Instantaneous braids and Dehn twists in topologically ordered states

A defining feature of topologically ordered states of matter is the existence of locally indistinguishable states on spaces with nontrivial topology. These degenerate states form a representation of the mapping class group (MCG) of the space, which is generated by braids of defects or anyons, and by Dehn twists along noncontractible cycles. These operations can be viewed as fault-tolerant logical gates in the context of topological quantum error correcting codes and topological quantum computation. Here, we show that braids and Dehn twists can in general be implemented by a constant depth quantum circuit, with a depth that is independent of code distance $d$ and system size. The circuit consists of a constant depth local quantum circuit (LQC) implementing a local geometry deformation of the quantum state, followed by a permutation on (relabeling of) the qubits. The permutation requires permuting qubits that are separated by a distance of order $d$; it can be implemented by collective classical motion of mobile qubits or as a constant depth circuit using long-range SWAP operations (with a range set by $d$) on immobile qubits. We further show that (i) applying a given braid or Dehn twist $k$ times can be achieved with $\mathcal{O}(logk)$ time overhead, independent of code distance and system size, which implies an exponential speedup for certain logical gate sequences by trading space for time, and (ii) an arbitrary element of the MCG can be implemented by a constant depth (independent of $d$) LQC followed by a permutation, where in this case the range of interactions of the LQC grows with the number of generators in the presentation of the group element. Applying these results to certain non-Abelian quantum error correcting codes demonstrates how universal logical gate sets can be implemented on encoded qubits using only constant depth unitary circuits.

Figure 1

Noncontractible cycles and braid operations on a genus $g=3$ surface with $p=4$ punctures.

Figure 2

Definition of Dehn twists, half-twists, and braids, as well as their relations.

Figure 3

The mathematical definition and implementation of Dehn twists $T$ and $U$ on a torus via Dehn surgery. The coordinate web represents the metric of the manifold in the continuous case or underlying lattice structure in the discrete case. The shear deformation (green arrows) in (b) implements a Dehn twist and is equivalent to Dehn surgery in (e): cut, twist, and glue. The local geometric deformation returns the metric/lattice to the original configuration.

Figure 4

The elementary move of adding (removing) a diagonal edge in the center of the plaquette, which splits (merges) the plaquette(s). The quantum circuit implementing such a move is composed of two CNOTs (blue arrows) targeting the qubits on the diagonal edge $e$, controlled by the other two qubits $a$ and $b$ on the same triangular plaquette ${\mathrm{\Delta }}_{abe}$.

Figure 5

Gadgets of local geometry deformation of the lattice, which will be used for braiding, and Dehn twists on an annulus and high-genus surfaces. The blue arrows represent the CNOT gates. The white (empty) circles represent the added ancilla qubits, which are initialized in $|0\rangle$ or $|+\rangle$. The yellow (filled) circles represent qubits to be removed from the code. Pink solid lines represent edges to be added to the code, while yellow dashed lines represent edges to be removed from the code.

Figure 6

Creating NNN slanted plaquettes through a constant depth circuit. The gray and brown shades represent stabilizers with triangular or parallelogram shape.

Figure 7

Definition and elementary move for the ${\mathbb{Z}}_N$ toric code. The edges of the lattice are directed (with arrows) to represent the Hamiltonian. (a) Two types of stabilizers. (b) Two types of anyons strings with corresponding anyon pairs in the ends. (c) Two types of logical string operators along two directions on a torus. (d) Elementary move for adding an edge. The single and double arrows represent the $CX$ and $C{X}^{†}$ gates, respectively. (e) Elementary move for removing an edge.

Figure 8

Implementing Dehn twist ${\mathcal{D}}_{\beta }$ on a torus for the ${\mathbb{Z}}_2$ toric code. The thick lines represent the noncontractible loops of two types $\overline{Z}$ (red) and $\overline{X}$ (blue). The thin lines represent local contractible loops of two types. The blue arrows in (b) and (c) represent CNOTs, in order to add qubits (white circles, initialized at $|0\rangle$) and edges (pink solid lines), or to remove qubits (yellow circles) and edges (yellow dashed lines). The green thick arrows in (e) represent permutation of qubits in a shearing pattern.

Figure 9

Dehn twist along the boundary of the inner defect (the $\beta$ loop) on an annulus via shearing of the defect. An overall $2\pi$ rotation (indicated by the the letters on the four corners) is induced by nine steps of shearing.

Figure 10

(a) A hole with perimeter $L=12$ in a square lattice. The qubits are on the edges but are not shown for simplicity. The sites are divided into rings surrounding the hole and are numbered clockwise. We start with the upper left corner, shown in red numbers, and move clockwise. (b) We shift the labels on the $n\mathrm{th}$ ring by $9n$ sites. The labels which originally corresponded to upper left corners are shown in red. (c) After shifting labels, we reconnect the sites as they were connected originally. For example, since the (0,0) site was connected to (1,1) and (1,19) in (a), it should be reconnected by the blue lines shown above. (d) The modified lattice after reconnecting the sites.

Figure 11

Performing Dehn twist around the $\beta$ loop in one shot. (a) A boundary defect with length $L=12$ in a square lattice. A string operator which terminates on the hole's boundary is also shown in red color. (b) How the lattice and the string look like after reconnecting the sites. (c) Following the permutation we recover the original lattice, but now the string operator goes around the hole defect.

Figure 12

Understanding the essence of braiding via the equivalence of local entanglement renormalization (its inverse) and manifold stretching (squeezing). (a) The global MERA circuit performing coarse graining (downward) or fine graining (upward). (b) Locally stretch the manifold in the region between the two punctures to enlarge their distance by a factor of 2. (c) Aside from stretching, also squeeze the upper and lower regions to preserve the shape of the manifold. (d) Stretch the puncture only in one direction to only move one puncture upward.

Figure 13

Braiding protocol via single-shot long-distance moving of the boundary defects. (a) Two pairs of boundary defects (X cut and Z cut). (b) Adding qubits (white circles) and edges (pink solid lines). (c) Removing qubits (yellow circles) and edges (yellow dashed lines). (d) Permutation of qubits in a shearing pattern ${\mathcal{P}}_{\sigma }$. (e) Effectively move the defect 1 with distance $\mathcal{O}\left(d\right)$ after the permutation. (f) A full braiding is implemented by 12 steps of $\mathcal{LU}$ and ${\mathcal{P}}_{\sigma }$.

Figure 14

Implementing braiding (full braiding) via half-twists (Dehn twists) through multiple steps of shearing operations. The parallelogram (purple dashed lines) surrounding both punctures is a guide to the eye. The red and blue dashed loops in (f) indicate gauge transformations, which shows that the half–self-twists around each puncture are trivial operations.

Figure 15

(a) Three types of noncontractible loops on a high-genus surface: $\alpha ,\beta$, and $\gamma$. Dehn twists along these loops generate the whole mapping class group. (b) The equivalent loops in a bilayer topological system coupled via “wormholes.” The equivalent $\beta$ loops on the upper and lower layers have opposite orientations.

Figure 16

Dehn twist along the $\alpha$ loop. (a) Adding qubits (white circles) and edges (red solid lines). (b) Removing qubits and edges (yellow dashed lines). (c) Obtain a lattice with a “gift-wrapping” (slanted) pattern in the solenoid region. (d) Shear permutation (green arrows) of the qubits in the solenoid region. (e) A Dehn twist is achieved and the lattice is mapped to the original geometry in (a).

Figure 17

Dehn twist along the $\gamma$ loop.

Figure 18

Dehn twist along the $\beta$ loop on a high-genus surface via nine steps of shearing on the top layers.

Figure 19

Dehn twist along the $\beta$ loop implemented by two half-twists of an annulus on both layers with five puncture-shearing steps in total.

Figure 20

Multiple Dehn twists implemented in a single shot on a ${\mathbb{Z}}_N$ toric code via changing the aspect ratio of the torus. The single and double arrows represent the $CX$ and $C{X}^{†}$ gates, respectively. The curved green arrows in (b) represent permutations.

Figure 21

Implementing multiple Dehn twists in a single shot via increasing the range of interaction on a ${\mathbb{Z}}_N$ toric code. In (b)–(d), the diagonal edges overpass the horizontal edges, i.e., equivalent to a bridge structure.

Figure 22

Definition of the Levin-Wen Hamiltonian and Turaev-Viro codes on a triangulated manifold (light gray lines indicate the triangulation $\mathrm{\Lambda }$) and the corresponding trivalent graph $\widehat{\mathrm{\Lambda }}$ (blue lines). The arrows on the lines specify the branching structure. The thin red lines represent the string nets. The thick blue and red lines illustrate the plaquette and vertex projectors, respectively.

Figure 23

The building block of the quantum computation scheme: the $F$ operation quantum circuit for the Fibonacci surface code.

Figure 24

Definition of the 2-2 Pachner move ($F$ move) on the triangulation grid and the corresponding trivalent graph. The pink edges represent the edges being switched during the moves.

Figure 25

Definition of the 1-3 Pachner move on the triangulation grid and the corresponding trivalent graph.

Figure 26

Implementing the 1-3 Pachner move with a unitary circuit by attaching/detaching a tadpole diagram in the center of an arbitrary plaquette, followed/preceded by two 2-2 Pachner moves.

Figure 27

Dehn twist on a torus for the Turev-Viro code. The upper rows [(b), (d), (f), (g)] show the protocol on the triangulation (light gray), while the lower rows [(c), (e), (g), (i)] illustrate the protocol on the dual trivalent graph (blue). The pink lines indicate the edges to be switched during the $F$ moves. The green dashed box specifies the four external legs of the $F$ moves on the trivalent graph. The green arrows represent qubit permutations, with the column numbers specified before and after the permutation.

Figure 28

Braiding of two punctures, e.g., non-Abelian anyons (blue and red circles) in Turaev-Viro codes, with constant depth circuit on a space with periodic boundary conditions (e.g., a genus $g>0$ surface). The scheme can also be applied to punctures on a bilayer system with holes and appropriate boundary conditions, which is a way to effectively obtain a single-layer system on a high-genus surface.

Figure 29

Gadgets for local geometry deformation in Turaev-Viro codes. The solid (dashed) purple lines represent added (removed) edges during the 1-3 Pachner moves. The pink line represents the switched edges in $F$ moves. The yellow arrow indicates the equivalence between two triangulations by locally shifting the positions of the edges, which can be physically implemented by local SWAPs.

Figure 30

Braiding of two non-Abelian anyons in Turaev-Viro codes with constant depth circuit on a disk subregion, keeping the boundary fixed. The yellow dashed lines in (b) and (c) show the equivalent edges before and after the permutation.

Figure 31

Implementation of Dehn twist ${\mathcal{D}}_{\beta }$ in Turaev-Viro codes, on an annulus along the $\beta$ loop encircling the boundary defect via a sequence of shearing operation. The red line with arrows represents Wilson-line operators connecting the inner and outer boundaries.

Figure 32

Implementation of Dehn twist ${\mathcal{D}}_{\beta }$ in Turaev-Viro codes on an annulus (a continuation of Fig. 31).

Figure 33

Implementation of Dehn twist ${\mathcal{D}}_{\beta }$ in Turaev-Viro codes on an annulus (a continuation of Figs. 31 and 32).

Figure 34

The gadget used for the alternative protocol of Dehn twist on an annulus in Turaev-Viro codes.

Figure 35

Alternative protocol of Dehn twist on an annulus for Turaev-Viro codes.

Figure 36

Implementation of Dehn twist along the $\alpha$ loop in Turaev-Viro code. The red line with arrows represents the Wilson loop. The pink thick lines represent the edges being switched during the $F$ moves, with the green dashed boxes specifying the four external legs. The dark green arrows represent qubit permutation with a shearing pattern.

Figure 37

Implementation of Dehn twist along the $\gamma$ loop in Turaev-Viro code.

Figure 38

Implementing multiple Dehn twists in Turev-Viro code in a single shot by choosing a particular aspect ratio of the torus.

Figure 39

Double the size of the torus along one cycle with by dividing a plaquette into two and long-range permutation. The yellow arrow in (e) indicates the equivalence of the triangulation in (e) and (f) by a local shift of the vertices. The curved green arrows in (h) represents long-range SWAPs.

Figure 40

Propagation of preexisting errors under the qubit permutation ${\mathcal{P}}_{\sigma }$ in the braiding scheme from Fig. 13. Five error strings before and after the permutation are illustrated.

Figure 41

Illustration of the propagation of error strings after a successive application of the instantaneous logical gates (Dehn twists) on a torus (for the ${\mathbb{Z}}_N$ toric code example). (a) A sequence of Dehn twists along the same $\alpha$ cycle (vertical) leads to a linear growing of error strings. (b) An alternating sequence of Dehn twists along both cycles leads to an exponential growing of error strings.