⚛️ quantum physics

Quantum Circuits as a Dynamical Resource to Learn Nonequilibrium Long-Range Order

This paper demonstrates that variational quantum circuits can engineer nonequilibrium long-range ordered states in one-dimensional systems at finite energy density, overcoming equilibrium constraints like the Mermin-Wagner theorem to create robust, symmetry-broken, and topological phases with enhanced metrological properties.

Original authors: Fabian Ballar Trigueros, Markus Heyl

Published 2026-02-20

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Fabian Ballar Trigueros, Markus Heyl

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 are trying to organize a massive, chaotic party in a small room.

In the world of physics, equilibrium (a state of rest) is like that party after everyone has been dancing for hours. The music stops, people are tired, and everyone just mingles randomly. In this state, physics has strict rules: in a small room (low dimensions), you can never get a perfect, synchronized dance line (long-range order) going. The famous Mermin-Wagner theorem is basically the bouncer saying, "No synchronized dancing allowed here; it's too chaotic."

Usually, if you look at a quantum system (like a group of atoms) with a lot of energy (a high-energy party), it's even worse. According to a rule called the Eigenstate Thermalization Hypothesis (ETH), every single person at the party is acting completely randomly and independently. If you look at any single atom, it looks like a featureless, boring blob. There is no order, no pattern, just noise.

The New Idea: The "Choreographer" Circuit

This paper introduces a brilliant new idea: What if we don't just let the party happen naturally? What if we hire a choreographer?

The authors use Quantum Circuits as this choreographer. Think of a quantum circuit not as a computer program, but as a set of instructions for how to move the atoms. But here's the twist: they don't just pick random moves. They use a Variational Quantum Circuit.

The Analogy: Imagine a dance instructor who doesn't know the perfect dance yet. They try a move, see if the group looks synchronized, and if not, they tweak the move slightly. They repeat this thousands of times (training) until the group is dancing in perfect unison, even though the room is small and the energy is high.

How They Did It

The Setup: They started with a "reference Hamiltonian" (the music of the party). In this music, every single dancer (energy state) is supposed to be random and chaotic.
The Training: They fed this chaotic music into their "choreographer" (the circuit). The circuit's job was to find a specific combination of moves (a superposition of many random states) that would make the whole group look like they were dancing in a line.
The Result: It worked! The circuit learned how to mix these chaotic, random states together in a very specific way. When you look at the whole group (the final state), they have Long-Range Order. They are synchronized across the entire room, defying the usual rules that say "this is impossible in a small room."

Two Types of "Dances"

The paper shows this works for two different kinds of order:

Symmetry Breaking (The "All Left" Dance): Imagine everyone suddenly deciding to face left. In a normal high-energy state, half would face left and half right, canceling each other out. The circuit learned to make everyone face left, creating a strong, unified signal.
Symmetry-Protected Topological Order (The "Secret Handshake"): This is a more complex order where the connection isn't visible if you just look at one person. It's like a secret handshake that only works if you look at the entire chain of people. The circuit learned to maintain this hidden connection even in a high-energy, chaotic environment.

Why This is a Big Deal (The "Unbreakable" Order)

Usually, when you have a perfect quantum state (like a GHZ state), it's incredibly fragile. If you look at just one person in the line (a local measurement), the whole synchronized dance collapses, and everyone goes back to being random. It's like a house of cards.

But the states created by this circuit are different.
The authors tested this by "measuring" (looking at) random people in the line.

The Result: The dance didn't collapse. The group remained synchronized even after people were observed.
The Metaphor: Imagine a choir singing a perfect harmony. If you ask one singer to stop, usually the whole song falls apart. But in this new state, if you ask one singer to stop, the rest of the choir instantly adjusts and keeps singing the perfect harmony. It is robust.

The Secret Sauce: "Learning" How to Not Thermalize

Why did this work? The circuit didn't just find a lucky state; it learned how to avoid the natural tendency of the universe to become chaotic (thermalize).

In the first case: The circuit found a "sweet spot" in the math where the system behaves almost like a perfect clock (near-integrable), avoiding the chaos.
In the second case: The circuit discovered a "hidden rule" (an emergent symmetry) that the original music didn't have, which kept the dancers connected.

The Takeaway

This paper proves that coherent quantum dynamics (carefully choreographed quantum movements) can be used as a tool to create states of matter that are impossible to find in nature under normal conditions.

It's like discovering a way to build a skyscraper in a swamp. The laws of physics say "you can't build a skyscraper here because the ground is too soft." But this paper says, "If you use the right dynamic foundation (quantum circuits), you can build a skyscraper that stands tall, even in the swamp."

This opens the door to creating new materials and quantum technologies that are stable, highly ordered, and incredibly useful for measuring things with extreme precision, all while operating at high energies where we previously thought only chaos could exist.

1. Problem Statement

The paper addresses a fundamental limitation in the study of quantum matter: equilibrium statistical ensembles impose strict constraints on the phases of quantum matter, particularly in low-dimensional systems.

The Mermin-Wagner Theorem: Prohibits spontaneous breaking of continuous symmetries (and thus long-range order) in 1D and 2D systems at non-zero temperatures.
Eigenstate Thermalization Hypothesis (ETH): In generic, non-integrable 1D systems, individual eigenstates at finite energy density are locally thermal and "featureless," lacking any long-range order or macroscopic correlations.
The Gap: While non-equilibrium classical systems exhibit diverse emergent phenomena (e.g., flocking), the landscape of non-equilibrium quantum phases remains under-explored. Specifically, there is no general framework to generate ordered phases in isolated quantum systems at high energy densities where equilibrium physics dictates disorder.

The central question is: Can coherent quantum dynamics be used as a resource to engineer states with long-range order at finite energy densities, effectively bypassing the constraints of ETH and equilibrium thermodynamics?

2. Methodology

The authors propose a Variational Quantum Circuit (VQC) framework constrained by global symmetries to prepare specific quantum states.

Ansatz Architecture:
- A system of $N$ qubits with a global symmetry group $G$ .
- A "brickwork" pattern of local $k$ -qubit gates with depth $d$ .
- Gates are parameterized as $U_G(\theta) = \prod U_k(\theta_k)$ , where parameters $\theta_k$ are optimized.
- Symmetry Constraint: The gates are chosen to form a unitary design within the symmetry sector, ensuring the circuit evolution strictly respects the symmetry (e.g., $Z_2$ or $Z_2 \times Z_2$ ).
Optimization Objective:
The circuit parameters are optimized to minimize a loss function $L(\theta)$ $L (θ)$ :
$L(\theta) = -\langle \chi \rangle_\theta + \sigma \langle H - E \rangle^2_\theta + \beta (\langle H^2 \rangle_\theta - \langle H \rangle^2_\theta)$
- $\chi$ : The target long-range order parameter (to be maximized).
- $H$ : A reference generic Hamiltonian (e.g., non-integrable Ising-type or Cluster-Ising model).
- $E$ : A target energy density in the middle of the spectrum (where ETH predicts disorder).
- The last two terms constrain the state to a narrow energy window and penalize energy fluctuations.
Initialization:
- Symmetry Breaking (Landau): Random product states on the $X-Z$ plane (unbiased with respect to symmetry).
- Symmetry Protected Topological (SPT): The ground state of a cluster model (which already possesses SPT order), used as a starting point to explore high-energy analogs.

3. Key Contributions

Demonstration of Non-Equilibrium Long-Range Order: The paper provides numerical evidence that VQCs can generate states with macroscopic long-range order at high energy densities in 1D systems, a regime where individual eigenstates are featureless.
Mechanism of Coherent Superposition: The authors show that the order does not arise from isolating specific eigenstates but from coherent superpositions of many ETH-compliant eigenstates. The circuit learns relative phases that amplify off-diagonal matrix elements to produce macroscopic order.
Robustness Against Local Measurements: Unlike fragile states like GHZ (which collapse to product states after a single local measurement), the learned states retain significant entanglement and order even after multiple local projective measurements.
Discovery of Non-Ergodic Dynamics: The optimization process drives the circuits toward non-thermal fixed points.
- In the symmetry-breaking case, the circuits approach near-integrable regions (deviating from Wigner-Dyson statistics).
- In the SPT case, the circuits spontaneously discover emergent hidden symmetries that stabilize topological order deep in the spectrum.

4. Key Results

A. Symmetry-Breaking (Landau) Order

Setup: $Z_2$ symmetry (parity) with a reference non-integrable Ising Hamiltonian.
Result: The optimized circuits produce states where the susceptibility $\chi$ saturates to a finite value as system size $N$ increases, signaling long-range order.
Spectral Analysis: The trained states are supported deep in the energy spectrum where individual eigenstates have $\chi \approx 0$ . The order arises entirely from the coherent interference of these states.
Robustness Test: When subjected to random local $Z$ -basis measurements, the post-measurement entanglement entropy remains high and decays slowly with the number of measurements. This contrasts sharply with GHZ states, which lose all entanglement after one measurement.

B. Symmetry-Protected Topological (SPT) Order

Setup: $Z_2 \times Z_2$ symmetry using a Cluster-Ising model.
Result: The circuits generate highly excited states that retain non-local string order (characteristic of SPT phases), defined by a non-local string operator $O_{ij}$ .
Mechanism: The circuit learns to stabilize string order in regions where ETH predicts it should vanish.
Emergent Symmetry: Analysis of the effective Hamiltonian ( $U = e^{-iH_{eff}}$ ) reveals systematic spectral degeneracies (4-fold degeneracies), indicating the emergence of a hidden conserved quantity that protects the topological order against thermalization.

C. Metrological Properties

The generated states exhibit Quantum Fisher Information (QFI) that scales extensively and approaches the Heisenberg limit (similar to GHZ states), indicating strong multipartite entanglement.
Crucially, unlike GHZ states, this metrological advantage is robust against local noise and measurements.

5. Significance and Conclusion

Redefining Quantum Phases: The work establishes that coherent quantum dynamics is a powerful resource for engineering phases of matter that are inaccessible to equilibrium statistical mechanics. It proves that thermalization is a dynamical barrier that can be circumvented through learned interference patterns.
Learning Non-Ergodicity: The variational training effectively "learns" how to avoid ergodicity. The circuits do not just prepare states; they discover dynamical regimes (near-integrable or symmetry-protected) where coherence persists across energy scales.
Future Directions: The authors suggest extending this framework to open quantum systems, measurement-induced phase transitions, and systems with continuous symmetries (where stability at finite energy density remains an open challenge).
Broader Impact: This opens a path toward a "dynamical scope" of quantum order, suggesting that the classification of quantum phases should include those accessible only through specific dynamical training protocols rather than just equilibrium or ground-state properties.

In summary, the paper demonstrates that by leveraging variational quantum circuits to learn specific interference patterns, one can construct robust, long-range ordered quantum states at high energy densities, effectively bypassing the prohibitions of the Mermin-Wagner theorem and the featureless nature of ETH eigenstates.