Adiabatic topological quantum computing

Topological quantum computing promises error-resistant quantum computation without active error correction. However, there is a worry that during the process of executing quantum gates by braiding anyons around each other, extra anyonic excitations will be created that will disorder the encoded quantum information. Here, we explore this question in detail by studying adiabatic code deformations on Hamiltonians based on topological codes, notably Kitaev's surface codes and the more recently discovered color codes. We develop protocols that enable universal quantum computing by adiabatic evolution in a way that keeps the energy gap of the system constant with respect to the computation size and introduces only simple local Hamiltonian interactions. This allows one to perform holonomic quantum computing with these topological quantum computing systems. The tools we develop allow one to go beyond numerical simulations and understand these processes analytically.

Figure 1

Stabilizer generators (checks) for the surface code. An example of a plaquette check $S_p$ and a vertex check $S_v$.

Figure 2

A smooth ($Z$-type) defect. A logical $Z$ operator is defined by a closed loop of $Z$'s on the lattice that surrounds the defect and a logical $X$ operator is defined by a connected path of $X$'s on the dual lattice from the defect to a smooth ($Z$-type) boundary. By removed, we mean that stabilizers in the gray region are removed from the stabilizer group. Also, the star operators on the boundary of the lattice or the defect have lower weight in general. Here, we depict the removed region by removing that part of the lattice; this simply indicates that the code of the system factors into a code in the drawn region and a code inside of the defect.

Figure 3

A rough ($X$-type) defect. A logical $X$ operator is defined by a closed loop of $X$'s on the dual lattice that surrounds the defect and a logical $Z$ operator is defined by a connected path of $Z$'s on the lattice from the defect to a rough ($X$-type) boundary.

Figure 4

A large array of qubits in the state $|0\rangle$, each protected by a Hamiltonian $H=-\mathrm{\Delta }Z$.

Figure 5

A large array of qubits, an eight-qubit region of which is now encoded in the surface code (shown in black). The boundaries of the code are chosen to be trivial so that the code space is nondegenerate.

Figure 6

Growth of a small surface code region that involves only the qubits labeled 1, 2, and 3.

Figure 7

Operators involved in creating the defect that includes $p_1$ and $p_2$. Note that the $X$ operations span to a nearby smooth boundary.

Figure 8

The four potential situations faced when growing a smooth defect. (a) Only one interior qubit. (b) Two interior qubits. (c) Three interior qubits. (d) Four interior qubits.

Figure 9

Growth of a smooth defect with only a single qubit on the interior after the procedure. (a) We wish to grow the defect to the indicated plaquette. (b) We adiabatically turn off the neighboring plaquette while (c) turning on a $-X$ Hamiltonian on the interior qubit. (d) This procedure causes modifications to the neighboring $X$ checks that can be performed simultaneously with steps (b) and (c). The initial and final Hamiltonians are $H_i=-\mathrm{\Delta }/2(Z_t{Z}_l{Z}_r{Z}_b+X_t{X}_1{X}_2+X_t{X}_3{X}_4{X}_5)$ and $H_f=-\mathrm{\Delta }/2(X_t+X_1{X}_2+X_3{X}_4{X}_5)$, where $Z_t, Z_r, Z_l, Z_b$ are the $Z$ operators for the top, right, left, and bottom spins on the relevant plaquette, and $X_i$ are $X$ operators for spins on the edges of the vertices adjacent to the top edge labeled in clockwise order.

Figure 10

Growth of a smooth defect with two qubits on the interior. (a) We wish to grow the defect to the indicated plaquette. (b) We adiabatically turn off the neighboring plaquette while (c) turning on two $-X$ Hamiltonians on the interior qubits. (d) This procedure causes modifications to the neighboring $X$ checks.

Figure 11

Growth of a smooth defect with three qubits on the interior. The process is essentially the same as the one depicted in Fig. 10.

Figure 12

Growth of a smooth defect with four qubits on the interior. The procedure is the same as the others, but there are no resulting $X$ check modifications.

Figure 13

The setup for pinching off a smooth defect from a smooth wall.

Figure 14

After the Hamiltonian deformation (or sequence of deformations), we are left with a surface code with trivial boundaries encoding a rough defect in the state $|+\rangle$.

Figure 15

The interior qubit is adiabatically dragged to the state $|\psi \rangle$, the desired magic state.

Figure 16

The missing $X$ checks are reintroduced to the code, causing neighboring $Z$ checks to be removed. This new defect is now encoded in the state $|\psi \rangle$ with an encircling $\overline{Z}$ and a single qubit $\overline{X}$.

Figure 17

Because $\overline{X}$ was only a single-qubit operator, the two removed faces are moved apart and grown to combat decoherence.

Figure 18

One of the defects is merged with the boundary to make the standard single defect.

Figure 19

Distillation circuit for $T|+\rangle$ states, constructed from the 15-qubit Reed-Muller code's encoding circuit.

Figure 20

A distillation protocol for $S|+\rangle$ states based on the encoding circuit for the $\left[\right[7,1,3\left]\right]$ quantum Steane code.

Figure 21

An example of measuring $\overline{Z}$ for a smooth qubit. It requires the preparation of a rough defect in a $+1$ eigenstate of $\overline{Z}$, as discussed in Sec. 4e.

Figure 22

Circuit used to apply one of the Pauli operators to a smooth defect qubit. The outcome of the $X$ measurement is $b\in \{0,1\}$ and the outcome of the $Z$ measurement is $a\in \{0,1\}$. The outcomes of the measurement all occur with equal probability and the final state depends on these outcomes as shown. If an undesired operator is applied, the ancilla qubit is reinitialized and the circuit is implemented again. However, now the appropriate operator is the one that undoes the operator applied in the first iteration and applies the desired operator. (This, of course, will just be a different one of the four operators ${\overline{X}}^a{\overline{Z}}^b$.)

Figure 23

Gate teleportation circuit using the $T|+\rangle$ state. The $S$ correction needs to be performed half of the time and can be implemented in the same way using the state $S|+\rangle =|+i\rangle$ instead of $T|+\rangle$ (and utilizing a $Z$ correction half of the time).

Figure 24

Circuit for applying the Hadamard gate with an ancilla state. The correction $A$ depends on the result of the measurement: if the measurement result is $+1$, then $A=X$, and if the measurement result is $-1$, then $A=Z$. The $S$ and $S^{†}$ gates can be performed using a circuit like the one in Fig. 23.

Figure 25

A lattice with colored plaquettes on which one can define the color codes.

Figure 26

Two adjacent green $Z$ plaquettes are turned off while turning on the $-XX$ Hamiltonian shown, creating a $Z$-type defect in the $+1$ eigenstate of $\overline{Z}$ (shown here as a light blue string encircling the defect). The green string depicts the associated $\overline{X}$ operator.

Figure 27

A $Z$-type red defect isolation procedure. The “drawbridge” in this case is the red plaquette adjacent to the yellow dots in the figure. The $Z$-type check on the red face is turned off while the two $-XX$ operators are turned on. The four-body $X$ operator that is the product of the two $-XX$ Hamiltonians is in the stabilizer group before the evolution, and it is trivially in the stabilizer group of the code after the evolution. The blue and green plaquettes adjacent to the yellow dots are modified to be four-body operators. (Also note that the $X$-type check on the red plaquette must also turned off to fully isolate the region, and two $-ZZ$ Hamiltonians are turned on.)

Figure 28

The creation of a defect region with both the $X$-type and $Z$-type green checks turned off. There are four interior qubits prepared in two Bell pairs by this procedure.

Figure 29

Four interior qubits are “exposed.”

Figure 30

The upper-right qubit is adiabatically dragged to the desired state. For instance, to inject $T|+\rangle$ states, $U=TH$.

Figure 31

The single-body terms in Fig. 30 are turned off while turning on the $X$-type and $Z$-type checks on the red plaquette.

Figure 32

$\overline{X}$ and $\overline{Z}$ after the reintroduction of the red plaquette in Fig. 31.

Figure 33

The arrangement of the defect after reintroducing the $X$-type green plaquettes. $\overline{X}$ is a string of Pauli $X$ operators connecting two blue faces and $\overline{Z}$ is a loop of Pauli $Z$ operators around a blue face.