Fault-tolerant fermionic quantum computation based on color code

An important approach to fault-tolerant quantum computation is protecting logical information using quantum error correction. Usually, logical information is in the form of logical qubits, which are encoded in physical qubits using quantum error correction codes. Compared with qubit quantum computation, fermionic quantum computation has advantages in quantum simulations of fermionic systems, e.g., molecules. In this paper, we show that fermionic quantum computation can be universal and fault tolerant if we encode logical Majorana fermions in physical Majorana fermions. We take a color code as an example to demonstrate the universal set of fault-tolerant operations on logical Majorana fermions, and we numerically find that the fault-tolerance threshold is about $0.8\%$.

Figure 1

Triangular code based on the 4.8.8 lattice. Each vertex represents a Majorana fermion mode, and each plaquette represents a stabilizer generator. The side length (the number of vertices on the side) of the lattice is $d=9$. Blue, squares; green, complete or incomplete hexagons with dotted perimeters; red, other complete or incomplete hexagons.

Figure 2

Circuits for Majorana fermions. Each horizontal line represents a Majorana fermion mode. (a) A phase gate $S_{a,b}$ is equivalent to two exchange gates $R_{a,b}$. (b) The state transfer operation ${\mathcal{T}}_{a,b,c}$ transfers the state of $a$ to $c$. Green boxes (opened to right) represent initialization operations, and red boxes (opened to left) represent measurement operations. The phase gate $S_{a,c}$ depends on the measurement outcome of $iab$ (indicated by the dotted line). (c) The exchange gate $R_{a_1,a_2}$ realized using a sequence of state transfer operations. (d) The gate $T_{a_1,a_2}$ realized using the magic state and state transfer operations. The magic state is the eigenstate with $i{b}_1{c}_{-}=1$ and $i{b}_2{c}_{+}=1$, where $c_{\pm }=\frac{1}{\sqrt{2}}(c_1\pm c_2)$. Outcome-dependent phase gates in ${\mathcal{T}}_{a_1,b_1,c_{-}}$ and ${\mathcal{T}}_{a_2,b_2,c_{+}}$ are equivalent to the outcome-dependent exchange gate $R_{c_2,c_1}$ (indicated by dotted lines), which can be realized using the circuit in (c). (e) The circuit for measuring the stabilizer operator of eight Majorana fermion modes $c_1,c_2,...,c_8$ using ancillary Majorana fermion modes $a_1,a_2,...,a_8$. Each pair of ancillary modes is initialized in the eigenstate $i{a}_{2i}{a}_{2i+1}=1$ ($a_9=a_1$). Parity projections are performed on each group of four Majorana fermion modes, and outcomes are $c_{2i-1}{c}_{2i}{a}_{2i-1}{a}_{2i}={\upsilon }_{2i-1,2i}$. The measurement outcome of the stabilizer operator is $c_1{c}_2\cdots c_8=-{\upsilon }_{1,2}{\upsilon }_{3,4}\cdots {\upsilon }_{7,8}$. After parity projections, ancillary modes are measured, and outcomes are $i{a}_{2i}{a}_{2i+1}={\eta }_{2i,2i+1}$. To complete the stabilizer measurement, phase gates need to be performed according to these outcomes as shown in Table 1.

Figure 3

(a) Array of logical Majorana fermions. Each right triangle represents a logical Majorana fermion. The red solid line denotes the type-I measurement, and the green dashed line denotes the type-II measurement. The parity projection (PP) can be performed on a group of four logical Majorana fermions (e.g., these four enveloped by the dotted line). The magic state can be prepared on such four logical Majorana fermions. The region in gray is filled by ancillary Majorana fermions. The dimension in the figure does not reflect the actual dimension on the color code lattice. (b) Circuits for the magic state distillation and duplication.

Figure 4

Operations on logical Majorana fermions. Each logical Majorana fermion mode is encoded in a triangular lattice, the boundary of which is marked by the dashed line. Ancillary Majorana fermion modes are covered by white bars. Each white bar covers two vertices and denotes $i{c}_i{c}_j$, where $c_i$ and $c_j$ are two corresponding Majorana fermion operators. At the beginning of the logical operation, ancillary modes are initialized according to white bars. Then stabilizer measurements are performed on the entire lattice to read eigenvalues of stabilizer operators. Stabilizer measurements are repeated for $\sim d$ rounds. Finally, ancillary modes are measured according to white bars. In (a) and (c), the separation between dashed lines is $\sim d$. In (b), the thickness of the region above the dashed line is $\sim d$. In (d), the lattice is a square with the side length $\sim d$. Here, $d$ is the side length of logical Majorana fermions. Blue, squares; green, complete or incomplete hexagons with dotted perimeters; red, other complete or incomplete hexagons.

Figure 5

Lattice formed by stabilizer operators and errors. Each circle denotes a stabilizer operator, and each edge denotes an error that flips two corresponding red and green stabilizer operators but does not flip any blue stabilizer operator. Blue, squares; green, complete or incomplete hexagons with dotted perimeters; red, other complete or incomplete hexagons.

Figure 6

The rate of logical errors as a function of the rate of errors in parity projections. On the left side of the vertical dashed line, the code distance is $d=5,9,13,17,21,25,29$ for the curves from top to bottom. Standard deviations of logical error rates are smaller than the size of circles.

Figure 7

The probability that two transformations are not parallelizable $P$ as a function of the total number of modes $N$.