Parafermions in a Kagome Lattice of Qubits for Topological Quantum Computation

Engineering complex non-Abelian anyon models with simple physical systems is crucial for topological quantum computation. Unfortunately, the simplest systems are typically restricted to Majorana zero modes (Ising anyons). Here, we go beyond this barrier, showing that the ${\mathbb{Z}}_4$ parafermion model of non-Abelian anyons can be realized on a qubit lattice. Our system additionally contains the Abelian $D({\mathbb{Z}}_4)$ anyons as low-energetic excitations. We show that braiding of these parafermions with each other and with the $D({\mathbb{Z}}_4)$ anyons allows the entire $d=4$ Clifford group to be generated. The error-correction problem for our model is also studied in detail, guaranteeing fault tolerance of the topological operations. Crucially, since the non-Abelian anyons are engineered through defect lines rather than as excitations, non-Abelian error correction is not required. Instead, the error-correction problem is performed on the underlying Abelian model, allowing high noise thresholds to be realized.

Popular Summary

Anyons are exotic quasiparticles (excitations) that can be found in certain physical systems. Unlike standard bosonic and fermionic particles, braiding anyons—and, specifically, non-Abelian anyons—around each other can lead to interesting effects. Non-Abelian anyons are ideal building blocks for quantum computation. However, as with all quantum computers, an error detection and removal process is critical. Only anyonic systems compatible with error correction will therefore be useful for building a truly scalable quantum computer. To this end, we focus on engineering anyons using the most experimentally well-developed quantum systems, known as qubits.

We also consider only those kinds of interactions that can realistically be engineered, i.e., those involving only two closely located qubits (spin 1/2) on a lattice. Thus far, theoretical proposals for systems that satisfy these constraints have only been able to offer a single kind of non-Abelian anyon (Majorana zero modes, otherwise known as Ising anyons). Although these kinds of anyons are compatible with quantum computation, high overheads are necessary to achieve fault tolerance. A search for more complex anyon models in realistic qubit systems is therefore required, which we present here. We propose a two-dimensional lattice of qubits with nearest-neighbor interactions. We show that this lattice supports a generalization of Majoranas known as parafermion modes. These parafermion modes can implement an important set of basic operations for quantum computation. We also study how to perform error correction in detail, and we design powerful methods to correct errors with high probability.

By harnessing the computational power of non-Abelian parafermions and still applying Abelian error correction, we expect that our findings will pave the way for future studies of feasible quantum computing.

Figure 1

Two qubits are located at each vertex of a Kagome lattice. Each pair of qubits hosts two ${\mathbb{Z}}_4$ parafermions. To unlock their potential for non-Abelian braiding, two such parafermions need to become unpaired, which is achieved by adding a defect line to the lattice. These are strings of strong local operators acting on qubit pairs (encircled). They create unpaired parafermion modes located at their ends (light pentagon-shaped regions consisting of a hexagon and a triangle).

Figure 2

Two sets of logical operators ${\tilde{X}}_1=X{X}^{†}X{X}^{†}\dots$, ${\tilde{Z}}_1=ZZZZ\dots$ (left figure) and ${\tilde{X}}_2=X{X}^{†}X{X}^{†}\dots$, ${\tilde{Z}}_2=Z{Z}^{†}Z{Z}^{†}\dots$ (right figure) that satisfy the commutation relations of four-dimensional generalized Pauli operators.

Figure 3

Phases obtained by braiding the $e$ and $m$ excitations of the $D({\mathbb{Z}}_4)$ model around each other (top), and by braiding the excitations $\psi$ and $r$ of the transformed stabilizers around each other (bottom).

Figure 4

Strings of alternating single-qudit operators of the form ${\omega }^{5/2}{Y}_i+\mathrm{H}.\mathrm{c}.$ or ${\omega }^{5/2}{X}_i{Z}_i^{†}+\mathrm{H}.\mathrm{c}.$ (encircled) are added to the Hamiltonian. These effectively eliminate the qudits on which they act from the code, leading to enlarged, pentagon-shaped stabilizers along the defect string. Parafermion modes reside on the pentagons at the ends of the defect strings (shaded areas). A possible choice for two logical operators ${\tilde{X}}_L$ (top panel) and ${\tilde{Z}}_L$ (bottom panel) satisfying ${\tilde{Z}}_L{\tilde{X}}_L=\omega {\tilde{X}}_L{\tilde{Z}}_L$ is illustrated.

Figure 5

All generators of the single-qudit Clifford group can be performed by braiding quasi-particles. The four circles correspond to the four parafermion modes which are used to store one logical qudit. The left part of the figure illustrates how to perform the logical operators $X$ and $Z$ by braiding the Abelian excitations of the $D({\mathbb{Z}}_4)$ model around the parafermions. The right part demonstrates a logical Hadamard gate $H$, which is performed by braiding the parafermion modes themselves.

Figure 6

Generators $S$ and $T$ of all gates that can be performed on a qudit stored in the fusion space of four parafermions by braiding them.

Figure 7

Performance of a controlled phase gate. Panel (a) shows a logical product state $|g,h⟩$ stored in the fusion space of eight parafermions. The defect line storing a mode ${\psi }_g$ can be split into two defect lines storing $D({\mathbb{Z}}_4)$ charges $e_g$ and $m_g$, respectively. Braiding both endpoints of one of these lines clockwise around the defect line storing the ${\psi }_h$ mode, as shown in (b), produces a phase ${\omega }^{gh}$, as required.

Figure 8

Error rates $p_L$ of the logical operators ${\tilde{X}}_1$ and ${\tilde{Z}}_1$ illustrated in Fig. 2 as a function of the qubit depolarization rate $p$ for code sizes $L=20$, 28, 36, 44, 52. Each data point represents ${10}^4$ logical errors, such that error bars are negligible. We recognize a threshold error rate $p_c\approx 24\%$ for ${\tilde{X}}_1$ and $p_c\approx 10\%$ for ${\tilde{Z}}_1$.

Figure 9

Average lifetimes $\tau$ of the logical qudit with logical operators ${\tilde{X}}_1$ and ${\tilde{Z}}_1$ as a function of code size $L$ for $\lambda =1$, 2, 3, 4, 5. Each data point represents ${10}^4$ experiments. The lifetime is defined as the number of Metropolis steps until the first logical operator detects an error, divided by the number of spins in the code.

Figure 10

Error rates $p_L$ of the logical operators ${\tilde{X}}_L$ and ${\tilde{Z}}_L$ illustrated in Fig. 4 as a function of the qubit depolarization rate $p$ for code sizes $L=20$, 28, 36, 44, 52. Each data point represents ${10}^4$ logical errors, such that error bars are negligible. We recognize a threshold error rate $p_c\approx 24\%$ for ${\tilde{X}}_L$ and $p_c\approx 10\%$ for ${\tilde{Z}}_L$.

Figure 11

Average lifetimes $\tau$ of the logical qudit stored in the defect logical operators $\tilde{X}$ and $\tilde{Z}$ illustrated in Fig. 4 as a function of code size $L$ for $\lambda =1$, 2, 3, 4, 5. Each data point represents ${10}^4$ experiments. The lifetime is defined as the number of Metropolis steps divided by the number of spins in the code.

Figure 12

A selection of double plaquettes used in a braiding operation. Each double plaquette corresponds to two parafermion modes. For a double plaquette $P$, the parafermion to the right is labeled $P_1$ and that to the left is $P_2$.

Figure 13

Double plaquettes used in the working example of a single exchange.

Figure 14

Red circles denote defects as in Eq. (13), with a pair of parafermions (purple) emerging at the ends of the defect line. Blue circles correspond to terms of the form $X_j+X_j^{†}$, while green circles correspond to terms of the form $Z_j+Z_j^{†}$. These grow the $e$ and $m$ parts of the defect line, respectively. Once both the blue and green defect lines have been added, the red defect line can be removed. The ${\psi }_g$ particle initially stored in the red defect line has then been split into its $e_g$ and $m_g$ components.