Fault-tolerant quantum gates with defects in topological stabilizer codes

Braiding defects in topological stabilizer codes has been widely studied as a promising approach to fault-tolerant quantum computing. Here, we explore the potential and limitations of such schemes in codes of all spatial dimensions. We prove that a universal gate set for quantum computing cannot be realized by supplementing locality-preserving logical operators with defect braiding, even in more than two dimensions. However, notwithstanding this no-go theorem, we demonstrate that higher-dimensional defect-braiding schemes have the potential to play an important role in realizing fault-tolerant quantum computing. Specifically, we present an approach to implement the full Clifford group via braiding in any code possessing twist defects on which a fermion can condense. We explore three such examples in higher-dimensional codes, specifically, in self-dual surface codes; the three-dimensional Levin-Wen fermion mode; and the checkerboard model. Finally, we show how our no-go theorems can be circumvented to provide a universal scheme in three-dimensional surface codes without magic-state distillation. Specifically, our scheme employs adaptive implementation of logical operators conditional on logical measurement outcomes to lift a combination of locality-preserving and braiding logical operators to universality.

Figure 1

An illustration of encoding a logical qubit in a pair of “rough boundary” holes in a 2D surface code. Blue lines represent the path of an excitation $e$ that can condense at the boundary of a hole. Red lines represent paths of excitation $m$. The Pauli logical operators for this encoded qubit are illustrated.

Figure 2

The condensation of excitations at domain walls with twists. The black line is a domain wall that transforms excitation $a$ to excitation $b$. Here, a (red) particle-antiparticle pair $a,a^{*}$ condenses in the vacuum. The excitation $a$ crosses the wall to become (blue) excitation $b$, while $a^{*}$ does not. Both then come together to give the excitation $a^{*}b$. Note that without the twists, such condensation would not be possible since excitations could then only come together if the number of times they crossed the domain wall was of the same parity.

Figure 3

An example of a logical qubit encoded in a pair of topological defects. The two tori are holes in the three-dimensional surface code at which magnetic fluxes may condense. The additional loop threaded through is a puncture (which may be viewed like a narrow tube) at which magnetic fluxes cannot condense. The computational basis states of the qubit correspond to the parity of fluxes at either of the toroidal holes, which is well defined since the threaded defect prevents these fluxes from being created from the vacuum. The $\overline{X}$ operator is a surface corresponding to propagating a flux between the holes. The logical $\overline{Z}$ is realized by braiding an electric charge around either hole.

Figure 4

An example of a logical qubit encoded in a single defect. The defect is a toroidal hole in a 3D toric code that can condense one-dimensional $m$ excitations. The logical $\overline{X}$ operator is implemented by a process by which an $m$ is created at a defect and then shrinks to be annihilated in the vacuum. This is equivalent to applying a membrane of $X$ operators that meets the torus in a topologically nontrivial loop. The logical $\overline{Z}$ is implemented by braiding an $e$ excitation around the defect. This is equivalent to a loop of $Z$ operators around the torus.

Figure 5

Examples of processes that can implement braiding logical operators. (a) One pointlike defect braided around another pointlike defect. (b) One pointlike defect is exchanged with another (identical) pointlike defect. (c) One looplike defect is braided around another looplike defect.

Figure 6

Examples of the action of braiding logical operators on TLPLOs. (a) Exchange of left and right of defects maps a TLPLO (red) enclosing the left defect to one enclosing the right defect. (b) Action of braiding of the top-right defect around the bottom right on TLPLO (blue). The TLPLO initially corresponds to transferring a topological excitation between the top defects. After the braid, it corresponds to this transferral along with the excitation being braided around the bottom-right defect.

Figure 7

A Venn diagram representing relevant groups of logical operators. The group of fault-tolerant logical operators (FTLOs) we consider is defined to be the group of operators implementable by products of locality-preserving logical operators (LPLOs) and braiding logical operators (BLOs). Lemma t2 shows that this group is contained in the normalizer of the group of TLPLOs. Lemma t1 shows that the set of TLPLOs is finite. Combined with the fact that the group of TLPLOs is nontrivial, this implies that the normalizer of the group of TLPLOs is not universal, and hence that the set of FTLOs cannot be universal, the result of Theorem t3. Similarly, for the standard encoding introduced in Sec. 3c, Lemma t5 shows that the group of TLPLOs is the logical Pauli group, which implies that the set of FTLOs is contained in the normalizer of the Pauli group (the Clifford group), the result of Theorem t6.

Figure 8

The logical operators acting on the logical qubit encoded in two-dimensional surface code twists. The Pauli logical operators are locality-preserving logical operators corresponding to paths of excitations of the code. Blue lines represent the path of an $m$ excitation and red lines represent the path of an $e$ excitation. The $\overline{Z}$ operator distinguishes the parity of $em$ excitations in the left domain wall and $\overline{X}$ transfers an $em$ excitation between the domain walls. The Clifford logical operators (orange and magenta) are implemented by braiding, specifically by exchanging twists.

Figure 9

A three-dimensional cross section of the twist setup in the four-dimensional surface code. The fourth dimension can be considered to be parametrized by an angle, in which all pictured objects except for the $\overline{X}$ operator are extended. Twists are tori, shown here as black circles. Domain walls are the volume between the pair of tori (shown here in gray). The logical $\overline{Z}$ and $\overline{X}$ operators are both tori, as shown in the top two pictures. Blue lines indicate paths of looplike $m$ excitations and red lines indicate paths of looplike $e$ excitations. The bottom picture shows the Clifford logical operators that can be implemented by braiding, with magenta and orange arrows indicating the movement of defects to implement $\overline{H}$ and $\overline{S}$, respectively.

Figure 10

The logical operators for a logical qubit encoded in the twists of the three-dimensional Levin-Wen fermion model. The twists are points at the end of one-dimensional domain walls in a three-dimensional space. The logical $\overline{Z}$ operator is a surface of $X$ operators across a sphere enclosing the left twists. The logical $\overline{X}$ operator is a surface of $X$ operators across a sphere enclosing the top twists with a line of $Z$ operators appended between the domain walls.

Figure 11

The two types of stabilizers of the checkerboard model. Each stabilizer acts on every cube in the lattice of one color. Cubes of the other color do not have corresponding stabilizers.

Figure 12

The logical operators for a logical qubit encoded in the twists of the checkerboard model (with periodic boundary conditions). The twists are the black (topologically nontrivial) loops at the boundaries of the gray domain walls. Pairs of blue lines represent paths of the planon $m_{z=j+1,j}$ and pairs of red lines represent paths of the planon $e_{z=j,j-1}$.

Figure 13

Illustration of two-dimensional case of general scheme for encoding with twists on domain walls that can condense a fermion $a$. Here, the domain walls map $b$ (blue) $\to a{b}^{*}$ (red), so that $a$ is represented by blue and red lines together. The logical $\overline{X}$ operator corresponds to transferring $a$ between the domain walls, and $\overline{Z}$ corresponds to transferring $a$ between twists on the same domain wall. Exchanging twists in this setup, in analogy to Fig. 8, will implement the full single-qubit Clifford group.

Figure 14

Defect setup for the universal scheme we present. All five defects are punctures in a three-dimensional color code (equivalent to three surface codes). $h_1$ and $h_2$ are rough with respect to surface code one and smooth with respect to surface codes two and three. $h_{\alpha }$ and $h_{\beta }$ are rough with respect to surface code two and smooth with respect to surface codes one and three. $h_3$ is threaded through all four other holes, and is rough with respect to surface codes one and two and smooth with respect to surface code three.

Figure 15

Logical Pauli operators of the encoded qubits. (a) Pauli operators for logical qubit 1 and ancilla qubit $a$. These are the two qubits encoded using excitations from code 1. Specifically, blue here is a path of $m_1$ and red is a path of $e_1$. ${\overline{X}}_1$ is a torus around $h_1$ and ${\overline{Z}}_1$ is a line between $h_1$ and $h_2$. ${\overline{X}}_a$ is a cylinder between $h_4$ and $h_5$ and ${\overline{Z}}_a$ is a loop around $h_4$. (b) Pauli operators for logical qubit 2 and ancilla qubit $b$. These are the two qubits encoded using excitations from code 2. Specifically, blue here is a path of $m_2$ and red is a path of $e_2$. ${\overline{X}}_2$ is a cylinder between $h_1$ and $h_2$ and ${\overline{Z}}_2$ is a loop around $h_1$. ${\overline{X}}_b$ is a torus around $h_4$ and ${\overline{Z}}_b$ is a line between $h_{\alpha }$ and $h_{\beta }$. (c) Pauli operators for logical qubit 3. This is the qubit encoded using excitations from code 3. Specifically, blue here is a path of $m_3$ and red is a path of $e_3$. ${\overline{X}}_3$ is a disk stretching out to $h_3$ and ${\overline{Z}}_3$ is a loop around $h_3$.

Figure 16

A state injection circuit allowing for a logical Hadamard operator to be implemented using only logical $\overline{\text{CZ}}$ and $\overline{X}$ conditional on logical $X$ measurement. The double line indicates a classical control. Specifically, $\overline{X}$ is applied if and only if the outcome of the logical measurement is $-1$.

Figure 17

Transversal logical $\overline{\text{CZ}}$ operators acting on the encoded qubits. Purple shows path of $s_{23}$, green shows path of $s_{31}$, and orange shows path of $s_{12}$. ${\overline{\text{CZ}}}_{12}{\overline{\text{CZ}}}_{ab}$ can be implemented by growing $s_{12}$ from the vacuum out to defect $h_3$. The propagation of $s_{ij}$ is realized by transversal implementation of ${\text{CZ}}_{ij}$. ${\overline{\text{CZ}}}_{13}$ may be implemented by a cylinder propagating $s_{31}$ between $h_1$ and $h_2$. ${\overline{\text{CZ}}}_{b3}$ may be implemented by a cylinder propagating $s_{23}$ between $h_{\alpha }$ and $h_{\beta }$. ${\overline{\text{CZ}}}_{23}$ may be implemented by a torus taking $s_{23}$ around $h_1$. ${\overline{\text{CZ}}}_{a3}$ may be implemented by a torus taking $s_{31}$ around $h_{\alpha }$.

Figure 18

Fault-tolerant circuit for measurement of $Z$ on the qubit (bottom wire) initially in state $|\psi \rangle$ in the case where the code distance is $d=3$. Logical measurement of $\overline{Z}$ on an encoded qubit can be performed by performing this measurement circuit on each physical qubit on a region that supports a logical $\overline{Z}$ operator. Fault-tolerant measurement of $X$ can be performed with the same circuit but with each CZ operator replaced with a CNOT operator whose target is the bottom qubit.

Figure 19

The gadget $H_{xy}$ which takes a logical state $|\overline{\psi }\rangle$ onto the state $\overline{H}|\overline{\psi }\rangle$ on an ancilla qubit. Note that this gadget is the same circuit as presented in the text in Fig. 16.

Figure 20

The gadget $I_{xy}$ which takes a logical state $|\overline{\psi }\rangle$ onto an ancilla qubit. It can be implemented by either of the above circuits, depending on which direction of $\overline{\text{CNOT}}$ gate is available for the encoded qubits $x$ and $y$.

Figure 21

Diagrammatic representation of how the sequence of gadgets $H_{b3}{H}_{ab}{H}_{3a}$ can be used to implement the logical operator ${\overline{H}}_3$.

Figure 22

Explicit circuit that can be used to implement ${\overline{H}}_3$. Double lines show the measurement outcome of logical measurements. The parameters specifying these outcomes are then used to control future logical operators.

Figure 23

Diagrammatic representation of how the sequence of gadgets $H_{a3}{H}_{31}{H}_{b3}{H}_{3a}{H}_{ab}{I}_{1a}$ can be used to implement the logical operator ${\overline{H}}_1$.

Figure 24

Diagrammatic representation of how the sequence of gadgets $H_{b3}{H}_{32}{H}_{a3}{H}_{3b}{H}_{ab}{I}_{2b}$ can be used to implement the logical operator ${\overline{H}}_2$.

Figure 25

Circuit that implements ${\overline{\text{CCZ}}}_{125}$ in the case of the scheme with five logical qubits.