Molecular Prototypes for Spin-Based CNOT and SWAP Quantum Gates

We show that a chemically engineered structural asymmetry in $[{\mathrm{Tb}}_2]$ molecular clusters renders the two weakly coupled ${\mathrm{Tb}}^{3+}$ spin qubits magnetically inequivalent. The magnetic energy level spectrum of these molecules meets then all conditions needed to realize a universal CNOT quantum gate. A proposal to realize a SWAP gate within the same molecule is also discussed. Electronic paramagnetic resonance experiments confirm that CNOT and SWAP transitions are not forbidden.

Figure 1

(a) Illustration of a quantum CNOT operation on two coupled spin qubits. (b) Synthesis [26] of an asymmetric $[{\mathrm{Tb}}_2]$ complex using three ${\mathrm{H}}_3\mathrm{L}$ ligands. (c) The resulting ${\mathrm{N}}_2{\mathrm{O}}_6\mathrm{Cl}$ and ${\mathrm{N}}_3{\mathrm{O}}_6$ coordination polyhedra around the two ${\mathrm{Tb}}^{3+}$ ions exhibit ${\mathrm{C}}_1$ symmetry and ${\mathrm{C}}_{4v}$ symmetry, respectively 28, which induces a misalignment between their anisotropy axes: left (purple) arrow, “control”; right (blue) arrow, “target”; large light (orange) ball, Cl; small dark (red) balls, O; small light (blue) balls, N.

Figure 2

(a) ac susceptibility of polycrystalline $[{\mathrm{Tb}}_2]$ at different frequencies. Main panel: ${\chi }^{\prime }T$ product (left axis) from which ${\mu }_{\mathrm{eff}}$ is determined (right axis). Inset: ${\chi }^{\prime }$ (solid symbols) and ${\chi }^{\prime \prime }$ (open symbols). dc-susceptibility data measured at ${\mu }_{0}H=0.1\mathrm{T}$ are also shown. The lines are least squares fits of the ac (solid lines) and dc (dotted lines) susceptibilities for collinear ($\delta =0$, red thin lines) and noncollinear ($\delta =66°$, blue thick lines) anisotropy axes. (b) Zero-field energy level structure of $[{\mathrm{Tb}}_2]$ derived from these fits.

Figure 3

(a) Magnetic heat capacity of a powdered sample of $[{\mathrm{Tb}}_2]$. (b) Theoretical predictions for noncollinear anisotropy axes.

Figure 4

Magnetization isotherms of $[{\mathrm{Tb}}_2]$. The data at $T=0.26\mathrm{K}$ are compared with calculations made for collinear ($\delta =0$, dotted line) and noncollinear ($\delta =66°$, solid line) anisotropy axes.

Figure 5

(a) $X$-band EPR spectrum of powdered $[{\mathrm{Tb}}_2]$. The solid line is a fit with two Lorentzian absorption contributions, shown by the dotted and dashed lines, respectively. (b) Field-dependent magnetic energy levels of $[{\mathrm{Tb}}_2]$ calculated by numerical diagonalization of Eq. (1) for $\delta =66°$. Only levels corresponding to magnetic nuclear states $m_{I,1}=m_{I,2}=-3/2$ are shown.