🔬 condensed matter

Unveiling a Hidden Percolation Transition in Monitored Clifford Circuits: Inroads from ZX-Calculus

By applying ZX-calculus simplification techniques to monitored Clifford circuits, this study reveals that the measurement-induced phase transition, previously thought to be distinct from classical percolation, is actually governed by a hidden classical percolation transition within the circuit's underlying structure.

Original authors: Einat Buznach, Debanjan Chowdhury, Jonathan Ruhman

Published 2026-07-01

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CC BY 4.0

Original authors: Einat Buznach, Debanjan Chowdhury, Jonathan Ruhman

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

The Big Picture: A Mystery in a Quantum Maze

Imagine you are watching a complex game played with a deck of cards (representing quantum bits, or "qubits"). In this game, you shuffle the cards (unitary evolution) and occasionally peek at them or remove some (measurements).

Physicists have been studying a specific version of this game called a Clifford Circuit. They noticed something strange: at a certain point, the game changes its behavior completely. Before this point, the cards are all tangled up together in a giant, messy knot (high "entanglement"). After this point, the knot untangles, and the cards become independent (low "entanglement").

This change is called a Measurement-Induced Phase Transition (MPT).

For a long time, scientists believed this specific quantum game was unique. They thought its rules were so different from the classical world that it didn't follow the standard rules of how things break apart or connect. They suspected it was a brand-new type of physics, unlike anything seen before.

The Old Way of Looking: The "Minimal Cut"

To understand why the cards are tangled, scientists usually look for the "weakest link." Imagine trying to cut a rope holding a heavy weight. If you cut the rope in the right place, the weight falls. In these quantum games, scientists tried to find the "minimal cut"—the smallest number of measurements needed to separate the tangled cards.

Using this old method, they concluded: "This quantum game is special. It doesn't act like a simple rope-cutting game (which is called classical percolation). It has its own secret, complex rules."

The New Tool: The "Magic Glasses" (ZX-Calculus)

In this paper, the authors (Einat Buznach Ahituv, Jonathan Ruhman, and Debanjan Chowdhury) decided to look at the game through a new pair of glasses called ZX-Calculus.

Think of ZX-Calculus as a special translator or a set of magic glasses. It takes the complex, messy quantum circuit and rewrites it into a simpler, cleaner diagram. It's like taking a tangled ball of yarn and using a special tool to untangle it until you see the simple, straight lines underneath.

The Big Discovery: The Hidden Percolation

When the authors used these "magic glasses" to simplify the quantum circuit, something surprising happened.

The Simplification: They applied a set of logical rules (like fusing shapes together or cutting connections) to the diagram.
The Result: The complex quantum diagram shrank down. In the "untangled" phase, the initial cards and the final cards became completely disconnected in the diagram. In the "tangled" phase, they stayed connected.
The Reveal: When they analyzed this simplified diagram, they found it wasn't following a new, mysterious rule at all. It was following the exact same rules as classical percolation.

The Analogy:
Imagine you are looking at a crowded city street from a helicopter. It looks chaotic and impossible to navigate (the complex quantum circuit). You think, "There must be a new law of traffic here!"

But then, you put on special glasses that remove all the cars, pedestrians, and noise, leaving only the road map. Suddenly, you see that the traffic flow is actually just following the simplest rule possible: "If a road is blocked, traffic stops."

The authors found that the "complex" quantum transition was just a classical percolation transition in disguise. The "new" physics was just the old physics wearing a mask.

What They Proved

The Coincidence: They calculated the exact point where the quantum game changes from "tangled" to "untangled" (using a measure called Mutual Information). Then, they calculated the point where the simplified diagram breaks apart (using percolation theory).
The Match: These two points were exactly the same.
The Conclusion: The mysterious "Quantum Phase Transition" in Clifford circuits is actually controlled by a hidden, classical "Percolation Transition." It's not a new universe of physics; it's the old one, just hidden behind a complex layer of quantum math.

Why This Matters (According to the Paper)

The paper suggests that for this specific type of quantum circuit, we don't need to invent a new theory to explain how it breaks down. We just need to find the right way to simplify the picture (using ZX-Calculus) to see the simple, classical rules underneath.

They also mention that this method helps settle a debate in the scientific community about whether these circuits are truly unique or not. By using this "simplification" tool, they showed that the complexity was an illusion created by looking at the circuit the wrong way.

In short: The authors used a special mathematical tool to strip away the complexity of a quantum game, revealing that its most dramatic moment is actually just a classic, everyday game of "connect the dots" that we already understand.

Technical Summary: Unveiling a Hidden Percolation Transition in Monitored Clifford Circuits

Problem Statement
Measurement-induced phase transitions (MPTs) in quantum systems undergoing unitary evolution interspersed with measurements represent a critical area of study in open quantum systems and quantum information. A central question in the field concerns the universality class of these transitions, particularly in Clifford circuits. While it is widely accepted that MPTs in Clifford circuits exhibit critical behavior distinct from classical percolation theory, the underlying mechanism remains debated. Specifically, the "minimal-cut picture," which successfully maps MPTs to classical percolation in the $q \to \infty$ limit, fails to capture the correct critical point for finite $q$ Clifford circuits. In these systems, the MPT (identified via mutual information or non-zero Rényi entropies) occurs at a measurement rate $p_q$ strictly lower than the classical percolation threshold $p_c$ (identified via the Hartley function), suggesting a novel universality class. The authors investigate whether this apparent distinction is fundamental or if a hidden classical percolation transition governs the MPT in a non-obvious representation.

Methodology
The authors employ ZX-calculus, a graphical language for representing and simplifying quantum circuits, to analyze a specific dynamical model.

Model: The system consists of $N$ qubits in a 1D chain with a brickwork pattern of two-site operations. The operations include CNOT gates, SWAP gates, Identity gates, and Bell-pair measurements, selected randomly with probabilities dependent on parameters $p$ (measurement rate) and $r$ (CNOT probability). The initial state is a tensor product of nearest-neighbor Bell pairs.
Simulation: The dynamics are simulated efficiently using the Gottesman-Knill theorem, which tracks the evolution of stabilizers (Pauli operators) with polynomial efficiency, allowing for large system sizes ( $N$ up to 1020).
Phase Identification: The MPT is characterized using the bipartite mutual information $I_2 = S_A + S_C - S_B$ between regions separated by a third region. The transition is identified by the crossing of $I_2$ curves for different system sizes, marking the shift from a volume-law (entangling) to an area-law (disentangling) phase.
ZX-Analysis: The authors translate the random Clifford circuits into ZX-diagrams. They apply local simplification rules—Spider fusion, Copy rule, and Hopf rule—to reduce the diagrams. The key metric derived from the simplified networks is the probability ( $P_{path}$ ) of finding a connected path between initial and final times.

Key Contributions and Results

Discovery of a Hidden Transition: By applying ZX-calculus simplification techniques, the authors reveal a "hidden" percolation transition within the circuit structure. In the simplified ZX-networks, the connectivity between initial and final times undergoes a classical percolation transition.
Coincidence of Critical Points: The study demonstrates that the critical measurement rate $p_c$ $p_{c}$ derived from the classical percolation transition in the ZX-simplified network coincides with the MPT critical point $p_q$ $p_{q}$ identified via mutual information in the original circuit.
- Numerical scaling analysis yields $p_c(N \to \infty) = 0.251 \pm 0.009$ from the path probability in ZX-simplified networks.
- This matches the value $p_q = 0.243 \pm 0.003$ obtained from the mutual information crossing in the original dynamical model.
Phase Diagram Reconstruction: The authors map the phase diagram in the $(r, p)$ plane. They confirm that the boundary between the volume-law and area-law phases, determined by mutual information, aligns with the percolation threshold of the simplified ZX-networks (represented as green dots in their phase diagram).
Resolution of the "Minimal-Cut" Discrepancy: The paper addresses why previous "minimal-cut" analyses failed to predict the correct MPT point. The authors argue that the standard minimal-cut picture overestimates entanglement in Clifford circuits because it does not account for the specific non-universal gate set (e.g., CNOTs can reduce entanglement). The ZX-simplification effectively reorganizes the circuit into a structure where the true connectivity (and thus the percolation transition) governing the MPT becomes visible.

Significance and Claims
The paper claims that the MPT in Clifford circuits is not a fundamentally new universality class distinct from classical percolation. Instead, the authors propose that the MPT is controlled by a classical percolation transition in disguise, which is only revealed through the specific non-local simplifications enabled by ZX-calculus.

Theoretical Implication: This finding suggests that the critical behavior of MPTs in Clifford circuits is governed by underlying classical percolation phenomena, provided the circuit is viewed through the appropriate graphical representation.
Methodological Impact: The work highlights the power of ZX-calculus not just as a verification tool, but as a method to uncover hidden structural properties of quantum circuits that are obscured in their standard gate-level representation.
Caveats: The authors note that the simplification process is non-local, meaning the resulting network statistics differ from the original brickwork circuit. Consequently, while the critical point coincides, the preservation of other critical properties through simplification is not guaranteed and requires careful analysis.

The authors conclude that their findings settle an important debate regarding the nature of the MPT in Clifford circuits, suggesting that the apparent deviation from classical percolation is an artifact of the representation rather than a fundamental difference in universality.