Universal quantum computation with ordered spin-chain networks

It is shown that anisotropic spin chains with gapped bulk excitations and magnetically ordered ground states offer a promising platform for quantum computation, which bridges the conventional single-spin-based qubit concept with recently developed topological Majorana-based proposals. We show how to realize the single-qubit Hadamard, phase, and $\pi /8$ gates as well as the two-qubit controlled-not (cnot) gate, which together form a fault-tolerant universal set of quantum gates. The gates are implemented by judiciously controlling Ising exchange and magnetic fields along a network of spin chains, with each individual qubit furnished by a spin-chain segment. A subset of single-qubit operations is geometric in nature, relying on control of anisotropy of spin interactions rather than their strength. We contrast topological aspects of the anisotropic spin-chain networks to those of $p$-wave superconducting wires discussed in the literature.

Figure 1

Initialization of an $N$-site ($N=3$ as drawn) chain in a Greenberger-Horne-Zeilinger state $|+\rangle$. See text for details.

Figure 2

(a) Application of the phase gate in the $|\pm \rangle$ basis to a three-site spin chain $|\Rightarrow \rangle$, as explained in the text. (b) Crude schematics for a longer chain: The blue (dark gray) regions show the “topological” Ising sections in the absence of magnetic field, with degenerate ground states. The red (light gray) regions are ordinary nondegenerate spin-polarized sections subjected to a large magnetic field in the $z$ direction. The white arrows show the direction of the Ising interaction axis at each stage. The topological region encoding a qubit is first pushed to the right by raising the magnetic field to its left and lowering it to its right. The key operation is rotating the Ising axis from the $x$ direction to $y$ in the second step. The Hamiltonian returns to its original form in the last step, while the qubit undergoes the phase-gate transformation (7). Note that there need not be a full control of individual sites, only anisotropies over the relevant segments of the entire chain.

Figure 3

(a) Two fused spin chains in the Majorana representation. The end Majorana fermions ${\gamma }_1$ and ${\tilde{\gamma }}_N$ form a fermionic zero mode $b=({\gamma }_1+i{\tilde{\gamma }}_N)/2$, while the junction Majorana fermions ${\tilde{\gamma }}_M$ and ${\gamma }_{M+1}$ fuse into an ordinary fermion $a=({\tilde{\gamma }}_M+i{\gamma }_{M+1})/2$, with energy $\epsilon =2\mathfrak{t}\cos \phi$. Here $\mathfrak{t}$ is the exchange coupling across the $\{M,M+1\}$ junction, and $\phi$ is the relative tilt of the Ising exchange interactions in the segments $1⩽i⩽M$ and $M+1⩽i⩽N$. (b) When $\phi =0$, occupying fermion state $a$ creates the spin domain wall sketched in the figure, while the domain wall corresponds to the empty fermion state when $\phi =\pi$.