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Effective Hamiltonian for two interacting double-dot exchange-only qubits and their controlled-NOT operations

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Double-dot exchange-only qubit represents a promising compromise between high speed and simple fabrication in solid-state implementations. A couple of interacting double-dot exchange-only qubits, each composed by three electrons distributed in a double quantum dot, is exploited to realize controlled-NOT (CNOT) operations. The effective Hamiltonian model of the composite system is expressed by only exchange interactions between pairs of spins. Consequently, the evolution operator has a simple form and represents the starting point for the research of sequences of operations that realize CNOT gates. Two different geometrical configurations of the pair are considered, and a numerical mixed simplex and genetic algorithm is used. We compare the nonphysical case in which all the interactions are controllable from the external and the realistic condition in which intra-dot interactions are fixed by the geometry of the system. In the latter case, we find the CNOT sequences for both the geometrical configurations and we considered a qubit system where electrons are electrostatically confined in two quantum dots in a silicon nanowire. The effects of the geometrical sizes of the nanowire and of the gates on the fundamental parameters controlling the qubit are studied by exploiting a spin-density-functional theory-based simulator. Consequently, CNOT gate performances are evaluated.

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Correspondence to E. Ferraro.

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E. Ferraro, M. De Michielis: Equally contributing authors.


Appendix 1: Details of the calculation of the effective Hamiltonians for the couple of interacting qubits

In this “Appendix,” we are going to report all the detailed expressions for the exchange coupling constants between pair of spins in both the configurations examined. The first (last) three indices inside parenthesis, \(0\le i\ne j\ne k\le 2\), assuming only integer values, denote the number of electrons in each level for qubit a (b).

1.1 Configuration A

The coupling constants for the configuration A are given by

$$\begin{aligned}&J_{1_q3_q}\simeq \frac{1}{\Delta E_{1_q}}4\left( t_{1_q3_q}-J^{\left( 1_q3_q\right) }_t\right) ^2 -2J^{\left( 1_q3_q\right) }_e\nonumber \\&J_{2_q3_q}\simeq \frac{1}{\Delta E_{2_q}}4\left( t_{2_q3_q}-J^{\left( 2_q3_q\right) }_t\right) ^2 -2J^{\left( 2_q3_q\right) }_e\nonumber \\&J_{1_q2_q}=\left( \frac{1}{\Delta E_{3_q}}+\frac{1}{\Delta E_{4_q}} \right) 4J^{\left( 1_q2_q\right) 2}_t-2J^{\left( 1_q2_q\right) }_e\nonumber \\&J_{3_a1_b}\simeq \frac{1}{\Delta E_5}4\left( t_{3_a1_b}-J^{\left( 3_a1_b\right) }_t\right) ^2 -2J^{\left( 3_a1_b\right) }_e\nonumber \\&J_{3_a2_b}\simeq \frac{1}{\Delta E_6}4\left( t_{3_a2_b}-J^{\left( 3_a2_b\right) }_t\right) ^2-2J^{\left( 3_a2_b\right) }_e. \end{aligned}$$

where the energy differences for qubits \(a\) and \(b\) are

$$\begin{aligned} \Delta E_{1_{a(b)}}&=E_{(012,111)}(E_{(111,012)})-E_{(111,111)}\nonumber \\ \Delta E_{2_{a(b)}}&=E_{(102,111)}(E_{(111,102)})-E_{(111,111)}\nonumber \\ \Delta E_{3_{a(b)}}&=E_{(201,111)}(E_{(111,201)})-E_{(111,111)}\nonumber \\ \Delta E_{4_{a(b)}}&=E_{(021,111)}(E_{(111,021)})-E_{(111,111)}\nonumber \\ \Delta E_5&=E_{(112,011)}-E_{(111,111)}\nonumber \\ \Delta E_6&=E_{(112,101)}-E_{(111,111)} \end{aligned}$$


$$\begin{aligned} E_{(ijk,111)}&=i\varepsilon _{1_a}+j\varepsilon _{2_a}+k\varepsilon _{3_a} +ijU_{1_a2_a}+ikU_{1_a3_a}+kjU_{2_a3_a}\nonumber \\&\qquad +\,\delta _{i2}U_{1_a}+\delta _{j2}U_{2_a}+\delta _{k2}U_{3_a}\nonumber \\&\qquad +\,\varepsilon _{1_b}+\varepsilon _{2_b}+\varepsilon _{3_b}+U_{1_b2_b} +U_{1_b3_b}+U_{2_b3_b}\nonumber \\&\qquad +\,kU_{3_a1_b}+kU_{3_a2_b}\end{aligned}$$
$$\begin{aligned} E_{(111,ijk)}&=\varepsilon _{1_a}+\varepsilon _{2_a}+\varepsilon _{3_a}+U_{1_a2_a} +U_{1_a3_a}+U_{2_a3_a}\nonumber \\&\qquad +\,i\varepsilon _{1_b}+j\varepsilon _{2_b}+k\varepsilon _{3_b}+ijU_{1_b2_b} +ikU_{1_b3_b}+kjU_{2_b3_b}\nonumber \\&\qquad +\,\delta _{i2}U_{1_b}+\delta _{j2}U_{2_b}+\delta _{k2}U_{3_b}\nonumber \\&\qquad +\,iU_{3_a1_b}+iU_{3_a2_b}\end{aligned}$$
$$\begin{aligned} E_{(111,111)}&=\varepsilon _{1_a}+\varepsilon _{2_a}+\varepsilon _{3_a} +U_{1_a2_a}+U_{1_a3_a}+U_{2_a3_a}\nonumber \\&\qquad +\varepsilon _{1_b}+\varepsilon _{2_b} +\varepsilon _{3_b}+U_{1_b2_b}\nonumber \\&\qquad +\,U_{1_b3_b}+U_{2_b3_b}\nonumber \\&\qquad +\,U_{3_a1_b}+U_{3_a2_b} \end{aligned}$$

1.2 Configuration B

The coupling constants for the configuration B are given by

$$\begin{aligned}&J_{1_q3_q}\simeq \frac{1}{\Delta E_{1_q}}4\left( t_{1_q3_q}-J^{\left( 1_q3_q\right) }_t\right) ^2 -2J^{\left( 1_q3_q\right) }_e\nonumber \\&J_{2_q3_q}\simeq \frac{1}{\Delta E_{2_q}}4\left( t_{2_q3_q}-J^{\left( 2_q3_q\right) }_t\right) ^2 -2J^{\left( 2_q3_q\right) }_e\nonumber \\&J_{1_q2_q}=\left( \frac{1}{\Delta E_{3_q}}+\frac{1}{\Delta E_{4_q}} \right) 4J^{\left( 1_q2_q\right) 2}_t-2J^{\left( 1_q2_q\right) }_e\nonumber \\&J_{1_a1_b}\simeq -2J^{\left( 1_A1_B\right) }_e\nonumber \\&J_{1_a2_b}=0\nonumber \\&J_{2_a1_b}=0\nonumber \\&J_{2_a2_b}\simeq -2J^{\left( 2_A2_B\right) }_e, \end{aligned}$$

where the energy differences for qubits \(a\) and \(b\) are

$$\begin{aligned}&\Delta E_{1_{a(b)}}=E_{(012,111)}(E_{(111,012)})-E_{(111,111)}\nonumber \\&\Delta E_{2_{a(b)}}=E_{(102,111)}(E_{(111,102)})-E_{(111,111)}\nonumber \\&\Delta E_{3_{a(b)}}=E_{(201,111)}(E_{(111,201)})-E_{(111,111)}\nonumber \\&\Delta E_{4_{a(b)}}=E_{(021,111)}(E_{(111,021)})-E_{(111,111)} \end{aligned}$$


$$\begin{aligned} E_{(ijk,111)}&=i\varepsilon _{1_a}+j\varepsilon _{2_a}+k\varepsilon _{3_a} +ijU_{1_a2_a}+ikU_{1_a3_a}+kjU_{2_a3_a}\nonumber \\&\qquad +\,\delta _{i2}U_{1_a}+\delta _{j2}U_{2_a}+\delta _{k2}U_{3_a}\nonumber \\&\qquad +\,\varepsilon _{1_b}+\varepsilon _{2_b}+\varepsilon _{3_b}+U_{1_b2_b} +U_{1_b3_b}+U_{2_b3_b}\nonumber \\&\qquad +\,iU_{1_a1_b}+iU_{1_a2_b}+jU_{2_a1_b}+jU_{2_a2_b}\end{aligned}$$
$$\begin{aligned} E_{(111,ijk)}&=\varepsilon _{1_a}+\varepsilon _{2_a}+\varepsilon _{3_a} +U_{1_a2_a}+U_{1_a3_a}+U_{2_a3_a}\nonumber \\&\qquad +\,i\varepsilon _{1_b}+j\varepsilon _{2_b}+k\varepsilon _{3_b}+ijU_{1_b2_b} +ikU_{1_b3_b}+kjU_{2_b3_b}\nonumber \\&\qquad +\,\delta _{i2}U_{1_b}+\delta _{j2}U_{2_b}+\delta _{k2}U_{3_b}\nonumber \\&\qquad +\,iU_{1_a1_b}+jU_{1_a2_b}+iU_{2_a1_b}+jU_{2_a2_b} \end{aligned}$$
$$\begin{aligned} E_{(111,111)}&=\varepsilon _{1_a}+\varepsilon _{2_a}+\varepsilon _{3_a} +U_{1_a2_a}+U_{1_a3_a}+U_{2_a3_a}+\varepsilon _{1_b}+\varepsilon _{2_b} +\varepsilon _{3_b}\nonumber \\&\qquad +\,U_{1_b2_b}+U_{1_b3_b}+U_{2_b3_b}\nonumber \\&\qquad +\,U_{1_a1_b}+U_{1_a2_b}+U_{2_a1_b}+U_{2_a2_b} \end{aligned}$$

Appendix 2: Mathematical background

In this “Appendix,” the fundamental mathematical tools used in the following to derive the sequences that realize the CNOT gates are presented.

Let us introduce the logical basis \(\{|0\rangle ,|1\rangle \}\) used hereafter for each qubit. It is composed by singlet and triplet states of a pair of spins, for example the pair in the left dot, in combination with the angular momentum of the third spin, localized in the right dot. This means that the logical states are finally expressed in this way

$$\begin{aligned} |0\rangle \equiv |S_0\rangle |\downarrow \rangle , \qquad |1\rangle \equiv \sqrt{\frac{1}{3}}|T_0\rangle |\downarrow \rangle -\sqrt{\frac{2}{3}}|T_-\rangle |\uparrow \rangle \end{aligned}$$

where \(|S_0\rangle ,\, |T_0\rangle \) and \(|T_{\pm }\rangle \) are, respectively, the singlet and triplet states, whose explicit form, in terms of the eigenstates of \(\sigma _z\), is here reported for completeness

$$\begin{aligned} |S_0\rangle =\frac{|\uparrow \downarrow \rangle -|\downarrow \uparrow \rangle }{\sqrt{2}}, \quad |T_0\rangle =\frac{|\uparrow \downarrow \rangle +|\downarrow \uparrow \rangle }{\sqrt{2}},\quad |T_-\rangle =|\downarrow \downarrow \rangle , \quad |T_+\rangle =|\uparrow \uparrow \rangle . \end{aligned}$$

The basis state of the Hilbert space containing three electron spins, representing one qubit, written in the computational basis via Clebsch–Gordan coefficients is given by:

$$\begin{aligned}&|1\rangle =|S_0\rangle |\uparrow \rangle \nonumber \\&|2\rangle =|S_0\rangle |\downarrow \rangle \nonumber \\&|3\rangle =\frac{1}{\sqrt{3}}\left( \sqrt{2}|T_+\rangle |\downarrow \rangle -|T_0\rangle |\uparrow \rangle \right) \nonumber \\&|4\rangle =\frac{1}{\sqrt{3}}\left( |T_0\rangle |\downarrow \rangle -\sqrt{2}|T_-\rangle |\uparrow \rangle \right) \nonumber \\&|5\rangle =|T_+\rangle |\uparrow \rangle \nonumber \\&|6\rangle =\frac{1}{\sqrt{3}}\left( |T_+\rangle |\downarrow \rangle +\sqrt{2}|T_0\rangle |\uparrow \rangle \right) \nonumber \\&|7\rangle =\frac{1}{\sqrt{3}}\left( \sqrt{2}|T_0\rangle |\downarrow \rangle +|T_-\rangle |\uparrow \rangle \right) \nonumber \\&|8\rangle =|T_-\rangle |\downarrow \rangle \end{aligned}$$

with \(|S_0\rangle ,\, |T_0\rangle \) and \(|T_{\pm }\rangle \) defined in Eqs. (22). On the other hand, the composite system of two qubits that is six electron spins is hereafter described by a nine-dimensional basis in the subspace with total angular momentum operator equal to \(S=1,\, S_z=-1\) obtained composing the one qubit states in Eq. (23) with appropriate Clebsch–Gordan coefficients:

$$\begin{aligned}&\left. \left| b_1^{(9)}\right. \right\rangle =|2\rangle |2\rangle \nonumber \\&\left. \left| b_2^{(9)}\right. \right\rangle =|2\rangle |4\rangle \nonumber \\&\left. \left| b_3^{(9)}\right. \right\rangle =|4\rangle |2\rangle \nonumber \\&\left. \left| b_4^{(9)}\right. \right\rangle =|4\rangle |4\rangle \nonumber \\&\left. \left| b_5^{(9)}\right. \right\rangle =\frac{\sqrt{3}}{2}\Big |1\Big \rangle \Big |8\Big \rangle -\frac{1}{2}\Big |2\Big \rangle \Big |7\Big \rangle \nonumber \\&\left. \left| b_6^{(9)}\right. \right\rangle =\frac{\sqrt{3}}{2}\Big |3\Big \rangle \Big |8\Big \rangle -\frac{1}{2}\Big |4\Big \rangle |7\rangle \nonumber \\&\left. \left| b_7^{(9)}\right. \right\rangle =-\frac{\sqrt{3}}{2}\Big |8\Big \rangle \Big |1\Big \rangle +\frac{1}{2}\Big |7\Big \rangle |2\rangle \nonumber \\&\left. \left| b_8^{(9)}\right. \right\rangle =-\frac{\sqrt{3}}{2}\Big |8\Big \rangle \Big |3\Big \rangle +\frac{1}{2}\Big |7\Big \rangle |4\rangle \nonumber \\&\left. \left| b_9^{(9)}\right. \right\rangle =\frac{1}{2}\sqrt{\frac{6}{5}} \left( \Big |6\Big \rangle \Big |8\Big \rangle +\Big |8\Big \rangle \Big |6\Big \rangle \right) -\sqrt{\frac{2}{5}}\Big |7\Big \rangle \Big |7\Big \rangle . \end{aligned}$$

Basis vectors \(|b_1^{(9)}\rangle - |b_4^{(9)}\rangle \) are valid encoded states and correspond exactly to the logical basis \(\{|00\rangle ,|01\rangle ,|10\rangle ,|11\rangle \}\) in which the CNOT gate has the usual form

$$\begin{aligned} \hbox {CNOT}=\left( \begin{array}{cccc} 1 &{} 0 &{} 0 &{} 0 \\ 0 &{} 1 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 1 \\ 0 &{} 0 &{} 1 &{} 0 \end{array}\right) , \end{aligned}$$

basis vectors \(\left. \left| b_5^{(9)}\right. \right\rangle - \left. \left| b_9^{(9)}\right. \right\rangle \) are leaked states.

The objective function used in the fixed-step genetic algorithm to derive single-qubit operation is:

$$\begin{aligned} f_{\mathrm{CNOT}}=\sqrt{1-\frac{1}{4}\left| U^{(9)}_{(1,1)}+U^{(9)}_{(2,2)} +U^{(9)}_{(3,4)}+U^{(9)}_{(4,3)}\right| }, \end{aligned}$$

where \(U^{(9)}_{(i,j)}\) are the matrix elements of the CNOT in the encoded states in the \(9\times 9\) subspace. The objective function is exactly equal to zero when all the \(U^{(9)}\) entering into Eq. (26) have modulus 1 and a common phase in each subspace.

To study the realistic situation in which intra-dot interactions are fixed by the geometry of the system, similarly to Ref. [27], a search algorithm with a variable number of time steps is developed and used. Following the procedure in Ref. [27], we adopt an objective function for the genetic algorithm that, due to the structures of exchange matrices, is confined into two subspaces for the total angular momentum operator of the composite system. The basis of the first \(5\times 5\) block with total angular momentum \(S=0,\, S_z=0\) is given by:

$$\begin{aligned}&\left. \left| b_1^{(5)}\right. \right\rangle =\frac{1}{\sqrt{2}}\left( |1\rangle |2\rangle -|2\rangle |1\rangle \right) \nonumber \\&\left. \left| b_2^{(5)}\right. \right\rangle =\frac{1}{\sqrt{2}}\left( |1\rangle |4\rangle -|2\rangle |3\rangle \right) \nonumber \\&\left. \left| b_3^{(5)}\right. \right\rangle =\frac{1}{\sqrt{2}}\left( |3\rangle |2\rangle -|4\rangle |1\rangle \right) \nonumber \\&\left. \left| b_4^{(5)}\right. \right\rangle =\frac{1}{\sqrt{2}}\left( |3\rangle |4\rangle -|4\rangle |3\rangle \right) \nonumber \\&\left. \left| b_5^{(5)}\right. \right\rangle =\frac{1}{2}\left( |5\rangle |8\rangle -|8\rangle |5\rangle +|7\rangle |6\rangle -|6\rangle |7\rangle \right) , \end{aligned}$$

where the states on the right of the equations are given in Eq. (23); the basis of the second block of dimension \(9\times 9\) with \(S=1,\, S_z=-1\) has been previously defined in Eq. (24). The three \(9\times 9\) blocks in correspondence for \(S=1\) are identical, so they need not being constrained separately. Moreover, the CNOT matrix on the subspaces in correspondence for \(S=2\) and \(S=3\) is completely unconstrained, which is automatically satisfied with the exchange gates. The objective function is finally defined by

$$\begin{aligned}&f_{\mathrm{CNOT}}=\nonumber \\&\sqrt{2-\frac{1}{4}\left| U^{(5)}_{(1,1)}+U^{(5)}_{(2,2)} +U^{(5)}_{(3,4)}+U^{(5)}_{(4,3)}\right| -\frac{1}{4}\left| U^{(9)}_{(1,1)} +U^{(9)}_{(2,2)}+U^{(9)}_{(3,4)}+U^{(9)}_{(4,3)}\right| },\nonumber \\ \end{aligned}$$

having the same meaning of the objective function in Eq. (26). In the case of the CNOT operation, the subspace with \(S=0\) and \(S=1\) has the same global phase.

Appendix 3: Toy model with all controllable interactions

In this “Appendix,” the CNOT sequence for the toy model with all controllable interactions is analyzed. In the first part, the uncorrectness of sequence reported in Ref. [19] is demonstrated, while in the second the sequence starting from our genetic algorithm is derived.

In the following, we present the sequence of exchange operations as done in Ref. [19] for configuration B, in which a locally equivalent CNOT gate is presented equipped with single-qubit operations to obtain an exact CNOT. In Fig. 8, we represent graphically the modulus and phase of the final transformation matrix presented in the supplemental material of Ref. [19] for the configuration B described by the Hamiltonian in Eq. (9). It includes the sequence to obtain a locally equivalent CNOT and single-qubit operations to obtain exactly a CNOT gate.

Fig. 8
figure 8

Graphical representation of modulus and phase of the complete transformation erroneously called exact CNOT gate in the supplemental material of Ref. [19] (Color figure online)

While the locally equivalent CNOT presented in the supplemental material of Ref. [19] returns \((0,1)\) Makhlin coefficients [27] as it should be, when the single-qubit operations proposed are applied before and after the central sequence, the \(4\times 4\) block, contrarily to they claim, does not represent an exact CNOT as Fig. 8 witnesses.

Starting from the central gate sequence of Ref. [19] for the local equivalent CNOT in configuration B and by adopting a genetic algorithm with a fixed number of time steps to find the single-qubit operations with the objective function (26), we obtain the exact CNOT shown in Fig. 9.

Fig. 9
figure 9

Graphical representation of modulus and phase of the final transformation matrix for the exact CNOT gate in the toy model (Color figure online)

The sequence of single-qubit interactions considered is reported in Table 3.

Table 3 Single-qubit operations transforming the locally equivalent CNOT (LECNOT) to an exact CNOT

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Ferraro, E., De Michielis, M., Fanciulli, M. et al. Effective Hamiltonian for two interacting double-dot exchange-only qubits and their controlled-NOT operations. Quantum Inf Process 14, 47–65 (2015).

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