Quantum computation with generalized dislocation codes

In this paper, we study quantum computing with twists. Twists are yet another form of defects in a lattice. They arise from dislocations in the lattice and can be used to encode and process quantum information. Surface codes with twists are also called dislocation codes. Hastings and Geller showed that dislocation codes could provide gains in space-time complexity of quantum computation. In this paper, we undertake a detailed study of generalized dislocation codes. We develop the theory of qubit dislocation codes over arbitrary four-valent and bicolorable lattices. We give a construction to introduce twists in such lattices and also study the structure of logical operators. We then study dislocation codes over odd prime dimensions in square lattices. Using the theory developed, we present protocols for implementing a universal gate set in qubit dislocation codes. We also show how to implement the generalized Clifford group in qudit dislocation codes in odd prime dimensions.

Figure 1

Surface code on a square lattice with boundary. Qubits are placed on vertices and stabilizers are attached to faces. The stabilizers will be of $Z$ type along the dark faces and $X$ type along the light faces. The weight-two stabilizers on the boundary are to avoid weight-one logical operators.

Figure 2

Creating a twist in a square lattice. The resulting lattice with twists ${\tau }_1$ and ${\tau }_2$ is shown on the right. The separation between the twists ${\tau }_1$ and ${\tau }_2$ in this lattice is two.

Figure 3

Twist creation. (a) Twists are introduced by merging and splitting three contiguous faces $f_1$, $f_2$, $f_3$. (b) First we merge $f_1$, $f_2$, and $f_3$ to create a combined face. (c) A pair of twists is created by adding an edge between $u_1$ and $v_2$. (d) Twists can also be introduced by adding an edge between $u_2$ and $v_1$.

Figure 4

Twist movement. (a) The direction of twist movement is from face ${\tau }_2$ to $f_4$; $f_4$ is not incident on the trivalent vertex of ${\tau }_2$. (b) To move the twist, the edge common to the twist and the face $f_4$ is shifted to the trivalent vertex. (c) The direction of twist movement is from ${\tau }_2$ to $f_4^{\prime }$. Here $f_4^{\prime }$ is incident on the trivalent vertex. (d) To move the twist, the common edge to faces $f_4^{\prime }$ and $f_5$ is shifted to trivalent vertex.

Figure 5

Twists in triangular-hexagonal lattice.

Figure 6

The path in the lattice along which faces of same color meet is shown as solid line in color. This path, shown as solid line, is referred to as the $T$ line. Also shown is the domain wall (dashed line) connecting the two trivalent vertices. The domain wall passes through the modified faces between the two twists.

Figure 7

Domain walls during twist creation and twist movement. (a) Domain wall when twists are just created. Domain wall shown as dashed line. Also shown is the $T$ line (solid line). (b) Domain wall grows as the twists are moved apart. Twist is moved to a face not incident on the trivalent vertex. (c) Domain wall when twist is moved through a face incident on a trivalent vertex.

Figure 8

Stabilizer assignment for normal, i.e., unmodified faces.

Figure 9

Illustrating the stabilizer assignment rules. On the left when one of the vertices is not shared with a normal face and on the right when one of the vertices is not shared with a normal face. Here $\tau$ denotes a twist face and $m$ and $m^{\prime }$ are modified faces. Faces $g$ and $h$ are normal faces.

Figure 10

(a) Commutation between stabilizers of a normal face $g$ and twist face $\tau$ not sharing a trivalent vertex. (b) Commutation between stabilizers of a twist face $\tau$ and a normal face $g$ sharing a trivalent vertex. (c) Commutation between stabilizers of two twist faces ${\tau }_1$ and ${\tau }_2$.

Figure 11

Stabilizer update during twist creation. (a) Stabilizer assignment before twist creation. Pauli operator assignment around the four-valent vertices alternates to avoid weight-one logical operator. Stabilizer assignment changes at the vertices in red. (b) Stabilizer assignment after twist creation. The stabilizer generators of just the faces involved in twist creation are updated. The path highlighted in blue is shifted from $f_2$ to the twist ${\tau }_1$.

Figure 12

Stabilizer update during twist movement. The Pauli operators around the vertices are indicated as ${\gamma }_1$, ${\gamma }_2$ and ${\delta }_1$, ${\delta }_2$ which can be either $X$ or $Z$ Pauli operators. During twist movement, these operators are reassigned around old and new trivalent vertices (shown in red circles). The new stabilizers so created commute with other stabilizer generators.

Figure 13

Representing single qubit Pauli operators as strings in the lattice. The ends of the string correspond to the faces where the error causes a nonzero syndrome.

Figure 14

Various types of closed strings in the lattice and the Pauli operators associated with them. Closed strings $\left(\mathrm{a}\right)$–$\left(\mathrm{e}\right)$ are stabilizers.

Figure 15

Logical operators $\overline{Z}$ (solid line) and $\overline{X}$ (dashed line). Logical $Z$ operator has Pauli operator $Z$ on all vertices in its support. Logical $X$ has Pauli operator $Z$ on vertices where the blue part of the string has support and Pauli operator $X$ where the string is red.

Figure 16

A closed nonintersecting string $\sigma$ (dashed) crosses another string ${\sigma }^{\prime }$ where they anticommute. (a) String ${\sigma }^{\prime }$ encircles twists created together. (b) String ${\sigma }^{\prime }$ encircles twists not created together.

Figure 17

One choice for logical operators when we encode $(t/2)-1$ qubits. Logical $Z$ operators correspond to blue strings while the logical $X$ operators correspond to the strings that change color. For clarity, some of them are shown with dashed and dash-dotted lines.

Figure 18

Alternate choice of logical operators. Closed blue strings are logical $Z$ operators and closed strings that change color are logical $X$ operators. Six twists are needed for encoding two logical qubits. For the same number of twists, this choice leads few encoded qubits compared to the one in Fig. 17. However, this leads to simpler quantum gates.

Figure 19

An equivalent logical $Z$ operator, obtained by deforming the operator in Fig. 15. This operator is supported on the $T$ line connecting the twists. The operators are equivalent up to a stabilizer.

Figure 20

An equivalent $X$ logical operator supported on a path connecting the trivalent vertices of a pair of twists. This operator is obtained by deforming the logical $X$ operator shown in Fig. 15. We can think of this path also as a $T$ line.

Figure 21

Stabilizer assignment for different faces.

Figure 22

$T$ lines and domain walls in qudit dislocation codes. The $T$ lines are shown as solid thick lines and the domain walls as dashed lines.

Figure 23

Stabilizer assignment during twist creation and twist movement.

Figure 24

Representing qudit Pauli operators as strings.

Figure 25

Closed strings in lattice of qudit code can produce both zero and nonzero syndrome. This is in contrast to the qubit case where closed strings always produce zero syndrome.

Figure 26

Logical $Z$ and $X$ operators shown with corresponding Pauli operators on their support. Observe that at exactly one vertex where logical $Z$ and $X$ intersect and have different colors, the Pauli operators are $Z^{†}$ and $X^{†}$, respectively. From Eq. (7), the strings reproduce the same commutation relations between $Z$ and $X$.

Figure 27

If the edge $(u,v)$ is in the interior of $\sigma$, then $\mathcal{O}\left(\sigma \right)$ on vertices $u$ and $v$ is ${\alpha }_3$ and ${\alpha }_4$ respectively. Otherwise, it is ${\alpha }_3^{†}$ and ${\alpha }_4^{†}$.

Figure 28

Various choices for logical operators for qudit dislocation codes.

Figure 29

Logical operators.

Figure 30

Evolution of logical operators during the implementation of the phase gate.

Figure 31

Evolution of logical operators during the implementation of Hadamard gate.

Figure 32

Implementation of CNOT gate on neighboring qubits up to phase gates on control and target qubits.

Figure 33

Choice of logical operators for CNOT gate between distant logical qubits. The qubits are labeled as shown above. Qubit 1 is control and qubit 3 is target qubit.

Figure 34

Control and target qubits for controlled-$Z$ gate. Also shown are the corresponding logical operators.

Figure 35

Implementation of SWAP gate by twist pair movement and braiding. This scheme is advantageous over doing three CNOTs, as twist-pair movements in steps $\left(\mathrm{a}\right)$–$\left(\mathrm{c}\right)$ can be done simultaneously.

Figure 36

Merging lattices along boundary having $X$ stabilizers. The new $X$ stabilizers are product of old $X$ stabilizers from the two lattices.

Figure 37

Logical operators used in implementing qudit gates.

Figure 38

Qudit phase gate by braiding.

Figure 39

Qudit DFT gate by braiding.

Figure 40

Qudit controlled phase gate.

Figure 41

Logical operators for implementing controlled phase gate on distant qudits.

Figure 42

Combining logical $X$ and $Z$ operators to get logical $Y$ of Fig. 29.

Figure 43

Deformation of logical operators to prove Eq. (26).