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Controlled-NOT gate sequences for mixed spin qubit architectures in a noisy environment

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Explicit controlled-NOT gate sequences between two qubits of different types are presented in view of applications for large-scale quantum computation. Here, the building blocks for such composite systems are qubits based on the electrostatically confined electronic spin in semiconductor quantum dots. For each system the effective Hamiltonian models expressed by only exchange interactions between pair of electrons are exploited in two different geometrical configurations. A numerical genetic algorithm that takes into account the realistic physical parameters involved is adopted. Gate operations are addressed by modulating the tunneling barriers and the energy offsets between different couple of quantum dots. Gate infidelities are calculated considering limitations due to unideal control of gate sequence pulses, hyperfine interaction and charge noise.

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This project has received funding from the European Union’s Horizon 2020 research and innovation programme under Grant Agreement No. 688539.

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Correspondence to E. Ferraro.


Appendix A: Effective exchange coupling constants

In this Appendix, following the same procedure already exploited in Refs. [24, 25], all the detailed expressions for the exchange coupling constants between pair of electrons in both the mixed architectures considered are reported. The Schrieffer–Wolff effective Hamiltonian models (5) and (11) are derived by combining a Hubbard-like model with a projector operator method [42]. As a result, the Hubbard-like Hamiltonian is transformed into an equivalent expression in terms of the exchange coupling interactions between pairs of electrons. Since the two dots composing the hybrid qubit are asymmetric, it follows that there are two different possible configurations for each architecture as shown in Figs. 1 and 4.

1.1 Quantum dot single-spin qubit and double quantum dot hybrid qubit

The expressions for the exchange coupling constants for the configuration A appearing in the effective Hamiltonian (5) are given by

$$\begin{aligned} J_{1_R2_R}&=\frac{1}{\Delta E_1}4\left( t_{1_R2_R}-J^{(1_R2_R)}_t\right) ^2 -2J^{(1_R2_R)}_e\nonumber \\ J_{2_R3_R}&=\frac{1}{\Delta E_2}4\left( t_{2_R3_R}-J^{(2_R3_R)}_t\right) ^2 -2J^{(2_R3_R)}_e\nonumber \\ J_{1_R3_R}&=\left( \frac{1}{\Delta E_3}+\frac{1}{\Delta E_4}\right) 4J^{(1_R3_R)2}_t-2J^{(1_R3_R)}_e\\ J_{1_L1_R}&=\frac{1}{\Delta E_5}4\left( t_{1_L1_R}-J^{(1_L1_R)}_t\right) ^2 -2J^{(1_L1_R)}_e\nonumber \\ J_{1_L3_R}&=\frac{1}{\Delta E_6}4\left( t_{1_L3_R}-J^{(1_L3_R)}_t\right) ^2 -2J^{(1_L3_R)}_e,\nonumber \end{aligned}$$

with the energy differences defined as

$$\begin{aligned} \Delta E_1&=E_{(1,012)}-E_{(1,111)}\nonumber \\ \Delta E_2&=E_{(1,102)}-E_{(1,111)}\nonumber \\ \Delta E_3&=E_{(1,201)}-E_{(1,111)}\nonumber \\ \Delta E_4&=E_{(1,021)}-E_{(1,111)}\nonumber \\ \Delta E_5&=E_{(0,211)}-E_{(1,111)}\nonumber \\ \Delta E_6&=E_{(0,121)}-E_{(1,111)} \end{aligned}$$


$$\begin{aligned} E_{(w,ijk)}&=w\varepsilon _{1_L}+i\varepsilon _{1_R}+j\varepsilon _{3_R} +k\varepsilon _{2_R}+ijU_{1_R3_R}+ikU_{1_R2_R}+kjU_{2_R3_R}+\delta _{i2}U_{1_R}\nonumber \\ {}&\quad +\delta _{j2}U_{3_R}+\delta _{k2}U_{2_R}+iwU_{1_L1_R}+jwU_{1_L3_R}. \end{aligned}$$

The first index in parenthesis \(w=0,1\) denotes the electron occupation for the single-spin qubit L, while the indices \(i,j,k=0,1,2\) denote the number of electrons in each level for the hybrid qubit R ordered as depicted in Fig. 1. The parameters involved are: the energy levels \(\varepsilon _i\), the tunneling coefficients between different dots \(t_{ij}\), the spin exchange \(J_e^{ij}\) and the occupation-modulation hopping terms \(J_t^{ij}\).

Analogously the exchange coupling constants for the configuration B are defined as in Eq. (18) with the new inter-qubit interaction

$$\begin{aligned} J_{1_L2_R}=\frac{1}{\Delta E_5}4\left( t_{1_L2_R}-J^{(1_L2_R)}_t\right) ^2-2J^{(1_L2_R)}_e \end{aligned}$$

where \(\Delta E_5=E_{(0,112)}-E_{(1,111)}\). The energies corresponding to each configurations are given for the configuration B by

$$\begin{aligned} E_{(w,ijk)}&=w\varepsilon _{1_L}+i\varepsilon _{1_R} +j\varepsilon _{3_R}+k\varepsilon _{2_R}+ijU_{1_R3_R}+ikU_{1_R2_R}+kjU_{2_R3_R}+\delta _{i2}U_{1_R}\nonumber \\ {}&\quad +\delta _{j2}U_{3_R} +\delta _{k2}U_{2_R}+kwU_{1_L2_R}. \end{aligned}$$

1.2 Double quantum dot singlet–triplet qubit and double quantum dot hybrid qubit

The exchange coupling constants for the double QD singlet–triplet and the double QD hybrid qubits in configuration A appearing in the effective Hamiltonian (11) are given by

$$\begin{aligned} J_{1_R2_R}&=\frac{1}{\Delta E_{1_R}}4\left( t_{1_R2_R} -J^{(1_R2_R)}_t\right) ^2-2J^{(1_R2_R)}_e\nonumber \\ J_{2_R3_R}&=\frac{1}{\Delta E_{2_R}}4\left( t_{2_R3_R} -J^{(2_R3_R)}_t\right) ^2-2J^{(2_R3_R)}_e\nonumber \\ J_{1_R3_R}&=\left( \frac{1}{\Delta E_{3_R}}+\frac{1}{\Delta E_{4_R}}\right) 4J^{(1_R3_R)2}_t-2J^{(1_R3_R)}_e\nonumber \\ J_{1_L2_L}&=\left( \frac{1}{\Delta E_{3_L}}+\frac{1}{\Delta E_{4_L}}\right) 4\left( t_{1_L2_L}-J^{(1_L2_L)}_t\right) ^2-2J^{(1_L2_L)}_e\nonumber \\ J_{2_L1_R}&=\frac{1}{\Delta E_5}4\left( t_{2_L1_R}-J^{(2_L1_R)}_t\right) ^2 -2J^{(2_L1_R)}_e\nonumber \\ J_{2_L3_R}&=\frac{1}{\Delta E_6}4\left( t_{2_L3_R}-J^{(2_L3_R)}_t\right) ^2 -2J^{(2_L3_R)}_e, \end{aligned}$$


$$\begin{aligned} \Delta E_{1_R}&=E_{(11,012)}-E_{(11,111)}\nonumber \\ \Delta E_{2_R}&=E_{(11,102)}-E_{(11,111)}\nonumber \\ \Delta E_{3_R}&=E_{(11,201)}-E_{(11,111)}\nonumber \\ \Delta E_{4_R}&=E_{(11,021)}-E_{(11,111)}\nonumber \\ \Delta E_{3_L}&=E_{(02,111)}-E_{(11,111)}\nonumber \\ \Delta E_{4_L}&=E_{(20,111)}-E_{(11,111)}\nonumber \\ \Delta E_5&=E_{(10,211)}-E_{(11,111)}\nonumber \\ \Delta E_6&=E_{(10,121)}-E_{(11,111)} \end{aligned}$$


$$\begin{aligned} E_{(wz,ijk)}&=w\varepsilon _{1_L}+z\varepsilon _{2_L}+wzU_{1_L2_L} +\delta _{w2}U_{1_L}+\delta _{z2}U_{2_L}+i\varepsilon _{1_R}+j\varepsilon _{3_R}+k\varepsilon _{2_R} \nonumber \\ {}&\quad +ijU_{1_R3_R}+ikU_{1_R2_R}+kjU_{2_R3_R}+\delta _{i2}U_{1_R}+\delta _{j2}U_{3_R} +\delta _{k2}U_{2_R}\nonumber \\ {}&\quad +izU_{2_L1_R}+jzU_{2_L3_R}. \end{aligned}$$

The first (last) indices inside parenthesis, assuming only integer values between 0 and 2, denote the number of electrons in each level for qubit L(R) ordered as depicted in Fig. 4.

Fig. 10
figure 10

Graphical representation of modulus and phase of the final transformation matrix for the CNOT gates. Top left (right): \(\text {single}+\text {hybrid}\) A (B); Bottom left (right): \(\text {singlet--triplet}+\text {hybrid}\) A (B)

For the configuration B the coupling constants are defined as in Eq. (23) with the new inter-qubit interaction

$$\begin{aligned} J_{2_L2_R}=\frac{1}{\Delta E_5}4\left( t_{2_L2_R}-J^{(2_L2_R)}_t\right) ^2-2J^{(2_L2_R)}_e \end{aligned}$$

where \(\Delta E_5=E_{(10,121)}-E_{(11,111)}\). The energies are now given by

$$\begin{aligned} E_{(wz,ijk)}&=w\varepsilon _{1_L}+z\varepsilon _{2_L}+wzU_{1_L2_L} +\delta _{w2}U_{1_L}+\delta _{z2}U_{2_L}+i\varepsilon _{1_R}+j\varepsilon _{3_R}+k\varepsilon _{2_R} \nonumber \\ {}&\quad +ijU_{1_R3_R}+ikU_{1_R2_R}+kjU_{2_R3_R}+\delta _{i2}U_{1_R}+\delta _{j2}U_{3_R} \nonumber \\ {}&\quad +\delta _{k2}U_{2_R}+kzU_{2_L2_R}. \end{aligned}$$

Appendix B: Graphical representation of CNOT gates

In this Appendix a graphical representation of modulus and phase (gray scale) of the final transformation matrix for the CNOT gates for the four mixed architectures studied is shown (Fig. 10). The resulting transformation matrices are obtained starting from the sequences reported in Figs. 2, 3, 5 and 6. The \(4\times 4\) block in the up left corner corresponds to the CNOT matrix reported in Eq. (9).

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Ferraro, E., Fanciulli, M. & De Michielis, M. Controlled-NOT gate sequences for mixed spin qubit architectures in a noisy environment. Quantum Inf Process 16, 277 (2017).

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