started a bit too far to the left; let's move you slightly right."
Repeat: They reset the system with the new, tweaked starting position and do it all over again.

By repeating this cycle, the system naturally "filters out" the chaotic paths. The chaotic paths get nudged away because they lose their balance quickly. The regular paths, however, are stable. The feedback loop keeps pushing the system back toward these stable paths until it locks onto a perfect, repeating rhythm.

What They Found

Using this method on a 24-qubit processor (which is like a tiny quantum computer with 24 switches), the team discovered something amazing. They found that the quantum system wasn't just a chaotic mess. It had a "mixed phase space."

The Map: They created a map (called a Poincaré section) that showed where the chaos was and where the order was. It looked like a sea of scattered points (chaos) with distinct islands of smooth curves (order).
The Discovery: They found that by starting in the right spot, they could make the system dance in a perfect, repeating loop for a very long time. Even better, their feedback robot could find these loops even if they started in the middle of the chaotic sea. The robot would nudge the system until it found the nearest "island of order" and then kept it there.
The Stability: They tested this by changing the strength of the interactions between the particles. They found that the islands of order didn't disappear; they just slowly changed shape, like a cloud drifting in the wind. This proved that the order wasn't a fluke; it was a robust feature of the system.

Why This Matters

This isn't just a cool trick with quantum computers. It shows us that even in complex, interacting quantum systems, there can be pockets of stability that resist the usual tendency to turn into chaos. This is a big step forward because it suggests that we might be able to control quantum systems better than we thought. If we can find and stabilize these "regular trajectories," we might be able to protect quantum information from the noise that usually destroys it.

The researchers showed that this "mixed phase space" isn't just a theoretical idea; it's real and can be observed and controlled. They didn't just simulate it; they built it, measured it, and stabilized it on a real quantum processor. While the system they studied is specific, the method they used—a hybrid loop of quantum evolution and classical optimization—could be a powerful tool for exploring other complex quantum systems. It's like finding a secret door in a chaotic maze that leads to a peaceful garden, and now we have a map and a key to find it again and again.

Technical Summary: Quantum Many-Body Mixed Phase Space Revealed by Hybrid Feedback Control

Problem Statement
Understanding the transition between order and chaos in complex systems is a central challenge in nonequilibrium physics. In classical mechanics, the Kolmogorov–Arnold–Moser (KAM) theorem describes how weak perturbations of integrable systems create a "mixed phase space" where regular (quasi-periodic) and chaotic trajectories coexist. While remnants of integrability exist in few-body quantum systems or under weak perturbations of Bethe-ansatz solvable models, identifying analogous mixed phase spaces in strongly interacting quantum many-body systems has remained elusive. Generic quantum systems typically thermalize according to the Eigenstate Thermalization Hypothesis (ETH), obscuring non-thermalizing states. Although systems with quantum many-body scars and Hilbert-space fragmentation offer a starting point, the specific initial states required to access them are exceedingly rare in the many-body Hilbert space, making them difficult to identify and stabilize experimentally.

Methodology
The authors develop and experimentally implement a hybrid quantum–classical feedback protocol designed to autonomously discover and stabilize long-lived regular trajectories within an interacting quantum many-body system. The approach integrates three core components:

Variational Manifold and TDVP Projection: The system is modeled using a shallow variational circuit ansatz (a two-layer entangling circuit) that maps the dynamics of an interacting Su–Schrieffer–Heeger (SSH) ladder onto a low-dimensional, low-entanglement manifold $\mathcal{M}$ . The Time-Dependent Variational Principle (TDVP) is used to project the exact Schrödinger evolution onto the tangent space of this manifold, yielding effective nonlinear classical equations of motion for the circuit parameters.
Experimental Observable (Imbalance): To experimentally access the phase space structure without full state tomography, the authors utilize a subsystem imbalance observable, $I_z$ . This quantity serves as a proxy for state fidelity, measuring the overlap between the time-evolved state and the variational ansatz. Regular trajectories exhibit long-lived coherent oscillations and revivals in $I_z(t)$ , while chaotic trajectories show rapid decay.
Hybrid Feedback Loop: The protocol alternates between short-time quantum evolution under the system Hamiltonian and a classical optimization step. After a short evolution time $\Delta t$ , an inverse variational circuit is applied, and the imbalance is measured. A classical optimizer adjusts the circuit parameters to maximize this imbalance, effectively projecting the state back onto the variational manifold $\mathcal{M}$ and filtering out the entanglement-generating components of chaotic evolution. This iterative process ( $| \psi^{(n+1)} \rangle = P_M \hat{U}(\Delta t) | \psi^{(n)} \rangle$ ) acts as a dynamical filter, converging to stable periodic orbits.

The experiments were conducted on a programmable superconducting qubit processor (24 qubits) realizing the SSH ladder Hamiltonian.

Key Results

Experimental Mapping of Mixed Phase Space: By scanning initial conditions and measuring the first revival peak of the imbalance ( $I_z^{\text{max}}$ ), the authors constructed a Poincaré-section-like map. The results reveal a clear coexistence of "islands" of regular motion (characterized by high revival fidelity) and a "sea" of chaotic motion (characterized by rapid thermalization). This structure matches numerical simulations of the TDVP equations and exact diagonalization.
Stabilization of Regular Trajectories: The hybrid feedback protocol successfully steers the system from chaotic initial conditions into stable, self-consistent periodic orbits within the variational manifold. The feedback loop "distills" coherence from chaotic motion, suppressing entropy growth and enhancing revival fidelity even in the presence of device noise and crosstalk.
Structural Stability: The authors demonstrate that the mixed phase space structure is robust. By varying the coupling strengths ( $J_e$ ) of the SSH ladder, the stable periodic orbits identified by the feedback protocol deform smoothly, confirming the structural stability characteristic of mixed phase spaces in classical mechanics.
Thermodynamic Limit: Numerical simulations using infinite matrix-product states (iMPS) indicate that this mixed phase space behavior persists in the thermodynamic limit ( $N \to \infty$ ), manifesting as a weak violation of the ETH where states at the same energy density exhibit distinct entanglement growth rates (slow/oscillatory vs. rapid/linear).

Significance and Claims
The paper claims to provide the first experimental evidence of a quantum many-body mixed phase space in an interacting system. Unlike previous observations of quantum scars which rely on specific, rare initial states, this work demonstrates a framework where regular dynamics emerge from the collective evolution of entangled degrees of freedom within a nonlinear variational manifold.

The authors assert that their hybrid feedback protocol establishes a versatile framework for the algorithmic discovery and control of coherent dynamics that were previously inaccessible to experiment. By combining short-time quantum evolution with classical optimization, the method bypasses the need for long coherent evolution times or full wavefunction reconstruction. The work bridges the gap between classical chaos theory (KAM tori) and quantum ergodicity breaking, suggesting that mixed phase spaces are a generic feature of nonlinear variational dynamics in quantum many-body systems, rather than an artifact of the classical limit ( $\hbar \to 0$ ). The study also highlights the potential of such protocols for preparing long-lived nonthermal states without prior knowledge of the model's specific scarred states.

The Dance of Order and Chaos

The Hybrid Feedback Robot

What They Found

Why This Matters

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