Exchange interaction between three and four coupled quantum dots: Theory and applications to quantum computing

Several prominent proposals have suggested that spins of localized electrons could serve as quantum computer qubits. The exchange interaction has been invoked as a means of implementing two qubit gates. In this paper, we analyze the strength and form of the exchange interaction under relevant conditions. We find that, when several spins are engaged in mutual interactions, the quantitative strengths or even qualitative forms of the interactions can change. A variety of interaction forms can arise depending on the symmetry of the system. It is shown that the changes can be dramatic within a Heitler-London model. Hund-Mülliken calculations are also presented, and support the qualitative conclusions from the Heitler-London model. The effects need to be considered in spin-based quantum computer designs, either as a source of gate error to be overcome or a new interaction to be exploited.

Figure 1

Matrix elements relevant to three-electron case. Arrows indicate transition from localized state on initial dot to localized state on final dot.

Figure 2

Selected matrix elements relevant to four-electron case. Arrows indicate transition from localized state on initial dot to localized state on final dot.

Figure 3

Plot of $K$ as a function of dimensionless tunneling barrier $x_b$ and Coulomb energy $x_c$ in the case of two interacting electrons.

Figure 4

Plot of $J$ as a function of $x_b$ and $x_c$ in the case of two interacting electrons.

Figure 5

Plot of $K$ as a function of $x_b$ and $x_c$ in the case of three mutually interacting electrons.

Figure 6

Plot of $\Delta K$ as a function of $x_b$ and $x_c$ in the case of three mutually interacting electrons. Axis directions are reversed from the preceding figure.

Figure 7

Plot of $J$ as a function of $x_b$ and $x_c$ in the case of three mutually interacting electrons.

Figure 8

Plot of $\Delta J$ as a function of $x_b$ and $x_c$ in the case of three mutually interacting electrons. Axis directions are reversed from the preceding figure.

Figure 9

Plot of $K$ as a function of $x_b$ and $x_c$ in the case of three mutually interacting electrons, computed within the HM approximation.

Figure 10

Plot of $J$ as a function of $x_b$ and $x_c$ in the case of three mutually interacting electrons, computed within the HM approximation.

Figure 11

Plot of $K$ as a function of $x_b$ and $x_c$ in the case of four mutually interacting electrons.

Figure 12

Plot of $\Delta K$ as a function of $x_b$ and $x_c$ in the case of four mutually interacting electrons. Axis directions are reversed from the preceding figure.

Figure 13

Plot of $J$ as a function of $x_b$ and $x_c$ in the case of four mutually interacting electrons.

Figure 14

Plot of $\Delta J$ as a function of $x_b$ and $x_c$ in the case of four mutually interacting electrons. Axis directions are reversed from the preceding figure.

Figure 15

Plot of $J^{\prime }$ as a function of $x_b$ and $x_c$ in the case of four mutually interacting electrons.

Figure 16

Plot of $\Delta J^{\prime }$ as a function of $x_b$ and $x_c$ in the case of four mutually interacting electrons. Axis directions are reversed from the preceding figure.