Quantum computing with Majorana fermion codes

We establish a unified framework for Majorana-based fault-tolerant quantum computation with Majorana surface codes and Majorana color codes. All logical Clifford gates are implemented with zero-time overhead. This is done by introducing a protocol for Pauli product measurements with tetrons and hexons which only requires local 4-Majorana parity measurements. An analogous protocol is used in the fault-tolerant setting, where tetrons and hexons are replaced by Majorana surface code patches, and parity measurements are replaced by lattice surgery, still only requiring local few-Majorana parity measurements. To this end, we discuss twist defects in Majorana fermion surface codes and adapt the technique of twist-based lattice surgery to fermionic codes. Moreover, we propose a family of codes that we refer to as Majorana color codes, which are obtained by concatenating Majorana surface codes with small Majorana fermion codes. Majorana surface and color codes can be used to decrease the space overhead and stabilizer weight compared to their bosonic counterparts.

Figure 1

Tetrons and hexons as the two smallest Majorana fermion codes, and their correspondence to 4 or 6 Majoranas in a fixed-parity sector, e.g., via the quantum-wire construction in Ref. [33].

Figure 2

The three uniform three-colorable tilings of the 2D plane (a), and three examples of nonuniform tilings (b).

Figure 3

Logical tetrons of the 4.8.8 (a), 6.6.6 (b), and 4.6.12 (c) Majorana surface code with code distances $d=1$, 3, and 5. The 6.6.6 Majorana surface code with $d=5$ shows an example of strings of red and blue edges through the bulk of the code. Products of Majoranas along the red and blue strings are equivalent to logical $X$ and $Z$ operators, respectively.

Figure 4

Logical hexons of the 4.8.8 (a), 6.6.6 (b), and 4.6.12 (c) Majorana surface code with code distances $d=1$, 3, and 5.

Figure 5

(a) Applying a $T$ gate to a qubit $\left|\psi \right.〉$ is equivalent to a CNOT between the qubit and a magic state $\left|m\right.〉=\frac{1}{\sqrt{2}}(\left|0\right.〉+e^{i\pi /4}\left|1\right.〉)$, a measurement of the magic state with outcome $m\in \{0,1\}$, and a corrective operation $S^m$ on the qubit. (b) Example of a circuit of Clifford gates and four $Z$ measurements that is compressed into four Pauli product measurements.

Figure 6

Array of tetrons and hexons, where hexons are used to encode data qubits, and tetrons are used as ancillas for Pauli product measurements.

Figure 7

Protocol for the measurement of the Pauli product operator $Z_1\otimes X_3\otimes Z_8\otimes Y_9$.

Figure 8

Lattice surgery between two 4.8.8 Majorana surface code tetrons with code distance $d=5$ measuring $Z\otimes Z$. The box in the bottom left corner shows the corresponding measurement with $d=1$, where $Z\otimes Z$ is the red 4-Majorana operator.

Figure 9

Lattice surgery protocols for the measurements of the operators $X\otimes X$ (a), $Z\otimes X$ (b), and $Y\otimes X$ (c) between a 4.8.8 Majorana surface code hexon and tetron. The insets show the equivalent operation with $d=1$.

Figure 10

Lattice surgery protocol for the measurement of $Y\otimes X$ between a 4.8.8 Majorana surface code hexon and tetron. The hexon is modified to feature hexons instead of tetrons at the bottom and top red boundaries. As a consequence, the weight of the twist defect (orange) reduces to 8 Majoranas.

Figure 11

Lattice surgery protocol for the measurement of $Y\otimes X$ between 6.6.6 (a) and 4.6.12 (b) Majorana surface code hexons and tetrons. The maximum weight of the operators during lattice surgery is 8 Majoranas for 6.6.6 codes and 6 Majoranas for 4.6.12 codes.

Figure 12

State injection protocols to convert a state encoded in a physical tetron into a logical 4.8.8 (a), 6.6.6 (b), or 4.6.12 (c) tetron for the example of $d=5$.

Figure 13

A bosonic surface code (a) concatenated with a [[4,2,2]] code yields a bosonic 4.8.8 code (b). This can also be drawn as a Majorana fermion code (c) by replacing each qubit with a tetron (yellow lines). This Majorana color code can also be obtained by replacing each tetron of the 4.8.8 Majorana surface code with a ${\left[[16,2,4]\right]}_m$ code. (e) Stabilizers and logical operators of the ${\left[[16,3,4]\right]}_m$ code presented in Ref. [24]. (f) Order-3 Majorana color code obtained by concatenating a 4.8.8 Majorana surface code with the ${\left[[16,3,4]\right]}_m$ code. Each yellow 16-Majorana box corresponds to a ${\left[[16,3,4]\right]}_m$ code. Since the stabilizers overlap, they are shown in two separate figures.

Figure 14

Scheme to obtain a 4.8.8 (${\left[[6,2,2]\right]}_m$) Majorana color code by concatenating the 4.8.8 Majorana surface code with a ${\left[[6,2,2]\right]}_m$ code, i.e., by stacking two surface codes and replacing each pair of tetrons with a hexon.

Figure 15

Scheme for quantum computation in an array of tetrons and hexons, where the hexons are used to encode qubits, and tetrons are used as ancillas for Pauli product measurements. One part of the array is used to encode the data qubits for quantum computation, whereas the rest of the quantum computer is used to distill magic states.

Figure 16

6.6.6 and 4.6.12 surface code tetrons in a square lattice of hexons and dodecons, respectively.

Figure 17

One choice of mapping of 4.8.8, 6.6.6, and 4.6.12 Majorana surface codes onto bosonic surface codes. For bosonic surface codes, the dark plaquettes are products of $X$ operators, and the light plaquettes are products of $Z$ operators.

Figure 18

A $T$ gate on a qubit $\left|\psi \right.〉$ is equivalent to a $Z\otimes Z$ measurement between $\left|\psi \right.〉$ and a magic state $\left|m\right.〉$ with outcome $m_1$, which results in a corrective $S^{m_1}$ operation on $\left|\psi \right.〉$. In order to disentangle $\left|m\right.〉$ from the qubit, it is measured in the $X$ basis, prompting a $Z^{m_2}$ Pauli correction on $\left|\psi \right.〉$.

Figure 19

A multitarget CNOT gate between a control qubit $\left|c\right.〉$ and multiple target qubits $\left|t_i\right.〉$ is equivalent to a $Z\otimes Z$ measurement between $\left|c\right.〉$ and an ancilla initialized in the $\left|+\right.〉$ state, followed by a $X^{\otimes n+1}$ measurement between the ancilla and the $n$ target qubits. Finally, the ancilla is measured in the $Z$ basis. The final Pauli corrections depend on the measurement results.

Figure 20

Pauli product measurement protocol from Fig. 7 realized with $d=3$ 4.8.8 Majorana surface codes. Only steps 1, 2, and 3 are shown.

Figure 21

Stabilizer configuration for $Z\otimes Z$ measurements between two 6.6.6 Majorana surface code tetrons, and for $X\otimes X$ and $Z\otimes X$ measurements between a 6.6.6 hexon and a 6.6.6 tetron.

Figure 22

Stabilizer configuration for $Z\otimes Z$ measurements between two 4.6.12 Majorana surface code tetrons, and for $X\otimes X$ and $Z\otimes X$ measurements between a 4.6.12 hexon and a 4.6.12 tetron.

Figure 23

Stabilizer configuration for $Y\otimes Z$ measurements between a 4.6.12 Majorana hexon and tetron.

Figure 24

Stabilizer configuration for $Y\otimes Z$ measurements between a 6.6.6 Majorana hexon and tetron. Some of the physical hexons located at the boundary have been replaced by octons. As a consequence, the maximum stabilizer weight reduces to 6 Majoranas.

Figure 25

Twist-based lattice surgery between a logical hexon and a logical tetron encoded in a $d=3$ bosonic subsystem surface code in order to measure $Y\otimes Z$. The Majorana fermion code obtained by replacing each qubit of a bosonic subsystem surface code tetron with 4 Majoranas in a fixed-parity sector resembles a 4.6.12 Majorana surface code.

Figure 26

Tunnel-coupling configurations for the Pauli product measurements protocol from Fig. 7 realized with the wire-based tetrons and hexons proposed in Ref. [33].

Figure 27

Nanowire-based implementations of 4.8.8 (a), 6.6.6 (b), and 4.6.12 (c), with examples of tunnel-coupling configurations for red and blue stabilizer measurements. Coherent links (d) may be necessary for the measurement of other operators.

Figure 28

4.8.8 (${\left[[6,2,2]\right]}_m$) Majorana color code in a 2D array of hexons.

Figure 29

Procedure to obtain a 4.8.8 $\left({\left[[8,3,2]\right]}_m\right)$ Majorana color code. In order to prevent stabilizer weights from increasing beyond 10 Majoranas, the tetrons of a 4.8.8 Majorana surface code are periodically labeled A, B, C, and D. Each tetron is replaced by a single octon, but the definition of the octon's logical operators depends on the label of the corresponding tetron. In the figure, the octon's $Z$ ($X$) operators correspond to products of blue (red) Majoranas. The resulting stabilizers have a maximum weight of 10 Majoranas, as shown for the three overlapping blue $Z$ type stabilizers.

Figure 30

Stabilizers ${\mathcal{O}}_1-{\mathcal{O}}_4$ and logical operators of the ${\left[[20,4,4]\right]}_m$ code presented in Ref. [24].

Figure 31

Lattice surgery protocol for the measurement of the operator $Z_1\otimes Z$ between a bosonic 4.8.8 color code tetron and a 4.8.8 Majorana surface code tetron. $Z_1$ is the $Z$ operator of the first qubit encoded in the bosonic 4.8.8 color code tetron.