Tomography scheme for two spin-$\frac{1}{2}$ qubits in a double quantum dot

We present a scheme for full quantum state tomography tailored for two spin qubits in a double quantum dot. A set of 15 quantum states allows to determine the density matrix in this two-qubit space by projective measurement. In this paper, we choose a set gained from mutually unbiased bases. We determine how those 15 projections can be represented by charge measurements after a spin-to-charge conversion and the application of quantum gates. The quantum gates include exchange-based gates as well as rotations by electron spin resonance (ESR). We assume the experimental realization of ESR operations to be more difficult than the exchange gate operation with respect to fidelity and speed. Therefore it is an important result that the ESR gates can be limited to a $\pi /2$ rotation for one of the electron spins per measurement.

Figure 1

(a) Spectrum of a two-electron double quantum dot with a magnetic field with $|h_z|\gg |\Delta h_z|\gg 4t^2/U$, $h_z$, $\Delta h_z<0$ for small values of $\varepsilon$ and $\varepsilon$ close to $U$. The avoided crossing between $|S\rangle$ and $|\uparrow \uparrow \rangle$ at the blue dotted line is induced by slightly different directions of the magnetic fields in the left and the right dot. (b) Three ways to tune $\varepsilon$ to achieve projection on $|\uparrow \downarrow \rangle$, dash-dotted (green) line, on $|S\rangle$, dashed (red) line, and on $|\uparrow \uparrow \rangle$, solid (cyan) line. Note that the $\varepsilon$ axis and the time axis are broken.

Figure 2

The quantum circuits used to realize the quorum (12). (a) represents the preparation of the state $|{\psi }_3\rangle$ by a SWAP gate starting from state $|{\psi }_2\rangle$. The circuits in (b) and (c) show the preparation of the state $|{\psi }_4\rangle$ and $|{\psi }_{10}\rangle$, respectively, starting from the singlet state $|S\rangle$. In (d), we show the construction of the remaining states of the quorum, $|{\psi }_j\rangle$ and $|{\psi }_{j+6}\rangle$ with $j=5,...,9$, by rotations $R_Z$ implemented using a static magnetic field, starting from $|{\psi }_4\rangle$ and $|{\psi }_{10}\rangle$ respectively. The angles ${\theta }_1^j$ and ${\theta }_2^j$ depend on the index $j$ of the final state but they are the same for the maps $|{\psi }_4\rangle$ to $|{\psi }_j\rangle$ and $|{\psi }_{10}\rangle$ to $|{\psi }_{j+6}\rangle$. The actual values of ${\theta }_1^j$ and ${\theta }_2^j$ can be found in Eq. (12), e.g., for $j=9$, they are given by ${\theta }_1^j=-{\theta }_2^j=-\pi /2$. Note that ESR is used to implement $R_X$ in the preparation of the states $|{\psi }_4\rangle$ and $|{\psi }_{10}\rangle$. The gate $R_X$ $\pi /2$ on the upper line in (b) denotes a rotation of the first qubit around the $x$ axis by the angle $\pi /2$, i.e., the operation $e^{i\frac{\pi }{4}{\sigma }_{1x}}$. The other rotations $R_X$ and $R_Z$ are defined accordingly.