🔬 condensed matter

Measurement-only circuit of perturbed toric code on triangular lattice: Topological entanglement, 1-form symmetry and logical qubits

This paper investigates the phase diagram of a measurement-only circuit based on the toric code on a triangular lattice, revealing that the system's lack of self-duality leads to distinct phase transitions with different critical exponents that are closely linked to two-dimensional percolation, thereby clarifying the complex interplay between topological entanglement entropy, 1-form symmetry breaking, and logical operator emergence.

Original authors: Keisuke Kataoka, Yoshihito Kuno, Takahiro Orito, Ikuo Ichinose

Published 2026-07-27

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CC BY 4.0

Original authors: Keisuke Kataoka, Yoshihito Kuno, Takahiro Orito, Ikuo Ichinose

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 a world where information isn't stored in bits like 0s and 1s on a hard drive, but is woven into the very fabric of space itself, like a knot in a piece of string that can't be untied no matter how much you pull. This is the realm of topological quantum computing, a branch of physics that hopes to build computers that are naturally immune to errors. In this world, scientists use a special kind of "quantum fabric" called the toric code. Think of the toric code as a magical grid where tiny particles dance in a way that creates a hidden, unbreakable shield around the information. Usually, to keep this shield intact, physicists have to constantly tweak a complex machine (a Hamiltonian) to hold the particles in place. But what if you could build this shield just by taking snapshots? That's the idea behind a Measurement-Only Circuit (MoC). Instead of pushing and pulling particles with forces, you simply "measure" them over and over. Surprisingly, if you measure them in the right pattern, the particles organize themselves into the perfect, error-proof state all on their own. It's like trying to organize a chaotic crowd of people into a perfect marching band; instead of shouting instructions, you just take photos of them, and somehow, the act of looking at them makes them fall into line.

This paper dives into a specific, tricky version of this experiment. The researchers decided to test their "measurement-only" magic not on a standard square grid, but on a triangular lattice—a grid made of triangles, which is much more tangled and less symmetrical than a square. They wanted to see if they could still create this magical, error-proof state just by measuring, and if so, what happens when they mess with the rules by adding "noise" (like random magnetic fields). They used powerful computer simulations to watch how the system evolved step-by-step, measuring things like how "entangled" the particles were (how deeply connected they are) and whether the hidden "logical" information (the actual qubits) survived the chaos.

Here is what they found: The triangular grid is a much tougher playground than the square one. Because the triangle grid isn't perfectly symmetrical, the different ways the system can break down don't happen at the same time. The researchers discovered that the system goes through several distinct phase transitions (sudden changes in behavior) at different points, rather than one big jump. They found that to get the system to settle into the desired "toric code" state, they needed to measure the stabilizer operators (the rules that keep the order) about 85% to 88% of the time, depending on the direction. When they measured less than that, the system collapsed into a boring, trivial state with no quantum magic left.

The team also looked at how the system behaves near the point where it changes from chaotic to ordered. They measured the "critical exponents," which are numbers that describe how sharp or smooth that change is. Their simulations suggest these transitions belong to a specific class of behavior with a critical exponent of roughly 1.7. This is interesting because it's different from the standard "percolation" (like water spreading through a sponge) we usually see in 2D, which has an exponent of about 1.33. This tells us that the triangular grid creates a unique kind of quantum phase transition that doesn't quite fit the old molds.

In short, the paper shows that you can build a topological quantum memory just by measuring particles on a triangular grid, but it's a delicate dance. The system doesn't behave like a simple mirror image of itself; instead, it has multiple "tipping points" where the magic either appears or vanishes. The researchers didn't just guess this; they ran thousands of simulation trajectories to map out exactly where these tipping points are and how the system reacts. They found that while the triangular lattice is more complex and less symmetrical than the square version, it offers a perfect testing ground to understand how topological order, symmetry, and logical information are all connected. The results confirm that with the right mix of measurements (roughly 85-88% stabilizer measurements), you can steer a chaotic quantum system into a stable, topologically protected state, even on a messy triangular grid.

Technical Summary: Measurement-only Circuit of Perturbed Toric Code on Triangular Lattice

Problem Statement
The Toric Code (TC), a prototypical topological stabilizer code, is traditionally studied via Hamiltonian formalism, where its ground states and phase transitions (e.g., between deconfined topological, Higgs, and confined phases) are analyzed under external fields. While the TC on a square lattice is well-understood and possesses self-duality, the TC on a triangular lattice lacks self-duality and bipartiteness. Consequently, the locations of phase transitions (such as the condensation of Wilson vs. 't Hooft loops) may differ, and the inter-relationship between topological entanglement entropy (TEE), 1-form symmetry breaking, and logical qubit emergence remains less clarified.

Furthermore, while Measurement-only Circuits (MoC)—where quantum states evolve solely through sequences of projective measurements—have been shown to generate topological order, a systematic understanding of phase transitions in MoCs, particularly on non-self-dual lattices like the triangular lattice, is lacking. Specifically, there is a need to determine if MoC-induced transitions belong to the same universality classes as their Hamiltonian counterparts and how observables like TEE and disorder parameters behave in this non-equilibrium setting.

Methodology
The authors investigate a MoC defined on a triangular lattice where the projective measurement operators consist of:

Stabilizers of the Toric Code: $A_s$ (star operators, products of $X_e$ ) and $B_p$ (plaquette operators, products of $Z_e$ ).
Competitive Local Pauli Operators: Single-site $X_e$ and $Z_e$ measurements, representing external magnetic and electric fields.

The study employs an efficient stabilizer formalism (Gottesman-Knill theorem) to simulate the time evolution of the quantum state. The circuit is constructed by sequentially applying these measurements with specific probabilities ( $p_x, p_z, p_g$ ), where $p_g$ is the probability of measuring stabilizers and $p_x + p_z + p_g = 1$ .

Key observables calculated include:

Topological Entanglement Entropy (TEE): Calculated using a specific partition of hexagonal subsystems to extract the universal constant $\gamma$ .
Disorder Parameters of 1-form Symmetries: Defined via Rényi-2 correlation functions of open string operators ( $W(r)$ for $Z$ -strings and $T(r)$ for $X$ -strings). These serve as diagnostics for anyon condensation and the spontaneous symmetry breaking (SSB) of electric and magnetic 1-form symmetries.
Logical Operators: Non-contractible loop operators ( $W_i, T_i$ ) that distinguish the four-fold degenerate ground states.

The authors utilize Finite-Size Scaling (FSS) analysis on system sizes up to $L=24$ to locate critical points ( $p_{gc}$ ) and extract critical exponents ( $\nu$ ).

Key Results

Phase Diagram and Transitions: The MoC exhibits distinct phases: a trivial product state (Higgs or confinement regimes depending on the dominant field) and the topologically ordered Toric Code phase. The transition from the trivial state to the TC phase occurs as the probability of measuring stabilizers ( $p_g$ ) increases.
Asymmetry in Critical Points: Unlike the square lattice TC, the triangular lattice system is not self-dual. Consequently, the critical probabilities for the transition from the confinement regime (dominated by $X$ $X$ -measurements) and the Higgs regime (dominated by $Z$ $Z$ -measurements) to the TC phase are distinct:
- Confinement $\to$ TC: $p_{gc}^x \approx 0.846 \pm 0.016$ .
- Higgs $\to$ TC: $p_{gc}^z \approx 0.876 \pm 0.048$ .
Critical Exponents: The correlation length critical exponent $\nu$ is found to be approximately $1.7 $($ \nu_x \approx 1.69, \nu_z \approx 1.74$) for both transitions. This value differs significantly from the standard 2D bond percolation universality class ( $\nu = 4/3 \approx 1.33$ ) and 3D percolation ( $\nu \approx 0.88$ ).
Decoupling of Observables: The study observes that the TEE, string disorder parameters, and logical operators exhibit transition-like behaviors at slightly different values of $p_g$ . This suggests that the "coincidence" of critical behaviors across different phases (e.g., TC to Higgs vs. TC to Confined) does not occur simultaneously due to the lack of self-duality.
Role of Percolation: While the system shares qualitative features with 2D bond percolation (due to the nature of stabilizer voids), the critical thresholds are distinct from pure percolation thresholds on triangular and honeycomb lattices. The authors attribute this to the competitive nature of the measurements and the trajectory-dependent evolution of the state.

Significance and Claims
The paper claims that the Toric Code MoC on a triangular lattice serves as a suitable "playground" to disentangle the mutual relationships between TEE, 1-form symmetry breaking, and the emergence of logical qubits, which are often conflated in self-dual systems.

The authors assert that their results demonstrate:

The existence of distinct phase transitions with different locations and critical exponents in a non-self-dual MoC.
That the critical behavior of these transitions belongs to a universality class distinct from standard 2D percolation, challenging simple mappings between MoC dynamics and classical percolation.
That the stabilizer formalism provides a precise and efficient method for calculating TEE and non-local correlation functions in 2D models, where Monte Carlo simulations often struggle.

The work does not claim to have identified the exact universality class definitively but provides quantitative evidence (via critical exponents) that strongly constrains its classification. It highlights that the triangular lattice TC MoC offers a unique perspective on the phase diagram of related stabilizer codes, such as the color code, by clarifying how partial anyon condensation affects topological order.

Technical Summary: Measurement-only Circuit of Perturbed Toric Code on Triangular Lattice

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