Antipodal Toric Code (ATC) State

• The antipodal toric code (ATC) state is a quantum state defined on a thin torus, where local stabilizers map to nonlocal antipodal Pauli strings.
• It exhibits dual entanglement behavior, with volume-law scaling in one context and complete separability in another, revealing its adaptable quantum nature.
• A 2-body parent Hamiltonian, constructed via factorizability of toric code stabilizers, underpins its identification as a nonthermal quantum many-body scar.

The antipodal toric code (ATC) state is a stabilizer quantum many-body scar constructed as a specific eigenstate of local Hamiltonians derived from the toric code model on a thin torus geometry. It exhibits either maximal volume-law entanglement or complete separability, depending on the context and parametrization, and is characterized by its mapping, stabilizer structure, parent Hamiltonian construction, entanglement scaling, and nontrivial circuit complexity. The ATC state is central to current research on quantum scars and topological phases, notably distinguished by its role in demonstrating ergodicity-breaking phenomena and explicit construction via factorizability of Pauli strings.

1. Lattice Architecture and Antipodal Mapping

The ATC state is defined on a square lattice of size $N_x \times N_y$ with periodic boundary conditions, forming a discrete torus. For the canonical ATC construction, a "thin torus" case is selected with $N_y = 2$ and $N_x$ odd. Qubits are positioned on the edges of the lattice. In the standard toric code, four-body vertex ($A_s$) and plaquette ($B_p$) operators are associated locally with the sites and plaquettes of the ladder, enforcing stabilizer constraints.

To construct the ATC state, the $2 \times N_x$ ladder is mapped onto a one-dimensional chain of $N = 2N_x$ sites via the transformation

$$n = x + y N_x, \quad x \in \{0, \dots, N_x - 1\},\ y \in \{0, 1\},$$

and the chain is closed with periodic boundary conditions ($n \equiv n + 2N_x$), generating a circular topology. Under this mapping, the originally local stabilizers transform into highly nonlocal, "antipodal" Pauli string operators wrapping around the circle (Dooley, 15 Jan 2026).

2. Stabilizer Group Construction and Projector Formalism

The ATC state $\lvert \mathrm{ATC} \rangle$ is uniquely defined as the common $+1$ eigenstate of all toric-code vertex and plaquette stabilizers on the thin torus, with fixed noncontractible Wilson-loop eigenvalues ($W_1 = W_2 = +1$). The explicit projector is

$$P_{\mathrm{ATC}} = \prod_{s}\frac{1 + A_s}{2} \times \prod_{p}\frac{1 + B_p}{2},$$

and the normalized state is

$$\lvert \mathrm{ATC} \rangle = \frac{P_{\mathrm{ATC}}\lvert 0 \rangle^{\otimes N}}{\Vert P_{\mathrm{ATC}}\lvert 0 \rangle^{\otimes N} \Vert}.$$

In chain coordinates, the stabilizer generators manifest as:

• Vertex operators (for $(x, y)$ with $x + y$ even):

$$A_{n} = \hat{\sigma}^X_{n} \hat{\sigma}^X_{n+1} \hat{\sigma}^X_{n+N_x} \hat{\sigma}^X_{n+N_x+1}$$

• Plaquette operators (for $(x, y)$ with $x + y$ odd):

$$B_{n} = \hat{\sigma}^Y_{n} \hat{\sigma}^Y_{n+1} \hat{\sigma}^Y_{n+N_x} \hat{\sigma}^Y_{n+N_x+1}$$

• Wilson-loop operators:

$$W_1 = \prod_{n=0}^{N_x-1}\hat{\sigma}^Z_n,\quad W_2 = \hat{\sigma}^Z_{n_0} \hat{\sigma}^Z_{n_0+N_x}$$

Fixing $W_1 = W_2 = +1$ designates a specific ATC ground-state sector (Dooley, 15 Jan 2026).

3. Factorizability and Two-Body Parent Hamiltonian

A central methodological advance is the explicit construction of a 2-body, 2-local parent Hamiltonian that annihilates the ATC state. This exploits the factorizability of the toric code stabilizers: if a stabilizer $P \in \mathcal{S}$ can be decomposed into two $\ell$-local, $b$-body factors ($P = P^{(1)}P^{(2)}$ with $\ell = b = 2$), then combinations of such factors serve as annihilators of the ATC eigenstate.

For the ATC state, the factorizations are:

• Vertex example:

$$A_n = (\hat{\sigma}^X_n\,\hat{\sigma}^X_{n+1}) \cdot (\hat{\sigma}^X_{n+N_x}\,\hat{\sigma}^X_{n+N_x+1}),$$

and also

$$A_n = (\hat{\sigma}^X_n\,\hat{\sigma}^X_{n+N_x}) \cdot (\hat{\sigma}^X_{n+1}\,\hat{\sigma}^X_{n+N_x+1}).$$

The resulting parent Hamiltonian for range-$\ell$ is

$$H_\ell^{\mathrm{ATC}} = \sum_{n=0}^{N_x-1} \sum_{\mu = X,Y,Z} J_\ell^\mu \left( \hat{\sigma}_n^\mu \hat{\sigma}_{n+\ell}^\mu - \hat{\sigma}_{n+N_x}^\mu \hat{\sigma}_{n+N_x+\ell}^\mu \right),$$

with arbitrary real couplings $J_\ell^\mu$. By summing over all distances $\ell = 1, \dots, N/2$, the full Hamiltonian is

$$H^{\rm ATC} = \sum_{\ell=1}^{N/2} H_\ell^{\rm ATC}.$$

This Hamiltonian is manifestly 2-local, 2-body, and annihilates the ATC state (Dooley, 15 Jan 2026).

4. Entanglement Structure and Scaling

The entanglement properties of the ATC state exhibit a duality dependent on both its stabilizer structure and its location on the Bloch sphere of the toric code manifold.

• On the thin torus (as stabilizer scar): Bipartitioning the $2N_x$ chain into equal halves $A$ and $B$, stabilizer subgroup-counting finds only four stabilizers are supported locally, while the remaining $2^{N-2}$ generate nonlocal "cut" correlations. The entanglement entropy then scales as

$S(A) = \frac{1}{2}(N-2)\ln 2 \propto |A|.$

This is volume-law scaling, placing the ATC state among the rare instances of volume-law entangled stabilizer scars (Dooley, 15 Jan 2026).

• On the Bloch sphere (as the antipode of the toric code ground state): The ATC state defined as $|ATC\rangle = |0\rangle^{\otimes n}$ is fully separable. Bipartite entropies $S_q(A)$ vanish identically for all subsystems, indicating zero entanglement for all Rényi indices (Liss et al., 2018). This distinction highlights sensitivity to the chosen lattice geometry and embedding.

5. Topological Signatures and Grover Connectivity

The ATC state as the product state $|0\rangle^{\otimes n}$ carries no topological order. Evaluated via the standard prescription for topological entanglement entropy,

$S_2(A) = \alpha |\partial A| + S_\gamma + o(1),$

the ATC point yields $S_\gamma = 0$, in contrast to the toric code ground state's $S_\gamma = -1$. Thus, the ATC state does not reside in the topological phase (Liss et al., 2018).

On the Bloch sphere spanned by the toric code ground state $|G\rangle$ and its closest product state $|P\rangle$, unitary Grover-type rotations connect the two poles:

• The Oracle $O$ and the mean-reflector $R$ yield a kernel $K = RO$.
• $O(\sqrt{|G|})$ steps bring one from $|G\rangle$ to $|P\rangle$ via repeated applications of $K$.
• As each step uses global projectors, the transformation path requires nonlocal operations, indicating substantial circuit complexity; a plausible implication is that preparing $|G\rangle$ from $|P\rangle$ is "hard" in the sense of nonlocality and circuit depth (Liss et al., 2018).

6. Numerical Spectroscopy and Scar Character

Exact diagonalization studies on $H^{\rm ATC}$ for systems up to $N=14$ sites establish that the ATC state sits at exactly zero energy, usually in the spectrum's center. The zero-energy subspace consists of four degenerate states indexed by Wilson-loop eigenvalues and uniformly exhibiting maximal volume-law entropy and zero stabilizer "magic." Removing symmetry sectors, level-spacing statistics match the GOE distribution, plainly distinguishing $H^{\rm ATC}$ as nonintegrable and confirming the ATC state as a nonthermal quantum many-body scar (Dooley, 15 Jan 2026).

7. Contextual Significance and Variants

The ATC state's construction and properties are integral to contemporary investigations into quantum many-body scars, weak ergodicity breaking, and stabilizer Hamiltonian engineering. Its highly entangled version is an archetype for volume-law entangled stabilizer eigenstates that remain exactly solvable. The dual realization (full separability or volume-law entanglement) depending on geometry and projective embedding exposes rich connections between topological coding, entanglement complexity, and circuit nonlocality. This suggests ongoing research directions in Hamiltonian design, computational complexity in topological systems, and connections to Grover-type global rotations (Dooley, 15 Jan 2026, Liss et al., 2018).