⇦Chinese Physics Letters, 2019, Vol. 36, No. 7, Article code 070302 Programmable Quantum Processor with Quantum Dot Qubits Yao Chen (陈垚), Fo-Liang Lin (林佛良), Xi Liang (梁喜), Nian-Quan Jiang (姜年权)** Affiliations College of Mathematics, Physics and Electronic Information Engineering, Wenzhou University, Wenzhou 325035 Received 18 February 2019, online 20 June 2019 **Corresponding author. Email: jiangnq@wzu.edu.cn Citation Text: Chen Y, Lin F L, Liang X and Jiang N Q 2019 Chin. Phys. Lett. 36 070302    Abstract The realization of controllable couplings between any two qubits and among any multiple qubits is the critical problem in building a programmable quantum processor (PQP). We present a design to implement these types of couplings in a double-dot molecule system, where all the qubits are connected directly with capacitors and the couplings between them are controlled via the voltage on the double-dot molecules. A general interaction Hamiltonian of $n$ qubits is presented, from which we can derive the Hamiltonians for performing operations needed in building a PQP, such as gate operations between arbitrary two qubits and parallel coupling operations for multigroup qubits. The scheme is realizable with current technology. DOI:10.1088/0256-307X/36/7/070302 PACS:03.67.Lx, 42.50.Dv, 42.50.Pq, 85.35.Be © 2019 Chinese Physics Society Article Text Semiconductor quantum dots (QDs) have been considered as one kind of the most promising systems for quantum computations.[1] Based on QDs, various qubits have been proposed in recent years.[2–11] Among them, singlet-triplet double-dot qubits have attracted much attention because they are robust against noise from quasistatic nuclear spins as well as low frequency fluctuation of electrostatic gates,[12–14] and tremendous advances have already been made for this type of qubit.[15–18] However, to scale up, there are still some challenges such as scalable design.[19] To solve this problem, a scheme was introduced to couple one double-dot qubit and a transmission line resonator (TLR),[14,20] then it was promoted[21] and a scalable architecture was presented, i.e., $n$ double-dot qubits are coupled with the TLR via capacitors, which allows for various major quantum operations such as nontrivial multiqubit quantum gates and entanglement generation. However, for a multiqubit system where all qubits are coupled to the same coupler via capacitors, the crosstalk will be caused by the cross talking capacitors, which is a large problem for performing programmable quantum computation (PQC). To realize universal PQC while reducing the disadvantage of crosstalk, we propose another type of scalable design in this Letter. In our scheme, we connect all the qubits directly with electrodes which extend to each of double quantum dots (DQDs),[22,23] and the electrodes provide capacitive interactions for all the qubits in the system. The couplings between qubits can be tuned on and off by the control of the voltage on the double-dot molecules. We present the general interaction Hamiltonian of $n$ double-dot qubits and the Hamiltonians for implementing all sorts of operations needed in PQC. With the design, a programmable quantum processor (PQP) can be achieved with current technology. Our scheme has the following advantages: (1) $n$ double-dot qubits in our system are coupled directly with cross talking capacitors, do not need the TLR coupler, thus the crosstalk is not a problem anymore because it is just the working way of coupling operations, and then a higher fidelity will be achievable. (2) Our design can be exploited not only to realize quantum gates between any qubits, but also to implement parallel operations for multipair qubits, multigroup qubits, etc., which are important for performing powerful quantum algorithms. Our scheme is implemented with the system in Fig. 1, in which $n$ double-dot qubits are interconnected directly with capacitors. Each capacitor is formed by one quantum dot of a qubit and an electrode which extends to the dot. All the electrodes are connected together with a metallic superconductor. The inductance of the connected electrodes is very small and can be neglected. In the following we treat the individual qubit as an independent system with the small coupling as perturbation.
cpl-36-7-070302-fig1.png
Fig. 1. Coupled system of $n$ DQDs. The DQDs are connected with each other via electrodes, the capacitance between each DQD $i$ and the electrode extended to one dot of the DQD is $C_{{\rm o}i}$, and all the electrodes are connected together with a center conductor which is denoted as o.
A double-dot molecule is shown in Fig. 2(a) along with an electrical circuit schematic in Fig. 2(b).[24] The left and right dots (LD, RD) are arranged in series with respect to the source and drain (S, D) and are realized on heterostructure with the two-dimensional electron gas (2DEG). An additional gate (RG) extends to the right quantum dot to make a strong coupling between this qubit and any other one in the system. The static potentials on the dots are tuned via the two plunger gates $V_{\rm LP}$ and $V_{\rm RP}$. To allow electron transport, the two dots are connected to each other and to the S and D contacts through tunnel barriers, tunable by $V_{\rm L}$, $V_{\rm C}$ and $V_{\rm R}$. With proper voltage biases, the molecule can be the state only two electrons with one single electron in each dot or two electrons in one dot. Due to the Coulomb interaction and Pauli principle, near the 'saddle' point the molecule can be reduced to an artificial three-level system with the Hamiltonian $$\begin{align} H=\,&E_{\rm T}|2\rangle\langle (1, 1)T_{0}|+E_{\rm S}|2\rangle\langle (1, 1)S|-\varepsilon|2\rangle\langle (0, 2)S|\\ &+T_{\rm C}[|2\rangle\langle (0, 2)S|+|2\rangle\langle (1, 1)S|],~~ \tag {1} \end{align} $$ where $n_{\rm l}$ and $n_{\rm r}$ denote the numbers of electrons in left and right dots respectively, $S$ represents spin singlet state, $T$ represents spin triplet state, $T_{\rm C}$ denotes tunneling amplitude between two dots, and energy detuning $\varepsilon ={\it \Delta}-E_{\rm c}/2$ with ${\it \Delta}$ the energy difference between two dots induced by external voltage bias and $E_{\rm c}$ the charging energy of a single dot. Choosing $T_{\rm C}$ to be real under an appropriate gauge and taking an effective approximation $E_{\rm T}=E_{\rm S}\approx0$,[22] then the eigenstates of Eq. (1) are $$\begin{align} |{\tilde{{G}}}\rangle =\,&\cos \theta|{(0,2)S}\rangle -\sin \theta|{(1,1)S}\rangle,~~ \tag {2} \end{align} $$ $$\begin{align} |{\tilde{{S}}}\rangle =\,&\sin \theta|{(0,2)S}\rangle +\cos \theta|{(1,1)S}\rangle,~~ \tag {3} \end{align} $$ where $|{\tilde{{G}}}\rangle$ is the higher energy state, and the energy difference between the two states is $\Delta E=\sqrt {4T_{\rm c}^{2} +\varepsilon^{2}} =\hslash \omega$. The parameter $\theta$ is the function of $\varepsilon$ and tunneling amplitude $T_{\rm C}$ with $\theta =\arctan [2T_{\rm c} /(\varepsilon -\sqrt {4T_{\rm c}^{2} +\varepsilon^{2}})]$, which can be quickly adjusted by varying the voltage applied to the metallic gates. At the point where $\theta =\pi /4$, the energy difference between the states is insensitive to the first-order fluctuation of $\varepsilon$[25] and the nuclear hyperfine field,[26] and the states in Eqs. (2) and (3) reduce to $|2\rangle =(|2\rangle \mp|2\rangle)/\sqrt 2$. The electric potential on the right dot is $$\begin{align} V=\frac{2e}{C_{{\it \Sigma}}}|2\rangle\langle (0, 2)S|+\frac{e}{C_{{\it \Sigma}}}|2\rangle\langle (1, 1)S|,~~ \tag {4} \end{align} $$ where $C_{{\it \Sigma}}$ is the total capacitance of the DQD. In the basis $\{|0\rangle,|1\rangle \}$, the electric potential in Eq. (4) and the Hamiltonian in Eq. (2) can be written as $V=e(3I+\hat{{\sigma}}^{x})/2C_{{\it \Sigma}}$ and $H=\hslash \omega \hat{{\sigma}}^{z}/2$, respectively, where $\hat{{\sigma}}^{x}$ and $\hat{{\sigma}}^{z}$ are Pauli matrices, and $I$ is the identity matrix.
cpl-36-7-070302-fig2.png
Fig. 2. A typical DQD schematic reproduced from Ref.  [24]. (a) Scanning electron micrograph of the gate structure defining the double quantum dots (LD, RD). RG marks the electrode which extends to the right dot and is connected to other modules, $V_{\rm L}$, $V_{\rm LP}$, $V_{\rm C}$, $V_{\rm RP}$ and $V_{\rm R}$ label top gate voltages, and S, D, the 2DEG source and drain. (b) Equivalent circuit diagram for (a).
cpl-36-7-070302-fig3.png
Fig. 3. Two qubits coupled system. One side of an electrode extends to one dot of qubit 1 and the other side extends to one dot of qubit 2, and the capacitances between the electrode and qubits 1, 2 are $C_{1}$ and $C_{2}$, respectively. Then the total capacitance between the two qubits is $C_{12}=C_{1}C_{2}/(C_{1}+C_{2})$.
As shown in Fig. 3, two qubits are coupled with a capacitor $C_{12}$, the Hamiltonian of the system is $$\begin{align} H=H_{1} +H_{2} +\frac{1}{2}C_{12} (V_{1}-V_{2})^{2},~~ \tag {5} \end{align} $$ where $H_{i} =\hslash \omega_{i} \hat{{\sigma}}_{i}^{z}/2$, $V_{i} =e(3I+\hat{{\sigma}}_{i}^{x})/2C_{{\it \Sigma}_{i}}$. Assuming $C_{{\it \Sigma}_{1}} =C_{{\it \Sigma}_{2}} =C_{{\it \Sigma}}$, the Hamiltonian (5) reduces to $H=\sum\limits_{i=1}^2 {\hslash \omega_{i} \hat{{\sigma}}_{i}^{z} /2} -\hslash g_{12} \hat{{\sigma}}_{1}^{x} \hat{{\sigma}}_{2}^{x}$ with $g_{12} =C_{12} (e/2C_{{\it \Sigma}})^{2}/\hslash$. When $|{\omega_{1} \pm \omega_{2}}|\gg g_{12}$, in the rotating wave approximation (RWA) the coupling between the two qubits is turned off. When $\omega_{1} =\omega_{2}$ and $2\omega_{1}\gg g_{12}$, in the RWA the coupling between the two qubits is turned on, and the interaction Hamiltonian reads $$\begin{align} H_{\rm int}=-\hslash g_{12} (\hat{{\sigma}}_{1}^{+} \hat{{\sigma}}_{2}^{-} +\hat{{\sigma}}_{1}^{-} \hat{{\sigma}}_{2}^{+}),~~ \tag {6} \end{align} $$ with $\hat{{\sigma}}_{i}^{+} =|1\rangle_{i} {}_{i}\langle 0|$, and $\hat{{\sigma}}_{i}^{-} =|0\rangle_{i} {}_{i}\langle 1|$. The corresponding evolution operator is $$ \hat{{U}}(t)=\left(\begin{matrix} 1& 0& 0& 0\\ 0& {\cos g_{12} t}& {-i\sin g_{12} t}& 0\\ 0& {-i\sin g_{12} t}& {\cos g_{12} t}& 0\\ 0& 0& 0& 1\\ \end{matrix}\right).~~ \tag {7} $$ After a period $t=\pi /2g_{12}$, $\hat{{U}}(t)$ will produce a swapping operation between $|2\rangle$ and $|2\rangle$, and when $t=\pi /4g_{12}$, it will be a universal two qubit gate operation: $|{00}\rangle \to|{00}\rangle$, $|{01}\rangle \to (|{01}\rangle -i|{10}\rangle)/\sqrt 2$, $|{10}\rangle \to (|{10}\rangle -i|{01}\rangle)/\sqrt 2$, $|{11}\rangle \to|{11}\rangle$. Considering the system with $n$ qubits coupled by capacitors as shown in Fig. 1, and assuming that each of the qubits is set at the avoided crossing where the detuning $\varepsilon =0$ and then $\theta =\pi /4$, the Hamiltonian of the system can be written as $H=\sum\nolimits_{i=1}^n {\hslash \omega_{i}} \hat{{\sigma}}_{i}^{z} /2+\sum\nolimits_{i=1}^n {C_{{\rm o}i} (V_{\rm o} -V_{i})^{2}/2}$, where $C_{{\rm o}i}$ is the capacitance between one dot of DQD $i$ and the electrode which extends to the dot, and $V_{i} =e(3I+\hat{{\sigma}}_{i}^{x})/2C_{{\it \Sigma}_{i}}$ is the electric potential of the dot with $C_{{\it \Sigma}_{i}}$ the total capacitance of DQD $i$. All the electrodes are connected together and remained isolated, and then the total net charges are zero, i.e., $\sum\nolimits_{i=1}^n {C_{{\rm o}i} (V_{\rm o} -V_{i})=0}$. For simplicity, we assume $C_{{\rm o}i} =C$ and $C_{{\it \Sigma}_{i}} =C_{{\it \Sigma}} (i=1, 2,\ldots, n)$, then the above Hamiltonian reduces to (constant terms are omitted) $$\begin{align} H=\frac{1}{2}\sum\limits_{i=1}^n {\hslash \omega_{i} \hat{{\sigma}}_{i}^{z}} +\sum\limits_{i\ne j;i,j=1}^n {\hslash g\hat{{\sigma}}_{i}^{x} \hat{{\sigma}}_{j}^{x}},~~ \tag {8} \end{align} $$ where the coupling strength $g=e^{2}C/(8\hslash nC_{{\it \Sigma}}^{2})$. Equation (8) shows that, when $|{\omega_{i} \pm \omega_{j}}|\gg g$, in the RWA the coupling between qubits $i$ and $j$ is turned off. However, when $\omega_{i}=\omega_{j}$ and $2\omega_{i} \gg g$, the coupling between them is turned on and the corresponding interaction Hamiltonian will be $-\hslash g(\hat{{\sigma}}_{i}^{+} \hat{{\sigma}}_{j}^{-} +\hat{{\sigma}}_{i}^{-} \hat{{\sigma}}_{j}^{+})$. After a period of evolving time $t=\pi /2g$, it will produce a swapping operation between $|{01}\rangle_{ij}$ and $|{10}\rangle_{ij}$, but when $t=\pi /4g$, the time evolving operator will provide a universal two-qubit gate operation, i.e., a $\sqrt {i{\rm SWAP}}$. In the following, based on the Hamiltonian (8), we demonstrate various operations needed for performing programmable universal quantum computations. In all of the following operations, we set all the qubits to be at the avoided crossing where $\theta =\pi /4$. Firstly, we show the coupling operation between arbitrary two qubits $i$ and $j$. As the frequency $\omega_{i}$ depends on the tunneling amplitude $T_{C_{i}}$ which can be tuned via gate voltages, we can change the frequency of any qubit by choosing proper voltage biases. When the frequencies of all the qubits are set to satisfy the conditions: $\omega_{i}=\omega_{j}$ and $2\omega_{i} \gg g$ with $i\ne j$, while $|{\omega_{j} -\omega_{k}}|\gg g$ and $|{\omega_{l} -\omega_{m}}|\gg g$ with $k, l, m\ne i,j$, $l\ne m$, and $k,l,m\in \{ {1,2,\ldots,n}\}$, then in the RWA the Hamiltonian in the interaction picture reads $$\begin{align} H_{\rm I} =-\hslash g(\hat{{\sigma}}_{i}^{+} \hat{{\sigma}}_{j}^{-} +\hat{{\sigma}}_{i}^{-} \hat{{\sigma}}_{j}^{+}).~~ \tag {9} \end{align} $$ After a period $t=\pi /2g$ or $t=\pi /4g$, the Hamiltonian (9) will produce a swapping operation or a $\sqrt {i{\rm SWAP}}$ universal gate operation on arbitrary two qubits $i$ and $j$, respectively, while the couplings between any other pairs of qubits are turned off. After that, all the qubits are set back to the idle states that the couplings between arbitrary qubits are turned off. To perform two-qubit universal gate operations on multiple pairs of qubits in parallel, for example, simultaneously couple $k$ pairs of qubits: $i_{1}$ and $j_{1}$, $i_{2}$ and $j_{2}$,$\ldots$, $i_{k}$ and $j_{k}$, we tune the voltage biases to satisfy the frequency conditions: $\omega_{i_{m}} =\omega_{j_{m}}$ and $2\omega_{i_{m}} \gg g$ ($m=1,2,\ldots,k$, $i_{m} \ne j_{m}$, $i_{m},j_{m} \in \{ {1,2,\ldots,n} \}$), and $|{\omega_{l} -\omega_{r}}|\gg g$ ($l\ne r$) for any two qubits $l$ and $r$ which are not both from the same pair of qubits $i_{m}$ and $j_{m}$. In the RWA, the Hamiltonian in the interaction picture is expressed as $$\begin{align} H_{\rm I} =-\sum\limits_{m=1}^k {\hslash g} (\hat{{\sigma}}_{i_{m}}^{+} \hat{{\sigma}}_{j_{m}}^{-} +\hat{{\sigma}}_{i_{m}}^{-} \hat{{\sigma}}_{j_{m}}^{+}).~~ \tag {10} \end{align} $$ After a period $t=\pi /2g$, the Hamiltonian (10) will produce $k$ pairs of swapping operations in parallel, i.e., $|2\rangle_{i_{m}j_{m}}\leftrightarrow |2\rangle_{i_{m}j_{m}}$ ($m=1, 2,\ldots, k$), while the couplings between any other two qubits beyond the above $k$ pairs are turned off. However, when $t=\pi /4g$, the Hamiltonian (10) will perform $k \sqrt {i{\rm SWAP}}$ gate operations on the above selected pairs in parallel, while the couplings between any other pair of qubits are turned off. After that, all the qubits are set back to the idle states and the couplings between arbitrary qubits are turned off. Coupling operations on a group of qubits (more than two qubits, that is, qubits 1–$k$, $k>2$) are also achievable. Tuning gate voltages of all qubits properly so that the frequencies satisfy the conditions $\omega_{i} =\omega_{0}$ ($i=1,2,\ldots,k$) and $2\omega_{0} \gg g$, while $|{\omega_{l} -\omega_{r}}|\gg g$ ($l\ne r$) for any two qubits $l$ and $r$ which are not both from the same selected group. In the RWA, the Hamiltonian in the interaction picture is governed by $$\begin{align} H_{\rm I} =-\sum\limits_{i < j; i,j=1}^k {\hslash g( {\hat{{\sigma}}_{i}^{+} \hat{{\sigma}}_{j}^{-} +\hat{{\sigma}}_{i}^{-} \hat{{\sigma}}_{j}^{+}})}.~~ \tag {11} \end{align} $$ After a period of evolving time $t=\pi /4g$, the Hamiltonian (11) should correspond to a series of $\sqrt {i{\rm SWAP}}$ operations on arbitrary two qubits in the selected group in parallel, while the couplings between any other two qubits in the system are turned off. After that, all the qubits are set back to idle states and the couplings are turned off. To couple multiple groups of qubits in parallel, for example, $l$ groups of qubits are selected to be operated and there are $k_{i}$ qubits in group $i$ ($i=1, 2,\ldots, l$), we tune the gate voltages to satisfy the frequency conditions: $\omega_{i_{m}} =\omega_{i_{0}}$ and $2\omega_{i_{0}} \gg g$ ($m=1,2,\ldots,k_{i}$, $i=1,2,\ldots,l$, $i_{m} \in \{1,2,\ldots,n\}$), while $|{\omega_{s} -\omega_{r}}|\gg g$ ($s\ne r$) for any two qubits $s$ and $r$, which are not both from any one selected group. In the RWA, the Hamiltonian of the system in the interaction picture is $$\begin{alignat}{1} \!\!\!\!\!\!\!\!H_{\rm I} =-\sum\limits_{i=1}^l { \sum\limits_{m < m'; m,m'=1}^{k_{i}} { \hslash g( {\hat{{\sigma}}_{i_{m}}^{+} \hat{{\sigma}}_{i_{m'}}^{-} +\hat{{\sigma}}_{i_{m}}^{-} \hat{{\sigma}}_{i_{m'}}^{+}})\,}}.~~ \tag {12} \end{alignat} $$ After the evolving time $t=\pi /4g$, any two qubits $i_{m}$ and $i_{m'}$ in group $i$ are coupled by a $\sqrt {i{\rm SWAP}}$ in parallel. However, the Hamiltonian (12) will not cause couplings between any two qubits which are not both from any one selected group. After that, the system is set back to the idle state where the interactions are turned off. Coupling multiple pairs of qubits and multiple groups of qubits in parallel can be performed as follows: when we intend to simultaneously operate two-qubit universal gates on $k$ pairs of qubits (e.g., qubits $i_{m}$ and $j_{m}$, $m=1,2,\ldots,k$, $i_{m} \ne j_{m}$, $i_{m},j_{m} \in \{1,2,\ldots,n\})$ and perform coupling operations on $l$ groups of qubits (there are $s_{r}$ qubits in group $r$ ($r=1, 2,\ldots, l))$, we tune the gate voltages so that the frequencies satisfy the conditions: $\omega_{i_{m}}=\omega_{j_{m}}$ and $2\omega_{i_{m}} \gg g$ ($i_{m}\ne j_{m}$), $\omega_{t_{r}}=\omega_{r}$ and $2\omega_{t_{r}} \gg g$ ($t=1,2,\ldots,s$, $r=1,2,\ldots,l$, $t_{r} \ne i_{m},j_{m}$), while $|{\omega_{p} -\omega_{q}}|\gg g$ ($p\ne q$) for any two qubits $p$ and $q$ which are not both from any one selected pair or group. In the RWA, the Hamiltonian in the interaction picture reads $$\begin{alignat}{1} H_{\rm I} =\,&-\sum\limits_{m=1}^k {\hslash g( {\hat{{\sigma}}_{i_{m}}^{+} \hat{{\sigma}}_{j_{m}}^{-} +\hat{{\sigma}}_{i_{m}}^{-} \hat{{\sigma}}_{j_{m}}^{+}})}\\ &-\sum\limits_{r=1}^l { \sum\limits_{t < t'; t,t'=1}^s { \hslash g( {\hat{{\sigma}}_{t_{r}}^{+} \hat{{\sigma}}_{{t}'_{r}}^{-} +\hat{{\sigma}}_{t_{r}}^{-} \hat{{\sigma}}_{{t}'_{r}}^{+}})\,}}.~~ \tag {13} \end{alignat} $$ After a period $t=\pi /4g$, arbitrary two qubits in any one selected pair or in any one selected group are coupled with a $\sqrt {i{\rm SWAP}}$ in parallel. However, the couplings between any other two qubits which are not both from the same selected pair or the same group are turned off. After that, the system is set back to the idle state where the interactions are turned off. As the RWA is used in turning on (off) the couplings between qubits in the design, we should discuss the conditions needed in the corresponding manipulation. With current technology, $2T_{C_{i}} /h\sim 10$ GHz ($h$ is Planck's constant) is achievable,[27] corresponding to $\omega_{i} \sim 60$ GHz. Applying the RWA requires $|{\omega_{i} -\omega_{j}}|\gg g$, thus we choose $| {\omega_{i} -\omega_{j}}|\geqslant 10^{2}g$. Then the coupling strength can be chosen as $g\sim 10^{2}$ MHz, which is easy to be realized with capacitive couplings.[27,28] Thus the time scale of a single operation will be $t\sim 0.01$ µs, which is much smaller than the typical relaxation time $T\sim 1$ µs for the charge system with qubits at the optimal point. In summary, we have demonstrated the scheme of performing programmable universal quantum computations with DQD architecture. By connecting all the qubits directly with capacitors and tuning the gate voltages of the DQDs, we have shown the implementing of various universal gate operations and coupling operations, including two-qubit gate operation between arbitrary two qubits, a series of two-qubit gate operations on multiple pairs of qubits in parallel, coupling operations between arbitrary qubits in parallel in a selected group, etc. These operations may enable us to realize a practical programmable quantum dot quantum processor. The design is achievable within current experimental technology. 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