Unfolding the Toric Code Model with Emergent Qubits

TL;DR

The paper develops an exact emergent-qubit framework that maps the toric code and related graph-based spin systems to independent emergent qubits using precise unitaries, yielding exact eigenstates and a density-matrix description. It starts with a one-dimensional trestle construction, then extends to arbitrary graphs, and finally unfolds the toric code on cylinder, sheet, and torus, revealing multi-qubit correlations and topological degeneracy encoded in emergent-qubit labels. The authors translate these mappings into concrete quantum circuits composed of Hadamard and CNOT gates, deriving circuit depths that range from sub-extensive on cylinder/sheet to extensive on torus, and provide explicit constructions for state preparation of any toric-code eigenstate (including two missing qubits on the torus that realize the fourfold degeneracy). The work offers a rigorous, executable route to preparing and manipulating toric-code eigenstates on quantum hardware, enabling experimental probes of topological order and emergent-qubit physics on near-term devices.

Abstract

We present the idea of emergent qubits by an exact model construction on a trestle, also generalized to arbitrary graphs. The corresponding eigenstates are quantum paramagnetic, with free multipolar moments. We rigorously transform the toric code model on a torus, cylinder and sheet into emergent qubits, writing all the eigenstates exactly. We devise exact quantum circuits for the toric code and other eigenstates described here. The depth of the circuit for toric code eigenstates on torus grows linearly with the total number of qubits, as compared to the sublinear growth on cylinder or sheet.

Figures (7)

• Figure 1: Model on a closed trestle with three-qubit interactions, Eq. \ref{['eq:model']}. The dark and light gray triangles denote $\hat{Z}_n$ and $\hat{X}_n$ operators, respectively.
• Figure 2: $(a)$ Model with exact emergent qubit eigenstates on arbitrary graph. $(b)$ Interaction operators involving qubits sitting on sites (filled circles) and bonds (empty circles). There are no isolated sites here; every site makes a bond with at least one other site.
• Figure 3: $(a)$ A toric code model on a cylinder; it is called $\hat{H}_3$ in the text. It is periodic along $n_1$ and open along $n_2$. Dark (light) gray squares (triangles) denote $\hat{Z} (\hat{X})$ interactions involving four (three) qubits. $(b)$ A planar toric code, termed $\hat{H}^\prime_3$. It has open boundaries along $n_1$ as well as $n_2$. Besides the four- and three-qubit interactions, it also has two two-qubit interactions of the $\hat{X}$ and $\hat{Z}$ type at the top-left and bottom-right corners shown respectively by a light and a dark gray thick line. The total number of qubits in both cases is $2N_1N_2$.
• Figure 4: $(a)$ The toric code model, $\hat{H}_4$. $(b)$ The transformed toric code, $\hat{U}^\dag_4\hat{H}_4\hat{U}_4$, with $\hat{U}_4$ defined in Eq. \ref{['eq:U4']}. The original interacting qubits (filled and empty circles) of $(a)$ transform into independent emergent qubits (vertical ovals denoting $\hat{\sigma}^x$ and horizontal denoting $\hat{\sigma}^z$) in $(b)$, and two missing qubits (colored circle on the red lines). Four (colored) reference lines at $n_{1(2)}^{\prime}$ and $n_{1(2)}^{\prime\prime}$ required for $\hat{U}_4$ are arbitrary; they mark the positions of two accumulation terms (colored plaquettes) in Eq. \ref{['eq:H4']} and the two missing qubits.
• Figure 5: Quantum circuit for generating the eigenstate, $\left|\{z\},\{x\}\right\rangle$, of the model on trestle. Here $\fbox{\sf H}$ stands for Hadamard gate, and a horizontal line connecting a control qubit ($\bullet$) with a target qubit ($\oplus$) denotes a CNOT gate. Input states of the target qubits are given independently by the quantum numbers $\{z\}$ and those of the control qubits by $\{x\}$. It produces the many-qubit state $\left|\{z\},\{x\}\right\rangle$ of Eq. \ref{['eq:ketzx']}.