Quantum Computing with Spin Cluster Qubits

We study the low energy states of finite spin chains with isotropic (Heisenberg) and anisotropic ($XY$ and Ising-like) antiferromagnetic exchange interaction with uniform and nonuniform coupling constants. We show that for an odd number of sites a spin cluster qubit can be defined in terms of the ground state doublet. This qubit is remarkably insensitive to the placement and coupling anisotropy of spins within the cluster. One- and two-qubit quantum gates can be generated by magnetic fields and intercluster exchange, and leakage during quantum gate operation is small. Spin cluster qubits inherit the long decoherence times and short gate operation times of single spins. Control of single spins is hence not necessary for the realization of universal quantum gates.

Figure 1

(a) The states of the spin cluster [Eq. (1)] define the spin cluster qubit. (b) $|0⟩$ and $|1⟩$ have a complicated representation in the single-spin product basis, as evidenced by the local spin density. (c) Quantum gates are generated by magnetic fields or $g$-factor engineering (one-qubit gates) and a switchable interqubit exchange $J_{*}(t)$ (two-qubit gates).

Figure 2

(a) Local spin density within a spin cluster qubit ($n_c=5$) as function of $\varphi \propto g{\mu }_B{B}_{x}t/\hslash$ obtained by integration of the full Schrödinger equation for homogeneous (solid line) and inhomogeneous $B_x$ (dashed and dashed-dotted lines). (b) For $B_x\ll \Delta /g{\mu }_B$ or homogeneous $B_x$, the state rotates coherently from $|0⟩$ to $|1⟩$. For a magnetic field acting only on the central spin of the cluster, the leakage increases to 7% for $g{\mu }_B{B}_x=0.5J$.

Figure 3

CNOT gate for two small spin cluster qubits ($n_c=3$) obtained by numerical integration of the Schrödinger equation [see Fig. 1c]. The plotted probabilities and the phases (not displayed) show that (a) $|00⟩\to |00⟩$ and (b) $|10⟩\to |11⟩$. We have chosen a pulse sequence [Eq. (4)] with instantaneous switching (at times $t_i$), $B=0.1J/g{\mu }_B$, and $J_{*}=0.1J$. Leakage due to instantaneous switching (0.7% for our parameters) can be reduced by decreasing $J_{*}$ and $B$.