Quantum computing with sine-Gordon qubits

A universal quantum computing scheme, with a universal set of logical gates, is proposed based on networks of one-dimensional (1D) quantum systems. The encoding of information is in terms of universal features of gapped phases, for which effective field theories such as sine-Gordon field theory can be employed to describe a qubit. Primary logical gates are from twist, pump, glue, and shuffle operations that can be realized in principle by tuning parameters of the systems. Our scheme demonstrates the power of 1D quantum systems for robust quantum computing.

Figure 1

Networks of sine-Gordon qubits. 1D systems (gray lines) are connected in networks forming square lattice (left) or triangular lattice (right). Each qubit is formed by edges of shaded plaquette. Unshaded plaquettes do not encode qubits. Single qubit gates are from operations on edges and plaquette, while entangling gates also involve glue operations on vertices. The (green) loops represent glued configurations of multiple qubits when the quantum switch at a shared corner converts the singlets from the initial ones (dashed) to the final ones (solid).

Figure 2

The singlet configurations at the corner of the 2D square lattice (a) and triangular lattice (b). A spin-$\frac{1}{2}$ is shown as a dot, and a singlet as a bar. The 0 (1) labels logical state $|0\rangle \left(|1\rangle \right)$ of a qubit. The state $|01\rangle$ is a rotated version of $|10\rangle$ (a), states $|001\rangle ,|010\rangle$ are rotated versions of $|100\rangle$, and $|011\rangle ,|011\rangle$ are rotated versions of $|110\rangle$ (b). The glued states are $|\mathrm{\Phi }\rangle$ (a) and $|\mathrm{\Psi }\rangle$ (b). The corner sites are labeled as u(p), d(own), l(eft), r(ight) (a), ul, ur, lu, ld, ru, and rd (b).