Fault-tolerant holonomic quantum computation in surface codes

We show that universal holonomic quantum computation can be achieved fault tolerantly by adiabatically deforming the gapped stabilizer Hamiltonian of the surface code, where quantum information is encoded in the degenerate ground space of the system Hamiltonian. We explicitly propose procedures to perform each logical operation, including logical state initialization, logical state measurement, logical controlled-not (cnot), state injection, distillation, etc. In particular, adiabatic braiding of different types of holes on the surface leads to a topologically protected, non-Abelian geometric logical cnot. Throughout the computation, quantum information is protected from both small perturbations and low-weight thermal excitations by a constant energy gap and is independent of the system size. Also, the Hamiltonian terms have weight at most four during the whole process. The effect of thermal error propagation is considered during the adiabatic code deformation. With the help of active error correction, this scheme is fault tolerant, in the sense that the computation time can be arbitrarily long for large-enough lattice size. It is shown that the frequency of error correction and the physical resources needed can be greatly reduced by the constant energy gap.

Figure 1

A surface code based on an $8\times 8$ lattice with 113 physical qubits on the edges. This code contains 1 logical qubit and has distance $d=L=8$, where $d$ is the distance of the code. The four-body (or three-body) plaquette stabilizer generator ($Z_p$) and vertex stabilizer generator ($X_s$) are indicated as cyan and yellow plaquettes, respectively, inside the lattice (or on the boundary). A particular choice of logical operators $X_L$ and $Z_L$ is shown. A number of qubits are affected by ${\sigma }_x$ (red dots) or ${\sigma }_z$ (purple dots) errors, leading to excited $Z_p$ operators (or $m$ anyons) and $X_s$ operators (or $e$ anyons). Measuring these operators yields the positions of the excited vertices and plaquettes but reveals no information about the actual physical errors which cause them. A minimum-weight matching error correction procedure applies ${\sigma }_x$ and ${\sigma }_z$ to the qubits marked by the larger red and purple circles. While the ${\sigma }_z$ errors are annihilated properly (up to a trivial loop of multiplication of $Z_p$ operators), the red pair underneath is connected by a topologically nontrivial path across the surface. This introduces a logical error in the state to be protected.

Figure 2

Four-body plaquette operator $Z_p$ (a) and vertex operator $X_s$ (b) as stabilizer generators of surface code inside the lattice. The black dot in the center of the plaquettes are syndrome qubits used to do stabilizer measurement.

Figure 3

An example of a double $X$-cut qubit, with $X_s$ operators turned off. Each $X$-cut hole forms a single $X$-cut logical qubit and there are two kinds of logical operators. $Z_{L_1}$ ($Z_{L_2}$) connects the left (right) hole with the $Z$ boundary on the top of lattice, while $X_{L_1}$ ($X_{L_2}$) are any loops encircling the left (right) hole. For a double $X$-cut qubit, a more convenient way to define the logical operators is to set $X_L=X_{L_1}$ and $Z_L=Z_{L_1}{Z}_{L_2}$. Note that $Z_L$ is equivalent to $Z_{L_1}{Z}_{L_2}$ up to multiplication by $Z_p$ operators inside the loop and has the effect of flip phases for both qubit holes.

Figure 4

Creation of $|+\rangle$ for an $X$-cut double qubit. System Hamiltonians before and after are shown in panels (a) and (b), respectively. Colored squares indicate that the corresponding $X_s$ (yellow) and $Z_p$ (cyan) stabilizer generators are turned on.

Figure 5

Enlarging a hole of an $X$-cut logical double qubit adiabatically. Colored squares indicate that the corresponding $X_s$ (yellow) and $Z_p$ (cyan) stabilizer generators are turned on. The yellow qubits in (b) and (c) indicate that ${\sigma }_x$ for that qubit is turned on in the Hamiltonian. Adiabatic evolution between (a) and (b) maps $X_L$ to $X_L^{\prime }$ and $Z_L$ to $Z_L^{\prime }$. Similarly, adiabatic evolution between (b) and (c) maps $X_L^{\prime }$ to $X_L^{\prime \prime }$ and $Z_L^{\prime }$ to $Z_L^{\prime \prime }$.

Figure 6

Error propagation during an adiabatic process to enlarge a hole of an $X$-cut logical qubit. Colored squares indicate that the corresponding $X_s$, ${\sigma }_x$ (yellow) and $Z_p$, ${\sigma }_z$ (cyan) operators are turned on. The purple circle around a qubit indicates that a ${\sigma }_z$ error occurs on that qubit. (a) A ${\sigma }_z$ error occurs on qubit 1. (b) Effective errors after the adiabatic process. (c) An additional ${\sigma }_z$ error occurs on qubit 4. (d) Effective errors after the adiabatic procedure to enlarge the hole will cause a logical error after decoding.

Figure 7

Adiabatic process for moving a $Z$-cut logical qubit hole horizontally right. Colored squares indicate that the corresponding $X_s$, ${\sigma }_x$ (yellow) and $Z_p$, ${\sigma }_z$ (cyan) operator are turned on. Logical operators of the qubit are $X_L$ and $Z_L$ in (a). An adiabatic process between (a) and (b) maps $X_L$ to $X_L^{\prime }$ and $Z_L$ to $Z_L^{\prime }$. Similarly, an adiabatic process between (c) and (d) maps $X_L^{\prime }$ to $X_L^{\prime \prime }$ and $Z_L^{\prime }$ to $Z_L^{\prime \prime }$.

Figure 8

Error propagation during adiabatic process to move a hole of an $X$-cut logical double qubit horizontally right. Colored squares and qubits indicate that the corresponding $X_s$, ${\sigma }_x$ (yellow) and $Z_p$, ${\sigma }_z$ (cyan) operators are turned on. The purple circle around a qubit indicates that a ${\sigma }_z$ error occurs on that qubit and a red one indicates that a ${\sigma }_x$ error occurs. (a) ${\sigma }_z$ errors occur on qubits 1 and 2. (b) Effective errors caused by ${\sigma }_{z_1}$ and ${\sigma }_{z_2}$ after an adiabatic process. (c) ${\sigma }_x$ errors occur on qubits 1 and 2. (d) Effective errors caused by ${\sigma }_{x_1}$ and ${\sigma }_{x_2}$ after an adiabatic process.

Figure 9

Scheme to fault-tolerantly detect errors occurring on the boundary. (a) Before the movement, an error occurs on the boundary. (b) After expanding the hole $d/8$ units rightward, the error propagates to a strip of errors. We measure all qubits in the dashed box and determine if the corresponding error happened on the boundary based on the majority vote of the measurement outcomes of each row of qubits.

Figure 10

Creation of $|0\rangle$ state for an $X$-cut double qubit. (a) Create a $|{+}_{DL}^X\rangle$ with two holes attached to each other. Measure ${\sigma }_{z_1}$, ${\sigma }_{z_2}$, and ${\sigma }_{z_3}$ and do a majority vote to determine whether $|0_{DL}^X\rangle$ or $|1_{DL}^X\rangle$ is prepared. (b) Move two holes apart to increase error correction ability of ${\sigma }_z$ errors of the logical qubit. Note that both ${\sigma }_z$ and ${\sigma }_x$ errors on qubits between two holes during the adiabatic movement have no uncorrectable effect on logical state $|0_{DL}^X\rangle$ or $|1_{DL}^X\rangle$ and can be left for future error correction.

Figure 11

Adiabatic braiding process of a $Z$-cut hole (dark blue) around an $X$-cut hole (orange). The operator $X_{L_1}$ has been stretched to multiply a loop of ${\sigma }_x$ operators which is equivalent to $X_{L_2}$ up to multiplication by $X_s$ stabilizer generators (yellow) inside the loop, while $X_{L_2}$ remains the same under the transformation.

Figure 12

Adiabatic braiding process of a $Z$-cut hole (dark blue) around an $X$-cut hole (orange). The operator $Z_{L_2}$ has been stretched to form a strip of ${\sigma }_z$ operators, which is equivalent to $Z_{L_1}$ up to multiplication by $Z_p$ stabilizer generators (cyan) inside the strip, while $Z_{L_1}$ remains the same under the transformation.

Figure 13

State injection for an $X$-cut qubit. Colored squares indicate that the corresponding $X_s$ (yellow) and $Z_p$ (cyan) operator is turned on.

Figure 14

Circuits for logical $|Y\rangle$ distillation from imperfect $|\tilde{Y}\rangle$ states.

Figure 15

Circuits for logical $|A\rangle$ distillation from imperfect $|\tilde{A}\rangle$ states.

Figure 16

A dislocation in the geometry of the Hamiltonian produced by shifting the stabilizer generators along a line between two twists. The stabilizer generators corresponding to two different parallelograms (yellow-cyan and cyan-yellow) and a pentagon (dark gray) are shown on the right side. A pair of anticommuting strings of Pauli operators $L_1$ (solid red) and $L_2$ (dashed blue) that commute with all stabilizer generators forms the logical operators of the extra qubit $\mathcal{F}$ attached to the pair of twists.

Figure 17

Adiabatically moving a pairs of holes of a $Z$-cut qubit (dark blue holes) across a twist on the surface to get a logical Hadamard gate. This process will transform a $Z$-cut qubit to an $X$-cut qubit (orange holes).

Figure 18

Logical error rate per $m$ time steps for various values of $m$ and $d$. The dashed lines are for $c\beta J=8$ and solid lines for $c\beta J=12$. The blue (top), green (second top), red (third), and yellow (bottom) lines are for $d=7$, $d=11$, $d=15$, and $d=19$, respectively.