Majorana Loop Stabilizer Codes for Error Mitigation in Fermionic Quantum Simulations

Fermion-to-qubit mappings that preserve geometric locality are especially useful for simulating lattice fermion models (e.g., the Hubbard model) on a quantum computer. They avoid the overhead associated with geometric nonlocal parity terms in mappings such as the Jordan-Wigner transformation and the Bravyi-Kitaev transformation. As a result, they often provide quantum circuits with lower depth and gate complexity. In such encodings, fermionic states are encoded in the common $+1$ eigenspace of a set of stabilizers, akin to stabilizer quantum error-correcting codes. Here, we discuss several known geometric locality-preserving mappings and their abilities to correct and detect single-qubit errors. We introduce a geometric locality-preserving map, whose stabilizers correspond to products of Majorana operators on closed paths of the fermionic hopping graph. We show that our code, which we refer to as the Majorana loop stabilizer code (MLSC) can correct all single-qubit errors on a two-dimensional square lattice, while previous geometric locality-preserving codes can only detect single-qubit errors on the same lattice. Compared to existing codes, the MLSC maps the relevant fermionic operators to lower-weight qubit operators despite having higher code distance. Going beyond lattice models, we demonstrate that the MLSC is compatible with state-of-the-art algorithms for simulating quantum chemistry, and can offer those simulations the same error-correction properties without additional asymptotic overhead. These properties make the MLSC a promising candidate for error-mitigated quantum simulations of fermions on near-term devices.

Figure 1

BKSF on a graph: the vertices (blue circles) of the graph represent fermionic modes; fermions can hop between neighboring vertices. One qubit (orange ellipses) is introduced for each edge of the graph. The fermionic state is mapped to a subspace of the qubits.

Figure 2

BKSF on a 2D lattice with a specific ordering of the incident edges. This code can detect all single-qubit errors.

Figure 3

Plaquette stabilizer for the BKSF code given the vertex and edge ordering in Fig. 2, up to an overall minus sign. From the chosen ordering, the above figure specifies the stabilizer that results from any generating plaquette in the graph. In the case of the infinite or periodic lattice, it is uniform. In the case of a finite lattice, the structure stays roughly the same; however, the dangling $Z$ operators are removed in the case that there is no edge in the graph, and the adjacent edges may change from $X$ to $Y$.

Figure 4

Structure of the code: each circle represents a vertex in the 2D lattice; its color denotes one of the four types of vertex. Similarly, the color of a plaquette indicates the type of the corresponding stabilizer.

Figure 5

(a) Logical operators associated with the red plaquette in Fig. 4. Similar to the BKSF, there is one physical qubit situated on each edge of the lattice, and $X$, $Y$, and $Z$ denote the corresponding Pauli operators. Each vertex operator $\eta$ is mapped to a product of three Pauli-$Z$ operators linked by a curved dashed line. Each edge operator $\xi$ is mapped to a product of single-qubit Pauli operators linked by a solid or dotted line, where the unique Pauli-$X$ operator acts on the corresponding edge of $\xi$. The two rightmost (leftmost) Pauli-$X$ operators are shared by the vertices $n$ and $q$ ($m$ and $p$) and the their right (left) neighbors, and similarly for the two top (bottom) Pauli-$X$ operators. This plaquette has a rotational symmetry of order $4$ (unchanged by a rotation of angle $\pi /2$). (b) Stabilizer operator $S_{mnqp}$ associated with the plaquette up to a minus sign.

Figure 6

(a) Logical operators of the green plaquette consisting of the vertices $l$, $m$, $o$, and $p$. This plaquette has a rotational symmetry of order $2$ (unchanged by a rotation of an angle $\pi$). (b) Stabilizer $S_{lmpo}$ of the plaquette up to a minus sign.

Figure 7

(a) Logical operators of the yellow plaquette consisting of the vertices $i$, $j$, $l$, $m$. (b) Corresponding stabilizer $S_{ijml}$ up to a minus sign.

Figure 8

(a) Logical operators of the blue plaquette consisting of vertices $j$, $k$, $m$, and $n$. These logical operators are identical to those in Fig. 7 by a $\pi$ rotation. (b) Corresponding stabilizer $S_{jknm}$ up to a minus sign, which is identical to the one in Fig. 7 by a $\pi$ rotation.

Figure 9

Stabilizers on the plaquettes for syndrome detection.

Figure 10

MLSC with open-boundary conditions: each vertex represents one fermionic mode (16 in total), each edge (solid black) in the bulk of the code is associated with a logical edge operator (24 in total), and each added dangling edge (dashed red) introduces an additional physical qubit at the boundaries (16 in total).

Figure 11

Edge operator associated with the dangling edge (dashed red) incident on vertex $j$.

Figure 12

Stabilizer $B_0$ associated with the green plaquette on the upper-left corner of the lattice.

Figure 13

Stabilizer $B_1$ associated with the red plaquette on the top of the lattice.

Figure 14

Stabilizer $B_2$ associated with the green plaquette on the top of the lattice.

Figure 15

Assign values to $z_{jk}$.