Resolving spurious topological entanglement entropy in stabilizer codes

This paper identifies the microscopic origin of spurious topological entanglement entropy in stabilizer codes and introduces a rigorous concave partition method to eliminate these artifacts, while also demonstrating topological frustration in bivariate bicycle codes through circumference-dependent entanglement entropy.

The Gist

The Big Picture: Measuring the "Secret Sauce" of Quantum Matter

Imagine you are trying to figure out how complex a quantum system is by measuring how "entangled" (interconnected) its parts are. In the world of quantum physics, there is a specific measurement called Topological Entanglement Entropy (TEE). Think of TEE as a "complexity score" that tells you if a material has a hidden, long-range order—like a secret code woven into the fabric of space itself.

Usually, this score is reliable. But, the authors of this paper discovered a glitch: sometimes, the measurement gives a false high score. They call this a "spurious" (fake) contribution. It's like a scale that says you weigh 200 pounds when you actually weigh 150, just because you forgot to take off your heavy winter coat.

The paper has two main goals:

1. Fix the scale: They found exactly why the scale is lying and invented a new way to measure that removes the "winter coat" (the fake data).
2. Test the new scale: They used a different type of quantum system to show that the new measurement is sensitive to the shape of the container, revealing hidden "frustration" in the quantum particles.

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Part 1: The "Winter Coat" Problem (Spurious TEE)

The Analogy: The Rectangular Room
Imagine you are trying to count how many people are in a large, crowded room (the quantum system) by looking at three sections: Left (A), Middle (B), and Right (C).

In the past, scientists used a standard rectangular partition to divide the room. They drew straight lines to separate A, B, and C.

• The Problem: In certain quantum systems (called stabilizer codes), the "people" (quantum particles) have special rules. Sometimes, a group of people standing near the corners of the room acts like a single unit, even though they are physically separated by the lines you drew.
• The Glitch: Because the standard rectangular lines cut right through these corner groups, the math gets confused. It thinks these corner groups are "extra" connections that shouldn't be there. This adds a fake number to the complexity score. The paper calls this spurious topological entanglement entropy.

The Solution: The "Concave" Cut
The authors realized that the problem was the shape of the cut.

• The Fix: Instead of drawing straight lines, they proposed drawing a concave shape (like a "C" or a bite taken out of the middle).
• How it works: By bending the boundary of the middle section (B) inward, they create a "nook" that swallows up those tricky corner groups. Now, the groups that were causing confusion are fully inside one section, not split across the lines.
• The Result: When they use this new "concave partition," the fake numbers disappear. The measurement now only counts the real complexity of the system.

The "Recipe" for Success
The paper proves mathematically that this works, but only if the room is big enough. They calculated a specific minimum size (a formula involving the size of the particles and the range of their interactions). If the room is larger than this "worst-case" size, the concave cut is guaranteed to remove all the fake data.

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Part 2: The "Rubber Band" Test (Topological Frustration)

After fixing the measurement, the authors looked at a different setup: an infinite cylinder (like a very long toilet paper roll).

The Analogy: The Rubber Band
Imagine you have a rubber band stretched around a cylinder.

• If the cylinder is very wide, the rubber band fits easily.
• If the cylinder is a specific width, the rubber band might get "stuck" or "frustrated" because it can't close perfectly without twisting.

The Discovery
The authors studied a specific type of quantum code (called bivariate bicycle codes) on this cylinder. They found that the entanglement entropy (the complexity score) changes depending on the circumference (width) of the cylinder.

• The Pattern: The score didn't just go up or down smoothly. It jumped between different levels based on how the width of the cylinder related to the number 12 (specifically, the greatest common divisor of the width and 12).
• What it means: This reveals topological frustration. The quantum particles (anyons) inside the cylinder are "frustrated" because the shape of the cylinder prevents them from arranging themselves in their preferred, smooth pattern. The measurement acts like a sensitive detector that "feels" this frustration.

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Summary of the Claims

1. The Glitch Exists: Standard rectangular measurements of quantum complexity often include fake numbers caused by the geometry of the cut, not the physics of the system.
2. The Fix: Using a concave partition (a bent, bite-shaped cut) eliminates these fake numbers for a wide class of quantum systems (translation-invariant stabilizer codes).
3. The Proof: They proved that if the system is large enough (based on a specific mathematical formula), the concave cut guarantees a "pure" measurement of the system's true topological order.
4. The Side Effect: When measuring these systems on a cylinder, the complexity score becomes highly sensitive to the cylinder's width, acting as a detector for "topological frustration" (particles being unable to settle comfortably due to the shape of the space).

What the paper does NOT claim:

• It does not claim this can be used to build a quantum computer today.
• It does not claim this solves problems in medicine or climate change.
• It does not claim the "concave partition" is the only way to measure these systems, just that it is a rigorous way to remove the specific "spurious" errors found in rectangular cuts.

In short, the authors built a better ruler for measuring quantum complexity, ensuring that what you measure is the real thing, not an artifact of how you drew the lines.

Technical Summary

1. Problem Statement

Topological Entanglement Entropy (TEE) is a fundamental diagnostic for identifying long-range entanglement and intrinsic topological order in 2D gapped quantum phases. It is typically extracted from the subleading constant term ($\gamma$) in the entanglement entropy scaling law $S_A = \alpha |\partial A| - \gamma$.

However, in stabilizer codes (particularly translation-invariant ones), the standard extraction methods often yield spurious topological entanglement entropy (sTEE). These are artificial contributions to $\gamma$ that arise not from intrinsic topological order but from:

• Subsystem symmetries.
• Specific geometric choices of the entanglement partition (e.g., standard rectangular cuts).
• Boundary gauge operators that are not fully captured by the bulk physics.

This spurious contribution leads to an overestimation of the total quantum dimension ($\mathcal{D}$), where the measured $\gamma$ satisfies $\gamma \geq \log \mathcal{D}$, obscuring the true topological nature of the phase. The paper addresses the need for a rigorous method to eliminate these spurious contributions to recover the genuine TEE.

2. Methodology

The authors develop a rigorous algebraic and geometric framework to analyze and eliminate sTEE in translation-invariant stabilizer codes defined on square lattices with prime-dimensional qudits ($\mathbb{Z}_p$).

A. The Concave Partition Strategy

The core methodological innovation is the introduction of a concave partition (Fig. 1b) for the Levin-Wen prescription, replacing the standard rectangular partition (Fig. 1a).

• Levin-Wen Formula: $\gamma = \frac{1}{2} I(A:C|B) = \frac{1}{2}(S_{AB} + S_{BC} - S_{ABC} - S_B)$.
• Geometric Deformation: By deforming the boundary of region $B$ to create a "concave" shape (indenting into region $D$), the authors ensure that specific boundary gauge operators, which cause spurious contributions in rectangular cuts, can be absorbed or canceled out.

B. Algebraic Tools

To prove the efficacy of this partition, the authors employ advanced algebraic techniques:

1. Gröbner Bases: They utilize Gröbner basis theory for modules over polynomial rings ($\mathbb{Z}_p[x, y]$) to analyze the structure of stabilizer generators. This allows them to bound the degree growth of generators when transforming between different bases (e.g., from local generators to bulk stabilizers).
2. Dubé's Theorem: They apply degree bounds derived from Dubé's theorem on Gröbner bases to establish rigorous limits on the spatial support (range) of bulk stabilizers and boundary gauge operators.
3. Anyon String Operators: The analysis relies on the properties of anyons and their creation via semi-infinite string operators. They distinguish between "primary" boundary gauge operators (truncations of bulk stabilizers) and "secondary" boundary gauge operators (which cannot be represented as such truncations).

C. Proof Structure

The proof proceeds by showing that any stabilizer $S$ contributing to sTEE must satisfy an "indivisibility condition" (it cannot be factorized into $S_{AB}S_{BC}$). The authors demonstrate that under the concave partition, any such stabilizer can be decomposed into factorizable parts by:

• Identifying residual support near the corners of the partition.
• Constructing "decorating" stabilizers (using boundary gauge operators) that shift these residuals entirely into region $B$.
• Proving that these decorations are finite in range due to the topological order condition and Gröbner basis bounds.

3. Key Contributions

1. Identification of the Microscopic Origin of sTEE

The paper rigorously identifies that sTEE arises specifically from secondary boundary gauge operators. In a standard rectangular partition, these operators cannot be absorbed into the central region $B$, leading to a non-zero conditional mutual information that mimics topological order. The concave geometry allows these operators to be "absorbed" into $B$, eliminating the spurious term.

2. Theorem 1: Spurious-Free TEE via Concave Partition

The authors prove Theorem 1, which states that for any translation-invariant topological stabilizer code on a square lattice of $\mathbb{Z}_p$ qudits:

• If the linear size $L$ of the concave partition satisfies a specific polynomial bound (Eq. 3):
  $$L \geq 32r^3q^4 + 2r^4 + 27r + 1$$
  (where $r$ is the stabilizer range and $q$ is the number of qudits per site),
• Then the calculated conditional mutual information $\gamma$ is guaranteed to be free of spurious contributions and equals the genuine TEE ($\log \mathcal{D}$).

3. Characterization of Spurious Contributions

They provide a necessary and sufficient condition for sTEE: it occurs if and only if there exist non-trivial secondary boundary gauge operators in the half-planes that cannot be represented as truncated stabilizers. They show that the magnitude of the spurious contribution is directly related to the number of such secondary operators.

4. Application to Quantum LDPC Codes

The paper extends the analysis to bivariate bicycle (BB) codes, a class of high-performance quantum low-density parity-check (qLDPC) codes.

• They study the bipartite entanglement entropy on an infinite cylinder.
• They demonstrate that the entanglement entropy depends sensitively on the cylinder circumference $l$.
• The constant offset in the entropy scales with the dimension of logical operators winding around the cylinder, revealing topological frustration of the underlying anyons. This provides a new entanglement-based signature for topological frustration in qLDPC codes.

4. Results

• Validation on Examples: The authors verify their theory on several models, including the shifted toric code, the 2D cluster state, and the (3,3)-BB code.
  • In the shifted toric code, the rectangular partition yields $\gamma = h+1$ (spurious), while the concave partition correctly yields $\gamma = 1$ (genuine).
  • In the (3,3)-BB code, the rectangular partition gives $\gamma=9$ (spurious), while the concave partition gives $\gamma=8$ (genuine, corresponding to 8 copies of the toric code).
• Quantitative Bounds: The paper provides explicit, albeit conservative, worst-case bounds for the system size required to eliminate sTEE. These bounds are derived from the degree growth of Gröbner bases ($O(r^3 q^4)$).
• Cylinder Entanglement: Numerical results for the BB code on a cylinder show that the entanglement entropy $S_A(l)$ follows linear branches $S_A(l) = 3l - k$, where the offset $k$ is determined by $\gcd(l, 12)$, directly linking entanglement to the logical dimension of the code on a torus.

5. Significance

• Resolution of a Long-Standing Issue: This work resolves the ambiguity in measuring TEE in stabilizer codes, a problem that has hindered the accurate characterization of topological phases in lattice models and quantum error-correcting codes.
• Rigorous Foundation for qLDPC Codes: As quantum LDPC codes (like BB codes) are central to fault-tolerant quantum computing, this paper provides a robust diagnostic tool to distinguish between intrinsic topological order and artifacts of code geometry or boundary conditions.
• New Diagnostic for Topological Frustration: The sensitivity of entanglement entropy to cylinder circumference offers a new probe for studying topological frustration and anyon statistics in systems with periodic boundary conditions.
• Algorithmic Implications: The constructive nature of the proof (using Gröbner bases) suggests algorithmic pathways to compute TEE and identify boundary gauge structures in complex stabilizer models, potentially aiding in the design of better quantum codes.

In summary, the paper establishes that the concave partition is the correct geometric tool for extracting genuine topological entanglement entropy in stabilizer codes, providing a rigorous mathematical proof and practical criteria to avoid the pitfalls of spurious contributions.