Toric code made subsystem: a framework for topological subsystem codes using anticommuting quantum spin liquids

Abstract: We introduce a framework of constructing topological subsystem codes based on the class of anticommuting quantum spin liquids described in [Phys. Rev. B 113, 064402 (2026)]. A canonical model from this class can be considered as a spatial modification of the toric code that voids its stabilizer code property. Rather, these models contain an extensive set of anticommuting local conserved operators that lead to an extensive ground state degeneracy. This degeneracy forms the basis of the subsystem degrees of freedom in the associated quantum error correcting code. The code inherits the many-body topological order of the quantum spin liquid, making it a topological subsystem code. We present two concrete and detailed examples for constructing these codes on a square lattice and a kagome lattice geometry, requiring weight-4 and weight-3 local check operator measurements respectively. In contrast to other subsystem codes, a unique property of these codes is the presence of an extensive number of local gauge qubits that are left undisturbed by the check operators apart from the logical qubits. Our construction provides a template for generating this new category of topological subsystem codes on different lattice or graph geometries, suitable for implementation on various quantum hardware platforms.

• The paper establishes a framework that converts anticommuting Z2 quantum spin liquids into anticommuting charge (ACC) topological subsystem codes, using local anticommuting charges as gauge degrees of freedom and nonlocal winding operators as logical qubits.
• The square-lattice and kagome constructions achieve [L², 2, L] and [3L², 2, L] codes with weight-4 and weight-3 checks, respectively, while offering reduced or comparable qubit overhead relative to existing topological subsystem codes.
• The framework enables flexible corner-sharing geometries and leaves an extensive set of gauge qubits undisturbed, but realistic-noise thresholds, practical uses for passive gauge qubits, and possible weight-2 variants remain open problems.

Overview and motivation

This paper establishes a construction framework that maps a class of anticommuting $\mathbb{Z}_2$ quantum spin liquids (ac-$\mathbb{Z}_2$ QSLs) onto topological subsystem quantum error-correcting codes, which the authors term anticommuting charge (ACC) codes (2606.26226). The central question addressed is whether these gapless spin liquids—whose ground states carry an extensive residual entropy rather than the topologically protected gap underlying toric-code fault tolerance—can serve as viable QEC codes. The answer given is affirmative: ac-$\mathbb{Z}_2$ QSL Hamiltonians with many-body topological order are realizations of topological subsystem codes.

The key structural observation is geometric. The canonical ac-$\mathbb{Z}_2$ model places qubits on vertices of a square lattice with four-spin plaquette terms of the form $$J_x \sum_{\boxed{x}} \prod \sigma^x_i + J_z \sum_{\boxed{z}} \prod \sigma^z_j$$. Unlike the toric code, whose plaquette operators share bonds and mutually commute, here neighboring $\boxed{x}$ and $\boxed{z}$ plaquettes share only a corner, so their Hamiltonian terms anticommute. This single change—from bond-sharing to site-sharing—destroys the stabilizer code property but instead produces an extensive set of anticommuting local conserved operators, an extensive ground state degeneracy, and many-body topological order. The degeneracy supplies the gauge degrees of freedom; the topological order supplies the non-local logical encoding.

Construction recipe

The general template proceeds in four steps:

1. Check operators: identify the minimum-weight local operators commuting with all local anticommuting conserved charges; these coincide naturally with the spin liquid Hamiltonian terms.
2. Code stabilizers: form mutually commuting products of check operators along lattice rows and columns.
3. Logical operators: identify non-local conserved operators winding around the lattice that commute with all checks and stabilizers; anticommuting pairs define logical qubits.
4. Gauge subsystem: the anticommuting checks plus the local conserved charges encode the redundant subsystem.

Because all ingredients derive from a topological spin liquid, the number of logical qubits is fixed by lattice topology by construction, and the framework applies to arbitrary corner-sharing graph geometries—a flexibility the authors contrast with earlier, more rigid topological subsystem code constructions such as Bombin's trivalent-lattice codes and Bravyi et al.'s subsystem surface code.

The weight-4 square lattice ACC code

On an $L \times L$ square lattice with one qubit per vertex ($L^2$ physical qubits), the check operators are weight-4 Pauli products: $C_X$ on $\mathbb{Z}_2$0 plaquettes, $\mathbb{Z}_2$1 on $\mathbb{Z}_2$2 plaquettes, and additional commuting $\mathbb{Z}_2$3 operators on white plaquettes. The $\mathbb{Z}_2$4 set plays a role analogous to toric-code stabilizers, while the anticommuting $\mathbb{Z}_2$5, $\mathbb{Z}_2$6 pairs play the Bacon-Shor-like gauge role. Stabilizers are row/column products $\mathbb{Z}_2$7 of weight $\mathbb{Z}_2$8, together with the local $\mathbb{Z}_2$9 generators; counting yields $\mathbb{Z}_2$0 independent stabilizers.

Two logical qubits are encoded via length-$\mathbb{Z}_2$1 winding string operators $\mathbb{Z}_2$2 and $\mathbb{Z}_2$3, so the code parameters are $\mathbb{Z}_2$4. Two quantitative claims stand out:

• Qubit economy: the physical qubit count is half that of the toric code and one-third that of the subsystem surface code for comparable distance and logical qubit count.
• Extensive undisturbed gauge sector: after stabilizers and logicals fix $\mathbb{Z}_2$5 degrees of freedom, $\mathbb{Z}_2$6 gauge qubits remain, and a subset corresponding directly to the local anticommuting conserved charges is left untouched even by check-operator measurements. This feature is generically absent in standard subsystem codes and constitutes the paper's most distinctive claim.

Error correction proceeds by sequential measurement of $\mathbb{Z}_2$7 then $\mathbb{Z}_2$8 (which do not commute when sharing corners), in parallel with $\mathbb{Z}_2$9. A single-qubit Pauli error is localized exactly via combined $\mathbb{Z}_2$0 and $\mathbb{Z}_2$1 syndromes; notably, correcting by applying the complementary three-qubit plaquette operator implements a conserved charge loop that acts trivially on the code space. Pauli strings are handled with minimum-weight perfect matching, giving distance $\mathbb{Z}_2$2 since strings longer than $\mathbb{Z}_2$3 can be misidentified as their complement.

The weight-3 kagome ACC code

The kagome realization demonstrates geometrical flexibility with lower-weight checks. The spin liquid Hamiltonian has XXX/ZZZ terms on up/down triangles, and hexagonal plaquettes support anticommuting conserved operators $\mathbb{Z}_2$4. Check operators are weight-3 triangle products; stabilizers are row/column products over alternating rows, giving $\mathbb{Z}_2$5 independent generators. With $\mathbb{Z}_2$6 physical qubits and two logical qubits from winding operators, the gauge sector contains $\mathbb{Z}_2$7 qubits, and the code is $\mathbb{Z}_2$8—the same parameters as the subsystem surface code but with weight-3 rather than higher-weight measurements.

Error correction here exhibits a genuinely new structure: the lattice decomposes into three site types with distinct syndrome signatures. Type-0 errors are localized exactly by intersecting row and column stabilizer syndromes; type-1 and type-2 errors are localized only up to a line of same-type sites, but any corrective choice works because two-qubit Pauli operators within such lines commute with all stabilizers and logicals. Even-length error strings on types 1 and 2 are invisible to all syndromes and hence harmless.

Relation to prior work and hardware relevance

Compared to Bombin's trivalent constructions (weight-2 checks) and the subsystem surface code, the ACC framework trades check weight for layout flexibility and reduced qubit count. The authors argue this suits specific hardware: they sketch a kagome-like corner-sharing graph overlaid on IBM Eagle's heavy-hex connectivity, and note Google Willow and Rigetti Cepheus as natural platforms for the square-lattice variant. These are schematic proposals; no device-level noise analysis or experimental demonstration is provided.

Limitations and open questions

Several caveats bear directly on the results. First, the paper does not compute error thresholds or decoding performance under realistic noise models; threshold characterization, including possible benefits of gauge fixing, is explicitly deferred as future work. Second, both examples require at least weight-3 check measurements, whereas Bombin's construction achieves weight-2; whether the ACC template admits weight-2 checks remains open—the authors survey candidate Hamiltonians (Kitaev honeycomb, quantum compass models) and find each falls short in some respect (no encoded logicals, no commuting stabilizers, or non-topological rigid geometry). Third, the undisturbed gauge qubits are passive in this work; whether they can store or process information or improve thresholds is unresolved. Fourth, connections to fracton physics, excitation structure analogous to toric-code $\mathbb{Z}_2$9/$$J_x \sum_{\boxed{x}} \prod \sigma^x_i + J_z \sum_{\boxed{z}} \prod \sigma^z_j$$0 anyons, non-local qLDPC-style generalizations, and measurement-induced phases of these codes are all posed as questions rather than answered.

Conclusion

The paper identifies a systematic correspondence between ac-$$J_x \sum_{\boxed{x}} \prod \sigma^x_i + J_z \sum_{\boxed{z}} \prod \sigma^z_j$$1 quantum spin liquids and topological subsystem codes, grounded in the observation that corner-sharing anticommuting plaquette terms simultaneously generate extensive degeneracy (gauge qubits) and topological order (logical encoding). Detailed square- and kagome-lattice constructions yield $$J_x \sum_{\boxed{x}} \prod \sigma^x_i + J_z \sum_{\boxed{z}} \prod \sigma^z_j$$2 and $$J_x \sum_{\boxed{x}} \prod \sigma^x_i + J_z \sum_{\boxed{z}} \prod \sigma^z_j$$3 codes with weight-4 and weight-3 checks respectively, halving or matching the qubit overhead of existing topological subsystem codes while introducing an extensive set of locally conserved gauge qubits untouched by syndrome measurement. The framework's principal contributions are its geometric flexibility and its template character; its principal gaps are the absence of threshold analysis and of weight-2 check-operator variants, both of which remain concrete open problems.