Boundaries and defects in the cubic code

Haah's cubic code is the prototypical type-II fracton topological order. It instantiates the no stringlike operator property that underlies the favorable scaling of its code distance and logical energy barrier. Previously, the cubic code was only explored in translation-invariant systems on infinite and periodic lattices. In these settings, the code distance scales superlinearly with the linear system size, while the number of logical qubits within the degenerate ground space exhibits a complicated functional dependence that undergoes large fluctuations within a linear envelope. Here, we extend the cubic code to systems with open boundary conditions and crystal lattice defects. We characterize the condensation of topological excitations in the vicinity of these boundaries and defects, finding that their inclusion can introduce local stringlike operators and enhance the mobility of otherwise fractonic excitations. Despite this, we use these boundaries and defects to define new encodings where the number of logical qubits scales linearly without fluctuations, and the code distance scales superlinearly, with the linear system size. These include a subsystem encoding with open boundary conditions and a subspace encoding using lattice defects.

Figure 1

Generators of the cubic code stabilizers, comprised of single-qubit Pauli operators. (Left) The $C_X$ operator. (Right) The $C_Z$ operator.

Figure 2

As per Table 3, each single-qubit Pauli operator creates 4 excitations of cubic code stabilizers, arranged in a tetrahedral shape. Transparency is used to show cubes hidden by the 3D perspective.

Figure 3

Number of encoded qubits, $k$, in the periodic cubic code model with linear system size $(L,L,L)$ in $x,y,z$, as per Eq. (6). The value is bounded by $2\le k\le 4L-2$.

Figure 4

Fractal pattern used to expand the separation between a set of 4 $m$ excitations in the bulk, using repeated $XI$ Pauli operators (purple circles). The shaded purple regions are used to highlight repeated applications of the leftmost operator, showing similarities to a Sierpinski pyramid.

Figure 5

The $F$ operators that can cascade excitations through the lattice, as in Fig. 6. (Left) $F_e^{xy}, F_e^{xz}$, and $F_e^{yz}$ operators. The labels indicate the plane and directions along which the excitations can move. (Right) $F_m^{\overline{x}\overline{y}},F_m^{\overline{x}\overline{z}}$, and $F_m^{\overline{y}\overline{z}}$ operators, which move excitations in the negative lattice directions.

Figure 6

The “cascading procedure”: The $F_m^{\overline{y}\overline{z}}$ operator in panel (a) can move excitations in the $-\widehat{z}$ direction. Starting with the excitation pattern in panel (a), we translate and reapply $F_m^{\overline{y}\overline{z}}$ so that the $★$ is aligned with the $★$ in panel (b). In doing so, the stabilizers marked by the black squares are flipped. Repeating this at the $★$ in panel (c) removes all excitations in the second row. We can repeat this process in each row to separate $m$ charges in the $-\widehat{z}$ direction. Larger separations $\mathrm{\Delta }$ create additional excitations and the weight of the operator (number of single-qubit Pauli operators) scales superlinearly with $\mathrm{\Delta }$.

Figure 7

Scaling properties of the $F$ operators as the separation between charges ($\mathrm{\Delta }$) is increased, computed by numerically continuing the process in Fig. 6. This demonstrates a superlinear upper bound on the weight and a near-linear polynomial upper bound on the energy barrier of a logical operator.

Figure 8

Using $F_e^{yz}$ and a new operator, $G$, to move excitations around a closed path (“cage”) in the $xy$ plane. (a) The $F_e^{yz}$ operator that we use to move excitations in the $\widehat{y}$ direction. (b) A new operator, which we call $G$ (see Fig. 25 in the Appendix), that can be used to move excitations in the $\widehat{x}+\widehat{y}$ direction. (c) Starting with $F_e^{yz}$, we move an excitation in the second column into the $\widehat{x}+\widehat{y}$ direction using the tip of the new operator, $G$. Wireframe cubes indicate the annihilated charges from the previous step. (d) We then repeat this, using $G$ to move the second excitation in the second column. (e) We now move all three excitations in the third column into the $\widehat{x}+\widehat{y}$ direction, using $G$. (f) We now use $F_e^{yz}$ to begin to move these charges back in the $-\widehat{y}$ direction. (g) We continue this process to move the remaining charges back in the $-\widehat{y}$ direction. To complete the loop, we would use $G$ once again. This process can be generalized to larger loops, in other planes, and also with $m$ charges.

Figure 9

From top to bottom: Truncated stabilizers on the edges, plaquettes, and vertices of a finite lattice, formed by taking a section of the bulk $C_X,C_Z$ stabilizers.

Figure 10

Positive (left) and negative (right), $X$-type (red) and $Z$-type (blue) boundaries, showing the excitation patterns for each combination of single-qubit Pauli operators.

Figure 11

An $X$-type positive boundary, colored to indicate three types of $m$ charges: $m_A,m_B$ and $m_C$. Each charge is mobile within its particular set of diagonals.

Figure 12

A hexagonal color code, with a qubit at each white circle. Overlayed is a rhomboidal lattice, with vertices corresponding to the lattice points in the boundary layer of the cubic code. As per the inset, at each vertex the adjacent two color code qubits $i,j$ are identified with the two qubits at each cubic code lattice site. To map onto the cubic code stabilizers a CNOT is used on each pair, controlled on qubit $j$.

Figure 13

Numerical calculation of $\tau (L_x,L_{\infty })\equiv \zeta (L_x/3)$ from Eq. (5) for increasing linear system sizes $L_x$ with $L_{\infty }\gg L_x$. There is a clear fractal nature to this behavior.

Figure 14

Anticommutation relations for plaquette operators on the boundary. (Left) When $P_Z$ is applied to the blue square on the positive faces of the lattice, the red squares on the same face correspond to $P_X$ that anticommute. (Right) When $P_X$ is applied to the red square on the negative faces of the lattice, the blue squares correspond to $P_Z$ that anticommute.

Figure 15

Notation used to specify the choice of boundaries, $(abc;def)$, where each letter is $e$ or $m$ (as defined in Table 4 and the $ABC$ subscript is implied), or $p$ for a periodic boundary.

Figure 16

Logical operators on the first tennis ball configuration, $(mem;mee)$. Solid shapes indicate repeated applications of the $F$ operators to cascade $m$ (yellow) and $e$ (cyan) charges. Red faces represent $X$, and blue for $Z$, stabilizer choices on the boundaries.

Figure 17

Two pairs of logical operators on the first tennis ball code $(mem;mee)$, for a linear system size $(11,11,L_z)$, considering a slice of the $xy$ plane at some $z$ far from the (001) and $\left(00\overline{1}\right)$ boundaries. These were calculated and plotted computationally; a slight jitter is applied to each point to distinguish when $XI$ and $IX$ occur on the same lattice point. Note that there are only two such pairs of logical operators that can be cleaned onto just this particular $xy$ plane. If we consider the plane at $z+1$, then we would find an analogous result of two new independent pairs of logical operators; in this way, the tennis ball 1 code has $k=2L_z$ logical pairs.

Figure 18

Edge dislocations on the cubic code contain twists (blue). An operator that moves a fracton around a closed cage (red loop) in the bulk will no longer be closed when encircling a twist. This translation potentially increases the mobility of fractons when near a defect.

Figure 19

Stabilizers for the twists occurring at the ends of edge dislocations. There are four types, corresponding to $T_X$ (left pane) and $T_Z$ (right pane), as well as at the bottom and top of the edge dislocation. The stabilizers were found by considering the commutation with the neighboring bulk $C_X$ and $C_Z$ stabilizers.

Figure 20

The two independent forms of screw dislocations in a ($3+1$)D lattice, such as the cubic code. The screws are labeled by the handedness of the Burgers vector when traversing around the dislocation. On the left is a right-handed screw, $\langle R\rangle$, and on the right is a left-handed screw, $\langle L\rangle$.

Figure 21

Pairs of edge dislocations support superlinear-weight logical operators.

Figure 22

Logical operators on the second tennis ball configuration, $(mee;mem)$. Solid shapes indicate repeated applications of the $F$ operators to cascade $m$ (yellow) and $e$ (cyan) charges. Red faces represent $X$, and blue for $Z$, stabilizer choices on the boundaries.

Figure 23

Two $\langle m\rangle$ vacancies of size $(w_{ix},w_{iy},w_{iz})$ for $i=1,2$ separated by a Manhattan distance $\mathrm{\Delta }$ in the bulk.

Figure 24

Numerical computation of the encoded qubits in a pair of $\langle m\rangle$ vacancies oriented along the $z$ direction, separated by distance $\mathrm{\Delta }$. The vacancy in the negative direction has size $(w_{1x},w_{1y},w_{1z})=(w_x,1,1)$, where $w_x$ is varied in the figure, and the positive vacancy has (1,1,1). We see the relationship in Eq. (B1), with the slight deviations when $w_x=1,4,7$ due to the nonuniformity of composing $F_m$ operators.

Figure 25

The operator $G$, constructed out of $IZ$ and $ZZ$ terms, which moves $e$ excitations along a line. In this representation, creating two $e_A$ in the top layer, for example, will also create an $e_A$ in the second and third layers. Considering the plane indicated by the dashed lines as a 2D grid of just the $A$ diagonal gives the representation on the right, where each square corresponds to a location along that diagonal. This representation is used in Fig. 26.

Figure 26

(a) Repeating the operator in Fig. 25 three times produces this excitation pattern. The resulting operator is named $O_3$. If the first and fourth columns are identified via a periodic boundary, then this commutes with all stabilizers in the first row. (b) Repeating the operator in panel (a) twice produces this excitation pattern, with a period of 6. (c) Applying $O_3$ onto the square indicated by the leftmost red circle annihilates two charges in the second layer, while flipping the squares indicated by the red outline. (d) Applying $O_3$ to each of the 6 charges to the left of the dashed line in panel (b) produces an excitation pattern that is equivalent to (a) but scaled by a factor of 2. This resulting operator is called $O_6$.

Figure 27

Two forms of the boundaries on the 6-6-6 color code: (a) Boundaries that condense a single color, in this case green. (b) Boundaries that condense a single Pauli charge, determined by which stabilizers are removed from the boundary group. Overlayed is the rhomboidal lattice used to construct the correspondence with the cubic code boundaries. Circles on the faces indicate charges, while colored circles on the vertices indicate acting with an $X$ (or equivalently $Z$) Pauli operator.

Figure 28

(a) A boundary on the (100) face of the cubic code, which condenses only one charge type, such as $m_A$. The excitation and coloring shown are equivalent to that in Fig. 27. (b) The two forms of the stabilizers that are placed along the boundary. Note that the two terms anticommute when on the same location, but commute when neighboring in the $x$ direction.