Charge qubits and limitations of electrostatic quantum gates

We investigate the characteristics of purely electrostatic interactions with external gates in constructing full single-qubit manipulations. The quantum bit is naturally encoded in the spatial wave function of the electron system. Single-electron transistor arrays based on quantum dots or insulating interfaces typically allow for electrostatic controls where the interisland tunneling is considered constant, e.g., determined by the thickness of an insulating layer. A representative array of $3\times 3$ quantum dots with two mobile electrons is analyzed using a Hubbard Hamiltonian and a capacitance matrix formalism. Our study shows that it is easy to realize the first quantum gate for single-qubit operations, but that a second quantum gate comes only at the cost of compromising the low-energy two-level system that encodes the qubit. We use perturbative arguments and the Feshbach formalism to show that this compromising of the two-level system is a rather general feature for electrostatically interacting qubits and is not just related to the specific details of the system chosen. We show further that full implementation requires tunable tunneling or external magnetic fields.

Figure 1

Model system, $3\times 3$ array: (a) Schematic layout: circles represent quantum dots and the three horizontal and three vertical bars represent the capacitively coupled gates which are connected to the outside world. (b) Same as panel (a) but drawn as a capacitor and tunnel junction network, where black circles represent the qudots and the box symbols in between the dots represent capacitive tunnel junctions.

Figure 2

Potential landscape for array in Fig. 1. (a) Single-particle potential on array with no gate voltages applied $(V_g=0)$. (b) Change of single-particle potential due to a specific set of applied gate voltages breaking the 90° symmetry of the system.

Figure 3

(Color online) (a) Energy level spectrum of the $3\times 3$ system and dependence on the symmetry breaking pattern of gate voltages in Fig. 2(b). Singlet states $\mid s,s_z⟩=\mid 0,0⟩$ are shown with solid lines (red), while triplet states $\mid s,s_z⟩=\mid 1,m=\{+1,0,-1\}⟩$ are shown as dashed (black). (b) and (c) Probability distribution over the $3\times 3$ array of the ground pair (qu2LS) for singlet states. Notice equal probability for spin up and spin down $({\mid {\psi }_{\uparrow }\mid }^2={\mid {\psi }_{\downarrow }\mid }^2)$. (d) and (e) Lowest triplet states. Case chosen $(s_z=+1)$ has only a spin up component (note, however, that the spatial probability distribution is the same for the threefold-degenerate triplet states).

Figure 4

(Color online) Coherent manipulation of the singlet state under gate action. (a) Evolution of the qubit in the Bloch sphere representation after projection onto the basis of the (initial) eigenspectrum at $V_g=0$. The Bloch sphere is shown in black, and the evolution of the Bloch vector in the qu2LS is shown by the solid line (blue). (b) Same as panel (a), but side view, showing the slight size reduction of the effective Bloch sphere due to the adiabatic interaction with the higher-lying reservoir of states. (c) Coherent rotation of the basis states for the sequence $A\to B\to A$ in (b) in the direction indicated with $t=10\mu \mathrm{eV}$. The probability distribution in real space is shown for the $3\times 3$ array over equally spaced time intervals in a total time window of $0.56\mathrm{ns}$, which corresponds to one period for this tunneling based action.

Figure 5

Numerical exploration of effective (pseudomagnetic) fields from a random sequence of gate voltages (4096 configurations for every $t$ value). (a) Energy level splitting between the lowest two eigenstates (the qu2LSs), $\delta$, as well as the level splitting between the second and third eigenstates, $\Delta$, shown with and without gate voltages applied. A well-behaved two-level system exists for $t\lesssim 0.5\times {10}^{-5}\mathrm{eV}$, while for larger $t$ higher-lying states cross over. (b) Sampling the gate voltages randomly, the resulting minimum and maximum pseudomagnetic fields are recorded ($H=a1+\vec{\mathcal{B}}\bullet \vec{\sigma }$, and thus $\vec{\mathcal{B}}$ has units of energy). Since ${\mathcal{B}}_{x,\mathit{min}}$ and ${\mathcal{B}}_{x,\mathit{max}}$ are very similar, the difference $\Delta {\mathcal{B}}_x$ is shown explicitly by the (blue) dashed line. ${\mathcal{B}}_{z,\mathit{min}∕\mathit{max}}$ values are clearly discernible. Note that ${\mathcal{B}}_x$ is directly related to the gap in the ground state [${\delta }_0$ in (a)].