Free-Fermion Subsystem Codes

We consider quantum error-correcting subsystem codes whose gauge generators realize a translation-invariant, free-fermion-solvable spin model. In this setting, errors are suppressed by a Hamiltonian whose terms are the gauge generators of the code and whose exact spectrum and eigenstates can be found via a generalized Jordan-Wigner transformation. Such solutions are characterized by the frustration graph of the Hamiltonian: the graph whose vertices are Hamiltonian terms, which are neighboring if the terms anticommute. We provide methods for embedding a given frustration graph in the anticommutation relations of a spin model and present the first known example of an exactly solvable spin model with a two-dimensional free-fermion description and exact topological qubits. This model can be viewed as a free-fermionized version of the two-dimensional Bacon-Shor code. Using graph-theoretic tools to study the unit cell, we give an efficient algorithm for deciding if a given translation-invariant spin model is solvable, and explicitly construct the solution. Further, we examine the energetics of these exactly solvable models from the graph-theoretic perspective and show that the relevant gaps of the spin model correspond to known graph-theoretic quantities: the skew energy and the median eigenvalue of an oriented graph. Finally, we numerically search for models that have large spectral gaps above the ground-state spin configuration and thus exhibit particularly robust thermal suppression of errors. These results suggest that optimal models will have low dimensionality and odd coordination numbers, and that the primary limit to energetic error suppression is the skew energy difference between different symmetry sectors rather than single-particle excitations of the free fermions.

Popular Summary

A fundamental obstacle to scalable quantum computation is the presence of noise. Error correction is therefore an essential requirement for any quantum computer. A natural solution is to use quantum materials that suppress errors passively, with minimal input from the user. However, determining how an arbitrary quantum many-body system will respond to noise is a difficult problem. In this work, we propose to implement quantum error correction in special materials whose behavior is easy to simulate. Drawing from earlier results for these models, we find specific examples, which are natural to quantum error correction, and we identify the key figures of merit, which are relevant to understanding their properties.

Quantum error-correcting codes are most commonly thought of as quantum spins situated on a lattice. However, the codes we consider admit an alternative description in terms of noninteracting particles called fermions. In this dual picture of free fermions, the behavior of the code can be exactly calculated. We utilize a surprising recent connection between these models and graph, or network, theory to give a framework for constructing and identifying these special codes. We then give novel examples of codes, which exactly encode logical quantum information. Using techniques from condensed-matter theory and graph theory, we analyze the energies of these codes and identify the bottleneck figure of merit for their error-correction behavior.

By showing that there do exist interesting quantum error-correction codes with exactly solvable behavior, we expect our results to inspire a new research direction in quantum error correction.

Figure 1

Translation-invariant comb graph. We denote the translation of a graphical structure by a subscript (e.g., the translation of vertex “1” by one unit cell in the $x$ direction by $1_x$, two unit cells by $1_{x^2}$, and so on).

Figure 2

The terms of the checkerboard-lattice code. (a),(b) The two inequivalent Hamiltonian terms of a fiducial bosonization $H_0$ of the line graph of the square lattice. The line graph is shown in light blue, with brown circles to indicate the vertices. Note that there is no vertex in the middle of the square plaquettes where the horizontal and vertical edges cross. The constituent single-qubit Pauli operators of each term are indicated in color-coded and labeled circles in the center of each edge. For ease of view, we indicate the identity by the absence of a label, and for the two edges that cross, we have displaced the drawn symbols to make it clear which edge they belong to. (c),(d) The two inequivalent terms of a second fiducial bosonization $H_1$ obtained by flipping all single-qubit Paulis. All terms in $H_0$ commute with all terms in $H_1$. (e)–(h) Four additional stabilizer terms per unit cell. Combining both Hamiltonians and the four stabilizers gives a frustration graph consisting of two independent copies of the line graph of the square lattice, and this produces two exact logical qubits, shown in Fig. 3.

Figure 3

Logical degrees of freedom of the checkerboard-lattice code. (a),(b) Stringlike logical $X$ and logical $Z$ operators that form an exact logical qubit and commute with all the Hamiltonian and stabilizer terms shown in Fig. 2. (c),(d) A second stringlike logical pair that commutes with the first. On a torus with even dimensions, these are the only independent operators that commute with all the Hamiltonian and stabilizer terms.

Figure 4

Building blocks of a triangle model. (a) Table showing the three effective qubits $Q_P$, $Q_S$, and $Q_F$ formed from three physical qubits on the edges of a triangular plaquette. Physical-qubit Pauli operators are indicated in color-coded and labeled circles in the center of the each edge. For ease of view, we indicate the identity by the absence of a label. (b),(c) Stabilizers of a Wen plaquette model constructed from the effective qubit $Q_S$. The loop stabilizer is the product $X^{(S)}{Y}^{(S)}{Z}^{(S)}{X}^{(S)}{Y}^{(S)}{Z}^{(S)}$, and the bond stabilizer is $Z^{(S)}{Z}^{(S)}$. Note that the definition of $Z^{(S)}$ is rotated by ${180}^{\circ }$ on the two inequivalent plaquettes of the $ZZ$ bond. (d)–(f) $ZZ$, $YY$, and $XX$ bonds, respectively, of a Kitaev honeycomb model defined using all the translates of $Q_F$. Despite being defined on the same set of physical qubits, every term in the Kitaev honeycomb model in $Q_F$ commutes with every term of the Wen plaquette model in $Q_S$, and all of the Paulis of $Q_P$.

Figure 5

An illustration of our line-graph recognition algorithm for translation-invariant graphs. (Top) The “triangle path graph.” We first perform line-graph recognition on the individual unit cells, partitioning edges into cliques (circled) such that every vertex belongs to exactly two cliques. (Middle) The root graphs of individual unit cells. We next identify vertices according to the connections between unit cells. In this case, we identify $K^{(1)}$ with $K_x^{(2)}$ to recover the clique $K$ shown in the top figure. (Bottom) After identification, we obtain the correct root graph.

Figure 6

Schematic diagram indicating which classes of lattices tend to exhibit large single-particle gaps at half-filling and where certain special examples fall. In general, higher-dimensional lattices tend to have fewer and smaller gaps due to their larger phase spaces. Lattices with odd coordination numbers often have very low densities of states near zero energy, making it much easier to find variations with large gaps or to introduce gaps by perturbations to the hopping coefficients. As a result, it is very difficult to open up a gap at zero energy for cases like the square lattice (2D, and even-coordinated), and significantly easier to do so for graphene (2D, odd-coordinated), whereas the largest single-particle gaps are found in cases like the hourglass ladder (see Fig. 8) and the ladder (see Fig. 9), which are quasi-one-dimensional and odd-coordinated.

Figure 7

Table of known properties for the four main examples in Fig. 6: (a) graphene, (b) the $n=2$, $m=0$ carbon nanotube, (c) the $n=1$, $m=1$ carbon nanotube, and (d) the square lattice. For each graph, the cycle-orientation, plaquette fluxes, and single-particle gap are tabulated for two key orientations: the elementary orientation and the ground state (if known). In the case of graphene, these two orientations are the same, whereas in the three others they are distinct. Both nanotubes have simpler realizations as a ladder or hourglass ladder, and both equivalent versions of the graph are shown. The two orientations of (b),(c) and the accompanying densities of states are shown in Figs. 8 and 9, respectively. Each example embodies the trends shown in Fig. 6 for creation of single-particle gaps, with the quasi-one-dimensional and $3$-coordinated nanotubes exhibiting the largest single-particle gaps.

Figure 8

$n=2$, $m=0$ nanotube. The unoriented version of this nanotube is one graph, which achieves the largest gap possible for any $3$-regular graph [51, 55]. The elementary orientation is shown in (a), with arrows on each edge indicating their orientation. The elementary orientation necessarily has the same DOS (b) as the unoriented graph. Despite the presence of a peak in the DOS at $-3$ and the absence of states in the interval $(-1,0]$, this orientation is not the ground state. (c),(d) The corresponding plots for a lower-energy orientation. This orientation has no eigenvalues near $-3$ and there are states above $-1$, but it does not have a flat band at $-1$ containing a quarter of all the states. As a result, the total energy per particle at half-filling is approximately $0.033$ lower than in the elementary orientation.

Figure 9

$n=1$, $m=1$ nanotube. This nanotube, or ladder, is an example of a graph where the exact ground-state orientation is known. The true ground-state orientation and the corresponding DOS are shown in (a),(b). This orientation has an extremely large single-particle gap: $2{\lambda }_1^{(\tau )}=2$. By contrast, the sector-energy gap is bounded above by approximately $0.068$ per particle. The corresponding plots for the elementary orientation are shown in c and d and exhibit no single-particle gap.