Error-correcting codes for fermionic quantum simulation

TL;DR

The symplectic automorphisms of the Pauli module over the Laurent polynomial ring enables us to systematically increase the code distances of stabilizer codes while fixing the rate between encoded logical fermions and physical qubits.

Abstract

Utilizing the framework of $\mathbb{Z}_2$ lattice gauge theories in the context of Pauli stabilizer codes, we present methodologies for simulating fermions via qubit systems on a two-dimensional square lattice. We investigate the symplectic automorphisms of the Pauli module over the Laurent polynomial ring. This enables us to systematically increase the code distances of stabilizer codes while fixing the rate between encoded logical fermions and physical qubits. We identify a family of stabilizer codes suitable for fermion simulation, achieving code distances of $d=2,3,4,5,6,7$, allowing correction of any $\lfloor \frac{d-1}{2} \rfloor$-qubit error. In contrast to the traditional code concatenation approach, our method can increase the code distances without decreasing the (fermionic) code rate. In particular, we explicitly show all stabilizers and logical operators for codes with code distances of $d=3,4,5$. We provide syndromes for all Pauli errors and invent a syndrome-matching algorithm to compute code distances numerically.

Figures (7)

• Figure 1: Bosonization on a square lattice CKR18. We put Pauli matrices $X_e$, $Y_e$, and $Z_e$ on each edge and one complex fermion $c_f, c_f^{\dagger}$ on each face. We will work in the Majorana basis $\gamma_f = c_f + c_f^\dagger$ and $\gamma'_f = -i (c_f - c_f^\dagger)$ for convenience.
• Figure 2: Syndromes of single-qubit errors for the bosonization with code distance $d=3$. The red vertices $v$ represent locations where the single-qubit error does not commute with the stabilizer $G_v$.
• Figure 3: Syndromes of single-qubit errors for the bosonization with code distance $d=4$. The red vertices $v$ represent locations where the single-qubit error does not commute with the stabilizer $G^{d=4}_v$.
• Figure 4: Syndromes of single-qubit errors for the $d=5$ bosonization.
• Figure 5: Examples of polynomial expressions for Pauli strings. The flux term (i.e., fermionic occupation) on a plaquette and the hopping term on an edge are both shown. The factors such as $x^2 y^2$ and $x^2$ represent the locations of the operators relative to the origin.