Topological Order, Quantum Codes, and Quantum Computation on Fractal Geometries

We investigate topological order on fractal geometries embedded in $n$ dimensions. We consider the $n$-dimensional lattice with holes at all length scales the corresponding fractal (Hausdorff) dimension of which is $D_H=n-\delta$. In particular, we diagnose the existence of the topological order through the lens of quantum information and geometry, i.e., via its equivalence to a quantum error-correcting code with a macroscopic code distance or the presence of macroscopic systoles in systolic geometry. We first prove a no-go theorem that ${\mathbb{Z}}_N$ topological order cannot survive on any fractal embedded in two spatial dimensions and with $D_H=2-\delta$. For fractal-lattice models embedded in three dimensions (3D) or higher spatial dimensions, ${\mathbb{Z}}_N$ topological order survives if the boundaries on the holes condense only loop or, more generally, $k$-dimensional membrane excitations ($k\ge 2$), thus predicting the existence of fractal topological quantum memories (at zero temperature) or topological codes that are embeddable in 3D. Moreover, for a class of models that contain only loop or membrane excitations and are hence self-correcting on an $n$-dimensional manifold, we prove that ${\mathbb{Z}}_N$ topological order survives on a large class of fractal geometries independent of the type of hole boundary and is hence extremely robust. We further construct fault-tolerant logical gates in the ${\mathbb{Z}}_2$ version of these fractal models, which we term fractal surface codes, using their connection to global and higher-form topological symmetries equivalent to sweeping the corresponding gapped domain walls. In particular, we discover a logical controlled-controlled-Z (ccz) gate corresponding to a global symmetry in a class of fractal codes embedded in 3D with Hausdorff dimension asymptotically approaching $D_H=2+ϵ$ for arbitrarily small $ϵ$, which hence only requires a space overhead $\mathrm{\Omega }(d^{2+ϵ})$, where $d$ is the code distance. This in turn leads to the surprising discovery of certain exotic gapped boundaries that only condense the combination of loop excitations and certain gapped domain walls. We further obtain logical $C^{p}Z$ gates with $p\le n-1$ on fractal codes embedded in $n$ dimensions. In particular, for the logical $C^{n-1}Z$ in the $n\mathrm{th}$ level of the Clifford hierarchy, we can reduce the space overhead to $\mathrm{\Omega }(d^{n-1+ϵ})$. On the mathematical side, our findings in this paper also lead to the discovery of macroscopic relative systoles in a class of fractal geometries.

Popular Summary

Fractals are geometric objects the characteristic dimension of which can be noninteger. This work investigates embedding topological order and quantum error-correcting codes into such objects. The motivation behind such embeddings is twofold: understanding the geometric aspects of topological order and reducing the physical footprint of the associated quantum codes. The resulting codes give rise to the first fractal topological quantum memory which can exist in nature and may significantly reduce the overhead required for universal quantum computation.

Figure 1

A summary of the motivation and scope of this paper and the correspondence among the three fields at the interface of fractal geometry. Items with the same color indicate the equivalence of concepts in different fields.

Figure 2

The 1-systole on a torus ${\mathrm{sys}}_1(T^2)=d$ is equivalent to the code distance of a 2D quantum error-correcting code or topological order defined on the torus, such as the toric code model.

Figure 3

The recursive definition of the Sierpiński carpet $SC(3,1)$. In the $l\mathrm{th}$ iteration, we punch level-$l$ holes with linear size $1/3^l$ and obtain the Sierpiński carpet at level $l$. One approaches the Sierpiński carpet as $l\to \mathrm{\infty }$.

Figure 4

(a) The surface code and two types of gapped boundaries: the horizontal (vertical) $e$-boundaries ($m$-boundaries) condense $e$-anyons ($m$-anyons) such that the logical $Z$-string ($X$-string) can terminate on them. (b) Two types of holes. The $e$-hole ($m$-hole) has $e$-boundaries ($m$-boundaries), which trap a logical $X$-loop ($Z$-loop) around it, and allows a logical $Z$-string ($X$-string) to terminate on it.

Figure 5

(a)–(d) The recursive definition of the fractal models with the external boundaries of the surface code and $m$-holes inside. (e)–(h) The logical string operators in each iteration. (f)–(h) Any representative of the macroscopic logical $Z$-string, such as $\overline{Z}$ illustrated in the figure, always intersects with a short logical $X$-string, which will have $O(1)$ length in the final iteration, as illustrated in (h) in this example. This leads to an $O(1)$ code distance.

Figure 6

(a) The exponential suppression of the ground-state degeneracy under a perturbation in the Hamiltonian of the conventional surface code defined on $SC(3,1,l=0)$. (b) On the Sierpiński-carpet model defined on $SC(3,1,l\to \mathrm{\infty })$, a perturbation in the Hamiltonian can lead to the splitting of the extensive ground-state degeneracy at a scale comparable to the gap.

Figure 7

(a)–(c) Three fractal models with only $e$-holes, both $e$- and $m$-holes, and holes with alternating $e$- and $m$-boundaries, respectively. In all three cases, a macroscopic logical string $\overline{X}$ or $\overline{Z}$ is intersected by a short dual logical string with $O(1)$ length, leading to an $O(1)$ code distance.

Figure 8

An illustration for the proof of Theorem 1 with a generic random fractal embedded in ${\mathbb{R}}_2$. (a) The holes are colored in gray and, in general, have possibly alternating types of boundary. The holes can also contain “islands” (white) within them, which can simply be ignored. A pair of macroscopic logical strings $\overline{X}$ and $\overline{Z}$ intersect at a single point $p$. (b) The different scenarios within the $O(1)$ radius from the intersection point $p$ with different configurations of hole boundaries.

Figure 9

The six-body vertex operators and three types of four-body plaquette stabilizers lying in the $y$-$z$, $x$-$y$, and $x$-$z$ planes, respectively, in the 3D toric code model defined on a cubic lattice.

Figure 10

An illustration of the ${\mathbb{Z}}_2$ fusion rules of the $e$-particles and $m$-strings, respectively. (a) A pair of $e$-particles can be created out of the vacuum $\mathbb{I}$ via a Wilson line $W^e$ in between, which corresponds to a $Z$-string operator $Z^{\otimes }$ in the 3D toric code model. (b) A pair of $m$-strings forming an $m$-loop can be created out of the vacuum $\mathbb{I}$ via a Wilson sheet $W^m$ in between, which corresponds to an $X$-brane operator $X^{\otimes }$ in the toric code.

Figure 11

The lattice realization of Fig. 10 with the 3D toric code model. A pair of $e$-particle excitations (red) are created from the ground state by a $Z$-string operator (four Pauli $Z$ operators on the red edges) and the vertex stabilizers occupied by the $e$-particles are violated. An m-loop excitation (blue) is created from the ground state by an $X$-brane operator (four Pauli $X$ operators on the blue edges) and the plaquettes (blue) penetrated by the $m$-loop are violated.

Figure 12

An illustration of the TQFT relations Eqs. ((9)–(14)) with the 3D toric code model of Eq. (6). (a) The bending of a $Z$-string by multiplying two plaquette $Z$ stabilizers (orange) corresponding to Eq. (9). The dashed lines indicate the previous $Z$-string location before bending. (b) A closed $Z$-loop can be shrunk to a logical identity by multiplying three plaquette $Z$ stabilizers (orange) enclosed by the loop corresponding to Eq. (11). (c) The recoupling of two $Z$-strings via multiplying a plaquette Z stabilizer in between corresponding to Eq. (13). (d) Local bending of the $X$-brane by multiplying four vertex $X$ stabilizers (blue hexagons) corresponding to Eq. (14). (e) A closed $X$-brane topologically equivalent to a sphere can be shrunk into a logical identity by multiplying four vertex $X$ stabilizers corresponding to Eq. (12). (f) The recoupling of two $X$-branes by multiplying vertex $X$ stabilizers corresponding to Eq. (14).

Figure 13

(a) An $e$-particle excitation can condense on the $e$-boundary when applying a Wilson-line operator $W^e$, which corresponds to a $Z$-string operator $Z^{\otimes }$ in the 3D toric code. (b) An $m$-string excitation can condense on the $m$-boundary when applying a Wilson-sheet operator $W^m$, which corresponds to an $X$-brane operator $X\otimes$ in the 3D toric code.

Figure 14

The logical $Z$-string and the logical $X$-brane operators are illustrated on (a) the discrete lattice and (b) the continuous TQFT pictures, respectively. They intersect on a single edge (purple) in the lattice picture or at a single point (yellow cross) in the TQFT picture. The dangling edges on the top and bottom faces (red) in (b) symbolize the rough boundaries and are sometimes omitted in the later figures for simplicity.

Figure 15

A 3D fractal surface code defined on the fractal-cube geometry $FC(3,1)$ (at levels $l=1$, 2, and 3). The illustrated model contains only $m$-holes (blue). The outer boundaries are the same as those of the 3D surface codes, with the top and bottom faces being $e$-boundaries, while the rest are $m$-boundaries.

Figure 16

A 3D fractal model defined on the fractal-cube geometry $FC(3,1)$ with only $e$-holes (red). A pair of macroscopic logical operators $\overline{X}$ and $\overline{Z}$ intersects at a single point (yellow cross). The macroscopic logical $\overline{X}$ is intersected by an $O(1)$-length dual logical string ${\overline{Z}}_i$, leading to an $O(1)$ code distance.

Figure 17

(a) A cubic hole is topologically equivalent to a 3-ball $D^3$ with its boundary being the 2-sphere $S^2$. (b) An $e$-hole can trap a logical $X$-brane and let a logical $Z$-string terminate on it due to condensation of $e$-anyons on its $e$-boundary. A pair of $e$-holes can encode a single logical qubit, similar to the situation of the 2D surface code in Fig. 4.

Figure 18

(a) A pair of macroscopic logical-operator representatives on the 3D fractal surface code defined on the fractal-cube geometry $FC(3,1)$ with $m$-holes: the logical string $\overline{Z}$ and the logical brane $\overline{X}$. (b) The minimal area representative of the logical brane $\overline{X}$ is interrupted by the $m$-holes and hence has the shape of a Sierpiński carpet with area scaling $O(L^{D_H})$, where $D_H=1.893$ is the Hausdorff dimension of the 2D Sierpiński carpet.

Figure 19

An illustration for Proof 3a. (a) A local $X$-brane connecting two $m$-holes. (b) The equivalence of the $m$-hole to a 3-ball $D^3$ with the boundary being a 2-sphere $S^2$. (c) The equivalence of the $X$-brane to a cylinder $S^1\times D^1$. (d) The filled (solid) cylinder is equivalent to a 3-ball $D^3$. (e) Gluing two 3-balls with a solid cylinder in between is equivalent to a solid cylinder, while its boundary is equivalent to a connected sum of three 2-spheres. Exclusion of the two $m$-boundaries leads to a 2-punctured sphere with two disks $D^2$ being removed, i.e., $S^2\setminus (D^2\cup D^2)$. (f) The intersection of the macroscopic logical string $\overline{Z}$ and the $X$-brane is equivalent to the intersection of the string with a 2-punctured sphere. The algebraic intersection is always zero since the string enters and leaves the interior of the punctured sphere the same number of times.

Figure 20

(a) The generic situation considered in Proofs 3a and 3b, where a local $X$-brane operator connects multiple $m$-holes and can be decomposed as a tensor product of several connected components. (b) The macroscopic logical string $\overline{Z}$ has zero algebraic intersection with the local $X$-brane, since the string enters and leaves the interior of the punctured closed 2-manifolds formed by the $X$-brane operator the same number of times.

Figure 21

(a)–(c) The iterative construction of a family of Sierpiński carpets $SC(p,q)$, shown via an illustration of $SC(6,4)$ and Hausdorff dimension $D_H=1.672$. (d)–(f) The iterative construction of a family of 3D fractal surface codes defined on a fractal-cube geometry $FC(p,q)$ and with $m$-holes inside, shown via an illustration of $FC(6,4)$ and Hausdorff dimension $D_H=2.804$.

Figure 22

The Hausdorff dimensions $D_H$ of a subfamily of fractal-cube geometries $FC(p,p-2)$.

Figure 23

The arrangement of a stack of three 3D surface codes for the application of a transversal logical ccz gate. (a) The pairs of $e$-boundaries are chosen to be perpendicular to the $z$, $x$, and $y$ directions, respectively. (b) The three copies are stacked together such that the corresponding qubits acted on by the transversal ccz gate in each copy are aligned with each other. (c) The alignment of the three pairs of logical operators. All logical $Z$-strings are perpendicular to each other and the same is true for the logical $X$-branes.

Figure 24

An illustration of the correspondence between a transversal logical gate (or a local constant-depth circuit) $U$ and domain-wall sweeping in 2D. The application of $U_{\mathcal{R}}$ in a region $\mathcal{R}$ is equivalent to sweeping the corresponding gapped domain wall $w$ to the boundary of the region $\mathcal{R}$. When passing through the codimension-1 domain wall $w$ across the entire system, the domain wall condenses on the external boundaries ${\mathcal{B}}_i$ and one effectively applies the global symmetry $U$ to the entire system corresponding to the transversal logical gate (or constant-depth circuit). The external boundaries ${\mathcal{B}}_i$ are required to be invariant under the action of $U$ according to Eq. (47).

Figure 25

(a) Both the particle and string (loop) excitations get transformed when passing the codimension-1 domain wall (green) in 3D. (b) A particle excitation cannot be transformed by the codimension-2 domain wall (pink) in 3D, since the particle can generically avoid such a wall. The string (loop) gets transformed by the wall only at the intersection point between the string and the wall, where a new particle (purple) is generated.

Figure 26

(a) Sweeping the cz domain wall $s_{1,2}^{(2)}$ (pink) across a noncontractible codimension-1 submanifold ${\mathcal{M}}^2$ (gray) gives rise to a transversal logical gate ${\overline{\mathrm{CZ}}}_{1,2}$ applied along this submanifold. When passing the $m_1$-string (loop) in copy 1 across the wall $s_{1,2}^{(2)}$, a new particle $e_2$ (orange) in copy 2 is generated at the intersection point. Therefore, a logical brane ${\overline{X}}_1$ is transformed into the product of itself and an additional string $Z_2^{\otimes }$ (yellow), which becomes the logical string ${\overline{Z}}_2$ when the sweeping is completed, as illustrated in the lower panel. (b) Sweeping the codimension-1 ccz domain wall $s_{1,2,3}^{(3)}$ across the system gives rise to the transversal logical gate ${\overline{\mathrm{CCZ}}}_{1,2,3}$. An $m_1$-string (loop) gets transformed to an $m_1{s}_{2,3}^{(2)}$-string (loop) when passing across the wall. Therefore, a logical brane ${\overline{X}}_1$ is transformed into the product of itself and an additional brane ${\mathrm{CZ}}_{2,3}^{\otimes }$ (pink), which becomes the logical brane ${\overline{\mathrm{CZ}}}_{2,3}$ when the sweeping is completed.

Figure 27

The existence of transversal logical cz and ccz gates in a stack of three 3D surface codes is ensured by the condensation of their corresponding domain walls $S_{a,c}^{(2)}$ and $S_{1,2,3}^{(3)}$ on the boundary $(m_a,m_b,e_c)$.

Figure 28

(a) The application of a transversal ccz effectively sweeps the domain wall $s_{1,2,3}^{(3)}$ across the system and attaches it to the $(m_1,m_2,m_3)$-boundary without being condensed, generating an exotic $(m_1{s}_{2,3}^{(2)},m_2{s}_{3,1}^{(2)},m_3{s}_{1,2}^{(2)})$-boundary (green). (b) The $e$-particle and $m$-string alone cannot condense on the $(m_1{s}_{2,3}^{(2)},m_2{s}_{3,1}^{(2)},m_3{s}_{1,2}^{(2)})$-boundary (shown on the right), which can be considered as a composite of the $s_{1,2,3}^{(3)}$-wall and the $(m_1,m_2,m_3)$-boundary (shown on the left). The $e_a$-particle and $m_a$-string get transformed into $e_a$ and $m_a{s}_{b,c}^2$ when passing the domain wall first and hence cannot condense on the remaining $(m_1,m_2,m_3)$-boundary. (c) The $m_a{s}_{b,c}^{(2)}$-string can condense on the exotic $(m_1{s}_{2,3}^{(2)},m_2{s}_{3,1}^{(2)},m_3{s}_{1,2}^{(2)})$-boundary, since it gets transformed to an $m_a$-string when passing the $S_{1,2,3}^3$-wall and can then condense on the remaining $(m_1,m_2,m_3)$-boundary. (d) The product of logical branes ${\overline{X}}_a{\overline{\mathrm{CZ}}}_{b,c}$ can terminate on the $(m_1{s}_{2,3}^{(2)},m_2{s}_{3,1}^{(2)},m_3{s}_{1,2}^{(2)})$-boundary, since it gets transformed to ${\overline{X}}_a$ when passing the $S_{1,2,3}^3$-wall and then terminates on the remaining $(m_1,m_2,m_3)$-boundary.

Figure 29

The application of a transversal cz along a codimension-1 submanifold (gray) effectively sweeps the $s_{1,2}^2$-domain wall across the submanifold and attaches the wall onto the $(m_1,m_2,m_3)$-boundary, generating a codimension-2 boundary $(m_1{e}_2,e_1{m}_2,m_3)$ nested on the $(m_1,m_2,m_3)$-boundary.

Figure 30

The application of a transversal ccz gate on a stack of three fractal surface codes effectively sweeps the $s_{1,2,3}^{(3)}$-domain wall across the system and attaches the wall onto all the $(m_1,m_2,m_3)$-holes (blue) in the bulk, producing new $(m_1{s}_{2,3}^{(2)},m_2{s}_{3,1}^{(2)},m_3{s}_{1,2}^{(2)})$-boundaries (green) on these holes, which couple the three codes together as previously illustrated in Fig. 28.

Figure 31

Logical operators in the new stack code with exotic hole boundaries $(m_1{s}_{2,3}^{(2)},m_2{s}_{3,1}^{(2)},m_3{s}_{1,2}^{(2)})$ (green). The logical $Z$-strings remain the same as those in the original stack code. A logical $X$-brane can only terminate on the outer boundaries. The combined logical brane ${\overline{X}}_1{\overline{\mathrm{CZ}}}_{2,3}$ can instead terminate on the exotic hole boundaries and its minimal support forms a Sierpiński carpet as shown.

Figure 32

The lattice-surgery protocol to transform the logical information inside one of the code copies within the coupled stack fractal code (top) to an ancilla fractal code block (bottom). (a) Initialization of the ancilla qubits (white balls) between the two codes in state $|0⟩$. (b) Lattice merging via $d$ rounds of syndrome measurement and subsequent decoding. Two types of new stabilizers are introduced at the interface: the four-body $Z$ stabilizers (red) with $+1$ eigenvalues and the six-body $X$ stabilizers (blue) with unknown eigenvalues and hence a random $\pm 1$ measurement outcome. The measurement outcome of the new interface $X$ stabilizers equivalently gives the measurement outcome of the joint parity of logical operators: ${\overline{X}}_A{\overline{X}}_B$ (supported on the highlighted blue vertical edges). (c) The merged code after $d$ rounds of measurement. (d) The lattice-splitting protocol. The unknown stabilizers at the interface are the three-body $Z$ stabilizers that have a random $\pm 1$ measurement outcome, which can be inferred by the projective measurement of the ancilla qubits in the $Z$ basis ($M_Z$). The $(-1)$-eigenvalue corresponds to the $m$-loop excitations, which can be corrected with a single-shot decoding process even in the presence of measurement error.

Figure 33

The lattice surgery performed between the new coupled stack fractal code with three ancilla fractal codes along three directions in order to transfer the three logical qubits encoded in the coupled stack fractal code into the three ancilla code blocks.

Figure 34

A transversal logical gate ${\overline{\mathrm{CZ}}}_{2,3}$ between copy $2$ and copy $3$ in the stack code hence be chosen to be applied on a brane perpendicular to the $z$ direction, avoiding all the holes. Similarly, ${\overline{\mathrm{CZ}}}_{1,2}$ and ${\overline{\mathrm{CZ}}}_{1,3}$ can be applied on a brane perpendicular to the $y$ and $x$ directions, respectively.

Figure 35

An illustration of an $(m_1,m_2,m_3)$-boundary in a stack of three copies of 3D toric codes. The boundary is on the right of the lattice, where the support of the stabilizers is reduced. The code the $X$ stabilizers of which are given by the yellow octahedra have the support of their stabilizers reduced to be weight-5, while the red and blue stabilizers, of codes 2 and 3, have weight-4 face stabilizers on the boundary. For illustrative purposes, we show to the right how the stabilizers would normally have larger support—these qubits are faded out. On the right side, after the application of ${\otimes }_j{\mathrm{CCZ}}_{j;1,2,3}$, the bulk $X$ stabilizers are preserved while those the supports of which were cut at the boundary of the hole are now transformed to be of the form explained in Fig. 36.

Figure 36

An example of the action of a transversal ccz gate on the $X$ stabilizers of a code. (a) We demonstrate the action of a transversal ccz gate on the yellow weight-6 stabilizers in code block 1. Under the action of ccz, Pauli $X$ on one qubit is mapped to a product of $X$ and cz on the other two code copies. This product is represented in green. We can break this product into two components, one comprised of the original $X$ stabilizer and a resulting tensor product of cz on the other two code blocks, which is represented in purple here. (b) The action of cz on the qubits in the octahedron will result in a logical identity in the bulk, as the support of this operator mutually overlaps with two qubits from the intersection of any two red and blue weight-12 stabilizers—these qubits are symbolized in pink. (c) On the boundary, the overlap of the weight-5 cz operator will only overlap with a single qubit from the intersection of the red and blue faces—as such, this does not satisfy the requirements to be a logical identity operator.

Figure 37

An illustration for the application of the Alexander duality: the first homology of the punctured 3-sphere (with interior holes homeomorphic to 3-balls $D^3$) is isomorphic to the first homology of the union of the 3-balls $D^3$.

Figure 38

An illustration of the homology decomposition using a connected sum. (a) An illustration of Eq. (C4) in the 2D case. The connected sum of a 2-torus $T^2$ and a 2-sphere $S^2$ is obtained by cutting out a 2-ball (disk) $D^2$ on each piece and gluing them together by identifying the boundaries of the 2-ball on each piece, i.e., a circle $S^1$. Note that the 2-sphere with a disk cut out is just a disk: $S^2\setminus D^2=D^2$. (b) An illustration of Eq. (C4) in the context of a 3D toric code, where the opposite sides of the cube are identified and hence represent a 3-torus. The connected sum of a 3-torus $T^3$ and a 3-sphere $S^3$ is obtained by cutting out a 3-ball $D^3$ on each piece and gluing them together by identifying the boundaries of the 3-ball on each piece, i.e., a 2-sphere $S^2$. Note that the 3-sphere with a 3-ball cut out is just a 3-ball: $D^3=S^3\setminus D^3$, which gets glued into the 3-torus. (c) An illustration of Eq. (C5), i.e., decomposing the homology of the punctured 3-torus $T^3\setminus {\cup }_j{D}_{(j)}^3$ to that of a connected sum of a punctured sphere $S^3\setminus {\cup }_j{D}_{(j)}^3$ and a 3-torus. (d) An illustration of Eq. (C6), i.e., decomposing the relative homology of the punctured fractal surface code to that of a connected sum of the 3D surface code and a punctured sphere. The relative first homology of the 3D surface code is equivalent to the absolute first homology of the 3D surface-code geometry, with its $e$-boundaries being identified into a single point: i.e., ${\mathcal{L}}_{\text{3DSF}}/{\mathcal{B}}_e$. The logical string $\overline{Z}$ (red) hence becomes a nontrivial absolute 1-cycle traveling around the entire complex.

Figure 39

An illustration of the decomposition of a punctured $n$-sphere $S^n$ with both $e$-holes and $m$-holes into a punctured $n$-sphere containing only $e$-holes and another containing only $m$-holes. One first constructs a spanning tree connecting all the $e$-holes and then thickens the spanning tree into a punctured $n$-ball. The entire punctured $n$-sphere is hence equivalent to the gluing of two punctured $n$-balls with only $e$-holes and $m$-holes, respectively, which is in turn equivalent to the connected sum of two punctured $n$-spheres.

Figure 40

(a)–(c) Disentangling a 2D color code defined on a hexagonal lattice (qubits located on the vertices) into a tensor product of two surface codes defined on triangular lattices (qubits located on the edges) by shrinking the color-$A$ faces and color-$B$ faces, respectively, into a single vertex. The color-code boundary with color $A$ ($B$) is turned into a rough (smooth) boundary condensing $e$-anyons ($m$-anyons) in the first (second) copy of the surface code. (d)–(f) The effective TQFT picture of the lattice model in (a)–(c), i.e., disentangling a square-patch color code into two square-patch surface codes. Two pairs of logical operators in the color code are mapped into the corresponding pair of logical operators within each surface code.

Figure 41

Disentangling a single 3D color code into three copies of 3D surface codes aligned in perpendicular directions. The correspondence of the logical operators in the color code and the three surface codes is shown.