Majorana Fermion Surface Code for Universal Quantum Computation

We introduce an exactly solvable model of interacting Majorana fermions realizing $Z_2$ topological order with a $Z_2$ fermion parity grading and lattice symmetries permuting the three fundamental anyon types. We propose a concrete physical realization by utilizing quantum phase slips in an array of Josephson-coupled mesoscopic topological superconductors, which can be implemented in a wide range of solid-state systems, including topological insulators, nanowires, or two-dimensional electron gases, proximitized by $s$-wave superconductors. Our model finds a natural application as a Majorana fermion surface code for universal quantum computation, with a single-step stabilizer measurement requiring no physical ancilla qubits, increased error tolerance, and simpler logical gates than a surface code with bosonic physical qubits. We thoroughly discuss protocols for stabilizer measurements, encoding and manipulating logical qubits, and gate implementations.

Popular Summary

Recent remarkable advances in superconducting qubit technology hold promise for building a quantum computer in the near future. Reliable quantum computing, however, depends crucially on the ability to correct for environmentally induced errors. Appealing schemes have been proposed to overcome this issue by tracking errors via measurements during information processing and readout. Here, we propose a practical blueprint for building a fault-tolerant fermionic quantum computer inspired by the “surface code” architecture. We propose a physical platform that is an array of Josephson-coupled topological superconductors that host spatially separated Majorana fermions—particles that are their own antiparticle and may be regarded as entangled “halves” of an electron. These fermions are arranged on a two-dimensional lattice, and they serve as physical qubits.

Our design for a quantum computer combines (i) engineering a strongly interacting system of Majorana fermions with a highly entangled ground state and (ii) measurement-based quantum information processing and error correction. Remarkably, we show that quantum-phase fluctuations induced by the charging energy of topological superconductor islands have the effect of virtually braiding the Majorana fermions and generating the desired multibody interaction that couples several neighboring Majorana fermions at once. The resulting interacting Majorana fermion system hosts anyon excitations, which we use to encode logical qubits. Our proposal is associated with significant error suppression and greater tolerance for measurement errors than conventional surface codes. We demonstrate an ability to detect errors via single-step stabilizer measurements, and we discuss protocols for the manipulation and readout of encoded qubits. Finally, we highlight the key advantages of using Majorana fermions as the underlying degrees of freedom in a quantum computer.

In light of the rapid progress being made in manipulating Majorana fermions in solid-state systems, we hope that our proposal will be experimentally pursued to build a fermionic quantum computer.

Figure 1

We consider a honeycomb lattice with (a) a single Majorana fermion on each lattice site, so the ${\mathcal{O}}_p$ operator is the product of the six Majorana fermions on the vertices of a hexagonal plaquette. The colored plaquettes in (b) correspond to the three distinct bosonic excitations that may be obtained by violating a plaquette constraint.

Figure 2

The action of the commuting Wilson loop operators $W_x$ and $W_y$ is shown above as the product of the Majorana fermions on the lattice sites intersected by the appropriate colored lines. The operator $W_{\tilde{y}}$ anticommutes with $W_x$ and takes the ground state between two topological sectors.

Figure 3

Array of hexagonal $s$-wave superconducting islands placed on a TI surface. Each arrow points in the direction of the relative phase of the associated island, with $\phi =0$, $\pm 2\pi /3$. This produces a honeycomb lattice of vortices (blue) and antivortices (red) at trijunctions, hosting Majorana fermions.

Figure 4

Schematic of a $2\pi$ phase slip on the central superconducting island in a hexagonal superconducting array on a TI surface, with the phase of the central island indicated in each panel. When the phase difference between neighboring islands is $\pi$, the pair of Majorana fermions on the shared edges couple [4] as indicated. The $2\pi$ phase slip permutes the Majorana fermions as shown, leading to the transformation in Eq. (12).

Figure 5

(a) Schematic of the harmonic oscillator energy levels of the effective Hamiltonian (17), centered at $\phi =2\pi n$, with the $2\pi$ and $4\pi$ phase-slip amplitudes for the lowest energy levels shown. In (b), we show a schematic plot of the two lowest harmonic oscillator levels as a function of the gate charge. The energy splittings ${\mathrm{\Delta }}_1$ and ${\mathrm{\Delta }}_2$ are between states with even ($\mathrm{\Gamma }=+1$) and odd fermion parity ($\mathrm{\Gamma }=-1$) within the first and second harmonic oscillator levels, respectively. Each level within a fixed fermion parity sector is nearly four-fold degenerate.

Figure 6

Logical qubits in the Majorana surface code. In (a) we stop the measurement of two plaquette operators in subsequent surface code cycles, increasing the ground-state degeneracy by a factor of 4. If we take ${\widehat{Z}}_1$ and ${\widehat{Z}}_2$ to be the logical $\widehat{Z}$ operators for the two encoded qubits, the corresponding ${\widehat{X}}_1$ and ${\widehat{X}}_2$ operators are given by Wilson lines connecting to the boundary. The two qubits may be coherently manipulated by applying the operator ${\widehat{X}}_{12}$ as shown. In practice, it is simpler to define logical qubits by stopping the measurement of pairs of plaquettes of a single type, with the logical $\widehat{X}$ and $\widehat{Z}$ defined as shown in (b). We may also consider a logical qubit made of several “holes,” as in (c), to minimize errors during qubit manipulation.

Figure 7

We may move a logical qubit defined by $\widehat{X}$ and $\widehat{Z}$ operators along a given sublattice. We first multiply the logical $\widehat{Z}$ by ${\widehat{\mathcal{O}}}_r$. After measuring $i\gamma \eta$ in the next code cycle, we extend the logical $\widehat{X}\to \widehat{X}\otimes i\gamma \eta$. Finally, we begin measuring $\widehat{Z}$ in the next surface code cycle and restore ${\widehat{\mathcal{O}}}_p$ and ${\widehat{\mathcal{O}}}_q$ to six-Majorana operators.

Figure 8

CNOT gate. Braiding two logical qubits to perform a logical CNOT. In (a), a possible trajectory for braiding the first qubit around the second is indicated by the dotted line. Since the two qubits live on distinct sublattices, the braiding procedure induces the transformation ${\widehat{X}}_A\to {\widehat{X}}_A\otimes {\widehat{X}}_B$ and ${\widehat{Z}}_B\to {\widehat{Z}}_A\otimes {\widehat{Z}}_B\otimes \widehat{\mathcal{O}}$, where $\widehat{\mathcal{O}}$ is the product of the colored plaquettes shown. As a result, this operation performs a CNOT transformation on the braided qubit.

Figure 9

Braiding processes that implement the transformation (a) ${\widehat{Z}}_a\to {\widehat{Z}}_a\otimes {\widehat{Z}}_c\otimes {\widehat{Z}}_t\otimes {\widehat{Z}}_{\mathrm{out}}$ up to an overall sign, as determined by the product of the appropriate plaquette operators enclosed by the path $\ell$, and (b) ${\widehat{Z}}_{\mathrm{out}}\to {\widehat{Z}}_{\mathrm{out}}\otimes {\widehat{Z}}_A$, ${\widehat{Z}}_t\to {\widehat{Z}}_t\otimes {\widehat{Z}}_A$, ${\widehat{Z}}_c\to {\widehat{Z}}_c\otimes {\widehat{Z}}_A$. The two braids are used to realize CNOT gates between two (a) $A$-type and (b) $B$-type logical qubits, respectively. By convention, we take the lowest qubit enclosed by the braiding trajectory to be the control for the logical CNOT.

Figure 10

Hadamard gate. A logical Hadamard is performed by transferring a qubit between distinct sublattices so that the logical $\widehat{X}$ and $\widehat{Z}$ operators are exchanged. We do this by taking the qubit in (a) and multiplying the logical $\widehat{X}$ by the plaquette operators $\{{\widehat{\mathcal{O}}}_{{\mu }_k}\}$ and the logical $\widehat{Z}$ by ${\widehat{\mathcal{O}}}_p$ and ceasing measurement of the fermion parity of plaquette $p$, yielding the operators shown in (b). Next, we measure the product $(i{\eta }_1{\eta }_2)(i{\eta }_3{\eta }_4)\dots$ and ${\widehat{\mathcal{O}}}_q$ and multiply by ${\widehat{Z}}^{\prime }$ and ${\widehat{X}}^{\prime }$, respectively. The final result is shown in (d).

Figure 11

$S$- and $T$-gate ancilla preparation. We create the $|{\phi }_S⟩$ and $|{\phi }_T⟩$ ancilla states, which are needed to realize logical $S$ and $T$ gates by preparing the short qubit [28] shown above. We cease stabilizer measurements on two adjacent plaquettes $p$ and $q$. In the next surface code cycle, we perform a rotation of the two-level system defined by $i\gamma \eta$. Finally, we enlarge the logical qubit by extending one end of the qubit to guarantee stability against noise.