Quantum Error Correction Codes for Truncated SU(2) Lattice Gauge Theories

TL;DR

The paper develops two quantum error correction codes for pure SU(2) lattice gauge theory with a $j_{ m max}=1/2$ truncation on lattices where each vertex has up to three links. Code I enforces Gauss's law at all vertices via stabilizers, using triple repetition to achieve a $[[9N+3,N,3]]$ code on plaquette chains and extending to 2D honeycomb and 3D lattices, with a logical Hamiltonian that matches known spin-Hamiltonians for gauge-singlet sectors. Code II revises the local vertex encoding to stabilize half-vertices, yielding a $[[4,2,2]]$ per vertex (carbon code when concatenated to $[[12,2,4]]$) and providing a more local error-detection framework, though it permits unphysical states unless supplemented. Periodic boundary conditions introduce topological flux degrees of freedom and corresponding logical operators, increasing the logical-qubit count in a controlled way. Overall, Gauss’s-law-based stabilizers reduce qubit overhead relative to generic codes while preserving gauge structure, with clear paths to higher $j_{ m max}$ and larger-distance codes for future gauge-theory quantum simulations.

Abstract

We construct two quantum error correction codes for pure SU(2) lattice gauge theory in the electric basis truncated at the electric flux $j_{\rm max}=1/2$, which are applicable on quasi-1D plaquette chains, 2D honeycomb and 3D triamond and hyperhoneycomb lattices. The first code converts Gauss's law at each vertex into a stabilizer while the second only uses half vertices and is locally the carbon code. Both codes are able to correct single-qubit errors. The electric and magnetic terms in the SU(2) Hamiltonian are expressed in terms of logical gates in both codes. The logical-gate Hamiltonian in the first code exactly matches the spin Hamiltonian for gauge singlet states found in previous work.

Figures (10)

• Figure 1: A five-plaquette chain. Positions of vertices can be specified by $(x,y)$ as shown. We use the bottom-left corner of a plaquette to specify its position.
• Figure 2: A vertex with three links joint. States in the full SU(2) lattice gauge theory are represented by the three $j$ quantum numbers $\ket{j_1j_2j_3}$. When the $j$ quantum number is truncated at $j_{\rm max}=1/2$, the local state on each link becomes a two-level system and can be represented by one qubit. Then the states associated with the vertex can be represented as $\ket{q_1q_2q_3}$. Physical states at $j_{\rm max}=1/2$ are listed in Table \ref{['tab:1']} for both ways of representing the states.
• Figure 3: A vertex repetition code designed to correct a phase-flip ($Z$) error on any of the nine physical qubits, but only detect a bit-flip ($X$) error, i.e., it is not able to identify the qubit where the $X$-error occurs. The stabilizers are $X_1X_{1'}$, $X_{1'}X_{1"}$, $X_2X_{2'}$, $X_{2'}X_{2"}$, $X_3X_{3'}$, $X_{3'}X_{3"}$ and $Z_1Z_{1'}Z_{1"}Z_2Z_{2'}Z_{2"}Z_3Z_{3'}Z_{3"}$, the last of which corresponds to the local Gauss's law constraint. The syndromes of the $X$-stabilizers for single-qubit $Z$-errors are listed in Table \ref{['tab:2']}.
• Figure 4: Quantum error correction code I for the SU(2) lattice gauge theory on a plaquette chain with aperiodic boundary conditions and $j_{\rm max}=1/2$. Each link is repeated thrice. As mentioned in the caption of Fig. \ref{['fig:qec_at_vertex']}, the code is able to correct any single-qubit $Z$-error. Using the $Z^{\otimes 9}$ stabilizers on neighboring vertices, one is also able to detect on which link a single-qubit $X$-error has happened, and thus able to correct it. It is worth noting that the code can identify the $X$-error link but cannot identify which of the three qubits has the error, but is still able to correct it. In other words, the code is degenerate.
• Figure 5: Encoding circuit in code I for the qubit $q_i$ on a link, where the state can be arbitrary $\ket{\psi}=c_0\ket{0}+c_1\ket{1}$. $H$ in the box stands for the Hadamard gate and the black solid point indicates the control qubit for the CNOT gate. The output state is $c_0\ket{\alpha}_i+c_1\ket{\beta}_i$.