Abstract
Universal multiplequbit gates can be implemented by a set of universal singlequbit gates and any one kind of entangling twoqubit gate, such as a controlledNOT gate. For semiconductor quantum dot qubits, twoqubit gate operations have so far only been demonstrated in individual electron spinbased quantum dot systems. Here we demonstrate the conditional rotation of two capacitively coupled charge qubits, each consisting of an electron confined in a GaAs/AlGaAs double quantum dot. Owing to the strong interqubit coupling strength, gate operations with a clock speed up to 6 GHz have been realized. A truth table measurement for controlledNOT operation shows comparable fidelities to that of spinbased twoqubit gates, although phase coherence is not explicitly measured. Our results suggest that semiconductor charge qubits have a considerable potential for scalable quantum computing and may stimulate the use of longrange Coulomb interaction for coherent quantum control in other devices.
Introduction
Semiconductor quantum dots (QDs), hailed for their potential scalability, are outstanding candidates for solid statebased quantum information processing^{1,2,3}. Qubits, encoded by the charge occupancy of a single electron in a double QD, have attracted considerable attention^{4,5,6,7,8,9} for number of reasons. First, speed of gate operation is primarily determined by the interdot tunnelling rate, which can be made to be extremely fast. Second, initialization, manipulation and readout are all intuitively simple in this allelectrical approach. Furthermore, the utilization of charge degree of freedom for quantum computation is compatible with today’s mainstream information processing technology, and is suitable for scaling up to largescale quantum circuits by taking advantage of the wealth of present semiconductor infrastructures.
One of the basic building blocks of universal quantum computation is a twoqubit gate. Previous research on two coupled semiconductor charge qubits have shown certain evidence of correlated oscillations^{10,11,12,13}. However, the implementation of a twoqubit gate operation in QD charge qubits has not been demonstrated to date, largely owing to the technical challenges of achieving strong coupling between qubits and the ability to control gate pulses in the subnanosecond time scales.
In the following, we report the coherent manipulation of a capacitively coupled qubit pair. We achieve a strong electrostatic dipole coupling between two charge qubits. The large coupling energy enables us to completely and coherently turn on/off the Larmor oscillations of one qubit by pulse driving the charge on the other qubit. Based on this effect, we demonstrated a controlledNOT (CNOT) operation, although phase coherence is not explicitly measured^{14,15,16,17,18,19}. In addition, we combined this CNOT operation and universal singlequbit gates by using Landau–Zener interferences to show the feasibility of achieving quantum twoqubit gates in this system. Our results also demonstrate that the fidelity of twoqubit operations for a QD charge qubit can be comparable to that of spinbased semiconductor qubits^{20,21,22}. For charge qubits, with a sufficiently large coupling energy, the fidelity of twoqubit operations is only limited by the fidelity of the single qubit. Thus, with the reduction of decoherence rate of single qubit using a more sophisticated double QD dispersion^{8}, the prospect of semiconductor charge qubits for scalable quantum computation can be considerably improved.
Results
Strong interqubit coupling
Figure 1a depicts our twoqubit system consisting of two coupled double QD (DQD)s and two quantum point contacts (QPCs). The Hamiltonian of this system is as follows:
Here, ɛ_{U} (ɛ_{L}) is the energy detuning, Δ_{U}=2t_{U} (Δ_{L}=2t_{L}) is twice the interdot tunnelling rate for the upper (lower) DQD, σ_{x} and σ_{z} are the Pauli matrixes, I is the identity matrix and J is the interqubit coupling energy. We denote the four eigenstates of the above Hamiltonian by 00>, 10>, 01> and 11>. Normally, the eigenstates are different from the charge states (RR>, LR>, LR> and LL>) that the QPCs can detect, except when the detuning of each qubit is far away from its balance point (ɛ_{U,L} ≫ 0). We always reset the qubits far from the balance points before applying any gate pulses, where the eigenstates coincide with the charge states. Therefore, in the later contexts, we will describe the evolution of the qubit states in the frame of eigenstates and ignore the unitary transformations between the frame of the eigenstates and that of the charge states.
The interqubit coupling energy J originates from the Coulomb repulsion between an electron in the upper DQD and another electron in the lower DQD. When the two electrons are closest to each other, the Coulomb interaction energy is higher, by an amount defined as J, than it is when the electrons are furthest apart from each other^{10,11,12,13}. This is illustrated in Fig. 1b: the abrupt energy shift from state 00> to state 11> is given by J. The origin and role of J are similar to those of the interqubit dipole–dipole interaction between two capacitively coupled semiconductor spin qubits^{22}. We will see that a sufficiently large J is the key to controlling the coherent rotations of one qubit by manipulating the state of the other qubit, and is therefore the key to realizing twoqubit gates such as CNOT gates.
In our experiment, we are able to achieve very high J (J/h≈29.0 GHz) compared with other characteristic parameters: Δ_{U}≈6.2 GHz and Δ_{L}≈6.0 GHz. Figure 1c presents the differential current of the upper QPC and Fig. 1d presents that of the lower QPC. Therefore, Fig. 1c records only the response to the upper detuning, ɛ_{U}, and Fig. 1d records only the response to the lower detuning, ɛ_{L}. The two figures together constitute a complete description of Fig. 1b. We find that J is equal to ∼119 μeV using the energy–voltage conversion factor obtained from transport measurements of 30 μeV per mV. We can deliberately tune the voltages on the two horizontal gates H_{1} and H_{2} to maximize the interqubit coupling energy J and simultaneously suppress the direct interqubit tunnelling.
Now, we apply a rectangular voltage pulse to one of the upper qubit’s gates, U_{1}. We initialize both the upper and lower qubits in state 0>, that is, −ɛ_{U,L} ≫ J ≫ Δ_{U,L}>0. Under these conditions, the two qubits are nearly uncorrelated, and only the upper qubit is affected by the voltage pulse. We choose the pulse amplitude such that it will drive the upper qubit exactly to its balance point, that is, from working point R_{a} to R_{b} as illustrated in Fig. 1c,d. The pulse’s combined rise and fall time is measured to be 130 ps on top of the refrigerator. When we sweep the pulse width to longer values, nonadiabatic evolution such as Larmor oscillations is observed. The qubit oscillates between states 0> and 1>, with a probability in each state as a cosine function of the pulse width W_{1}. In a Bloch sphere, the upper qubit rotates around the x axis by an angle proportional to W_{1} (refs 4, 5). More information can be found in Supplementary Fig. 1 and Supplementary Note 1.
These are simply regular Larmor oscillations for a single qubit. However, we will observe a difference if we change the state of the lower qubit to 1>, that is, ɛ_{L} ≫ J ≫ Δ_{L}>0, while keeping the rest of the system unchanged. When the lower qubit switches from 0> state to 1> state, the upper qubit’s balance point shifts toward higher energies by an amount J. As a result, the pulse now drives the system from point T_{a} to T_{b} as illustrated in Fig. 1c,d, meaning the upper qubit is unable to reach its balance point now. We thus expect the Larmor oscillations to disappear.
The experiment clearly demonstrates the above effect. In Fig. 1e,f, we present the Larmor oscillations of the upper qubit conditional on the lower qubit’s state. The x axis corresponds to the pulse width, W_{1}. The y axis corresponds to the lower qubit detuning, ɛ_{L}. Figure 1e presents the differential current of the upper qubit and Fig. 1f presents that of the lower qubit. Figure 1f reveals that the lower qubit switches between states as the line V_{L4}≈−0.525 V is crossed. When V_{L4}<<−0.525 V, the lower qubit is in the 0> state and Fig. 1e presents the Larmor oscillations of the upper qubit with a frequency equal to Δ_{U}=6.2 GHz. When V_{L4}>>−0.525 V, the lower qubit switches to the 1> state, and in Fig. 1e, it is evident that the Larmor oscillations of the upper qubit disappear. Near the balance point, that is, when −0.526 V<V_{L4}<−0.524 V, the two qubits should rotate as an entangled state, exhibiting Larmor oscillations at two frequencies, which is irrelevant to our CNOT gate and will be addressed elsewhere.
Figure 1e,f demonstrates that we can completely suppress the upper qubit’s Larmor oscillations by switching the lower qubit from the 0> state to the 1> state. A CNOT gate, the logical operation of which is to flip the upper qubit if the lower qubit is in state 0> and to do nothing if the lower qubit is in state 1> can thus obtain its maximum fidelity. We perform theoretical simulations by numerically solving the master equations. Details are provided in Supplementary Fig. 2 and Supplementary Note 2. The simulation successfully reproduces phenomena such as those observed in Fig. 1e,f when the experimentally obtained parameters were used, including J=119 μeV. We need to point out that experimentally we present the QPC differential current because of its better signaltonoise ratio. Our simulation focuses on the state probability, which is directly related to the QPC current. However, the simulation reflects the same features as those of the experimental result.
If we reduce J, for instance to 25 μeV, which is the same magnitude as Δ_{U} and Δ_{L}, our simulation indicates that in this case, the upper qubit’s Larmor oscillations cannot be completely suppressed. There are leakage Larmor oscillations with a 55% amplitude when the lower qubit is switched from state 0> to state 1>. Therefore, a CNOT gate for J=25 μeV will achieve a low fidelity. These leakage Larmor oscillations at low J occur because the two balance lines have finite line widths, as shown in Fig. 1c. If J is smaller than or comparable to this line width, the two balance lines are smeared out, and the same voltage pulse can drive the upper qubit to its balance point regardless of whether the lower qubit is in state 0> or state 1>. Only if J is much larger than this line width will there be no overlap between these two balance lines, and thus no leakage Larmor oscillations. The J value required to completely separate the two balance lines is therefore the threshold value for the CNOT gate to achieve maximum fidelity.
We can calculate the dependence of the upper bound of the CNOT gate fidelity on the interqubit coupling strength J through simulations. Details are provided in the Supplementary Note 2. We present the processindependent fidelity without the dephasing effect as the red solid curve in Fig. 1g. Two important features are apparent: the fidelity increases with increasing J and eventually saturates. In our case, J=119 μeV, we should, in principle, achieve 97% fidelity for the CNOT gate. However, this estimation is excessively idealistic. It assumes an infinitely long dephasing time, 100% fidelity for the singlequbit gates, and perfect pulse shaping for the twoqubit gates. After considering an inhomogeneous decoherence time 1.2 ns, we simulated the fidelity for CNOT gate operated with 3π rotating pulses, shown as the greed dashed curve in Fig. 1g. The fidelity for J=119 μeV drops to 0.89. And as we will see, experimentally, we measured the truth table of a CNOT operation and achieved 68% fidelity. Nonetheless, Fig. 1g strongly indicates that J is not the major limiting factor in preventing the achievement of perfect fidelity in our experiment. The interqubit coupling strength in our device has already been necessarily large to achieve a satisfactory CNOT gate. This is the greatest advancement of this study with respect to earlier experiments.
For simplicity, in this paper, we present the details only for the control of the upper qubit through the manipulations of the lower qubit. Considering the symmetric design, the opposite would certainly be possible, that is, controlling the lower qubit by manipulating the upper qubit.
Pulse timing
Additional experimental challenges remain in the development of a functional CNOT gate. The voltage pulses required for the implementation of a singlecharge qubit gate are already very short (200–500 ps). To demonstrate a CNOT gate, we require up to three sequential ultrashort pulses. These pulses must be carefully synchronized and aligned. Here we demonstrate how we manipulate two pulses, one on the lower qubit and the other on the upper qubit, to coherently rotate the lower qubit and thus control the state of the rotation of the upper qubit. Further details regarding pulse timing are presented in Supplementary Figs 3 and 4, and Supplementary Notes 3 and 4.
A schematic description of the manipulation process is presented in Fig. 2a. Both qubits are initialized in state 0>, that is, working point R_{a} as illustrated in Fig. 1b,c. In addition to a rectangular pulse of width W_{1} on the upper qubit, as described above, another rectangular pulse of width W_{2} is applied to one gate of the lower qubit, L_{5}. The lower pulse (W_{2}) first drives the lower qubit to its balance point, that is, from working point R_{a} to S_{a}. After a delay time (∼100 ps) that is much shorter than the dephasing time T_{2}* (∼1,200 ps) and the relaxing time T_{1} (∼19 ns), the upper pulse (W_{1}) follows and drives the upper qubit to its balance point, that is, from working point R_{a} to R_{b}. If the lower pulse is terminated at 2nπ, then the lower qubit will remain in the 0> state. The upper pulse will then rotate the upper qubit by an angle proportional to W_{1}. By contrast, if the lower pulse is terminated at (2n+1)π, then the lower qubit will enter the 1> state. Consequently, the upper pulse will have no effect, and the upper qubit will remain in the 0> state regardless of W_{1}.
Generally, we assume that the pulse of width W_{2} rotates the lower qubit by an angleβ, and that the pulse of width W_{1} rotates the upper qubit by an angleα when the lower qubit is in state 0>. Then, the two qubits will end up in the following entangled state: cosα cosβ 00>+sinα cosβ 10>+sinβ 01>. The probability of finding the upper qubit in state 0> is P_{U}^{0}=1−sin^{2}α cos^{2}β, and the probability of finding the lower qubit in state 0> is P_{L}^{0}=cos^{2}β. Therefore, we predict that P_{U}^{0} should oscillate with both W_{1} and W_{2}, whereas P_{L}^{0} should oscillate only with W_{2}. Moreover, the dependence of P_{U}^{0} on W_{2} is out of phase by π compared with P_{L}^{0}. We simulate this process by solving the master equations, as shown in Fig. 2b,c (ref. 16).
Experimentally, we observe the predicted pattern shown in Fig. 2d,e. The QPC differential current for the lower qubit periodically oscillates only along the W_{2} axis, whereas that for the upper qubit exhibits oscillations along both the W_{1} and W_{2} axes. The oscillation frequencies along the W_{1} and W_{2} axes are Δ_{U} and Δ_{L}, respectively. In addition, the dependence on W_{2} is out of phase by approximately π between the upper and lower qubits, which is as predicted. This finding demonstrates that we can indeed coherently control the Larmor oscillations of the upper qubit.
In addition, the QPC differential current for the lower qubit is invariant with respect to the upper pulse of width W_{1}. This observation serves as a proof that there is no observable crosstalk between the two qubits.
CNOT operation truth table
Based on the achievement of sufficiently high J and proper pulse timing, we now test the CNOT operation and perform truth table measurements to determine its fidelity^{8,14,15,16,17,18,19}. Figure 3a presents the process flowchart. In the initialization process, we reset the two qubits to the 00> state, that is, working point R_{a}. Then, in the input preparation process, we apply certain pulses to both the upper and lower qubits to obtain different input states. By tuning the pulse widths W_{2} and W_{3}, we prepare four input states: 00>, 10>, 01> and 11>. Finally, these input states are fed into the CNOT gate, which consists of a πpulse on the upper qubit. The logic of the CNOT gate means that the upper qubit undergoes a πrotation if the lower qubit is in the 0> state and no rotation if the lower qubit is in the 1> state. Therefore, after passing through the CNOT gate, the four input states will be transformed into the 10>, 00>, 01> and 11> states, respectively.
For the first input state 00>, the initial state is directly sent to the CNOT gate without any preparatory pulse. To prepare a 10> input state, a π pulse is applied to the upper qubit before the CNOT gate. A πpulse on the lower qubit will yield the 01> input state. Finally, we apply a πpulse to the lower qubit followed by a πpulse with an elevated amplitude on the upper qubit to obtain a 11> input state. A regular πpulse drives the working point from R_{a} to R_{b} and this pulse drives from R_{a} to R_{c}. The purpose is to force the upper qubit to rotate after the lower qubit has already been switched into the 1> state. Experimentally, because πpulses are too short to be well controlled, we use 3πpulses instead (360 ps for the upper qubit and 390 ps for the lower qubit in our experiment).
The output of the CNOT gate for each of the four input states is read through the QPC current. We use the pulsemodulation technique developed in previous studies to convert the QPC current into a state probability^{8}. Further details are provided in Supplementary Fig. 1 and Supplementary Note 1. In Fig. 3b,c, we sweep the pulse width of the CNOT gate (W_{1}) and measure the probabilities P_{U}^{0} and P_{L}^{0} after generating each of the four input states. As expected, P_{U}^{0} exhibits Larmor oscillations for input 00>. For input 10>, the Larmor oscillations are shifted by a phase of π. For inputs 01> and 11>, P_{U}^{0} exhibits essentially no oscillation because the lower qubit has been switched into state 1>. The difference between the two inputs is that P_{U}^{0} remains at a high level for input 01> and at a low level for input 11>. P_{L}^{0} exhibits essentially no dependence on W_{1} because the upper pulse does not affect the lower qubit.
From Fig. 3b,c, we extract the values of P_{U}^{0} and P_{L}^{0} at W_{1}=360 ps, which corresponds to a 3πpulse on the upper qubit and therefore a CNOT gate. Based on these values, we obtain the truth table for the CNOT gate, as illustrated in Fig. 3d. For comparison, we simulated the truth table for an ideal CNOT gate and for a CNOT gate with an inhomogeneous dephasing time of 1,200 ps, as shown in Fig. 3e,f, respectively. The detailed values of these truth tables are provided below (ref. 15):
The measured fidelity is 0.68. We must reiterate that the interqubit coupling strength J is not the limiting factor responsible for this imperfect fidelity because J is already sufficiently large to allow us to completely switch the rotations of the upper qubit on and off by manipulating the state of the lower qubit. We believe that there are two main factors that account for the deviation of the measured fidelity from 1. First, the relatively short dephasing time of a singlecharge qubit causes errors in singlequbit operations, and these errors are carried over into the twoqubit operations. Once the lower qubit suffers dephasing, the lower qubit state will contain a 0> component even when it should be in the 1> state. This leakage to the 0> state will cause the suppression of the upper qubit’s Larmor oscillations to be incomplete. In combination with the dephasing of the upper qubit, this effect will degrade the ultimate CNOT gate fidelity. By comparing Fig. 3e,f, our simulation reveals that the predicted CNOT gate fidelity decreases to 89% when a qubit dephasing time of 1,200 ps is considered^{5}.
Second, the pulse shaping of multiple ultrashort pulses is extremely challenging and cannot be made ideal. The relatively short dephasing time in our system requires us to complete gate operations as quickly as possible. Although this forces us to boost the gate operation speed, which proves to be helpful, it also increases the risk of poor pulse shaping. In particular, we observe that the finite nature of the pulse rising and falling times makes the precise tuning of the pulse width difficult. In addition, we observe that the increase in the pulse amplitude with increasing pulse width makes it difficult to accurately control the amplitude of each pulse and to align the amplitudes of multiple pulses. The accumulation of all these errors gives rise to deviation between the final output qubit state and the desired qubit state. This might be the primary reason for the CNOT fidelity to drop to ∼68%.
Although this fidelity does not contain the part of phase rotation, its value is almost as high as that of spinbased semiconductor qubits. The relatively short dephasing time of charge qubits has always been an obstacle preventing the serious consideration of the possibility of multiplecharge qubits. However, the large intraqubit coupling and interqubit coupling originating from the direct Coulomb interaction between electron charges enable us to operate the CNOT gate at a very high clock speed (a few GHz) and to maintain the gate fidelity at a satisfactory level. Coulomb interactions are significant in various types of multiqubit systems. For example, spin qubits in silicon (Si)based QDs have recently demonstrated remarkably long coherence times^{23,24,25,26}. Highquality singleelectron spin gates have been demonstrated using Si. However, longrange interqubit coupling still originates from the Coulomb interaction. Twoqubit gates using Si still suffer from high charge noise and therefore still require further development. We hope that our demonstration of twoqubit gates based on the Coulomb interaction may offer inspiration for the investigation of semiconductor multiqubits.
There is still a room to improve the fidelity of our electron charge twoqubit gates. The errors originating from imperfect pulse shaping are deterministic and could be corrected with further progress in highfrequency technology. We hope that the advancement of picosecond pulse generators and the incorporation of onchip transmission lines will help us to improve the fidelity of our single and doublequbit gates. Moreover, the qubit dephasing effect is intrinsic, and new materials or architectures will be necessary to achieve significant improvements. Recent progress in the field of hybrid qubits has demonstrated that fast operating speeds and long coherence times can be simultaneously achieved in electron charge qubits by engineering an energy structure with certain excited states^{8}. If similar schemes can be applied to increase T_{2}* in our system by perhaps 1 to 2 orders of magnitude, the fidelity of the twoqubit CNOT gate could markedly increase.
Prospective quantum logic gates
In the CNOT truth table measurement, we considered only the amplitudes of the quantum states. However, Fig. 2 previously illustrated the quantum nature of the CNOT gate, which has no counterpart among classical gates. The amplitudes of the quantum states of both the upper and lower qubits can be set to any arbitrary superposition value, corresponding to rotations by arbitrary angles around the x axis in each Bloch sphere.
Furthermore, we will show that we can vary both the phase and amplitude of the qubit’s states while preserving the CNOT logic. As illustrated in Fig. 4a, a pulse with a fixed width of 100 ps is applied to the lower qubit. This pulse width is set to be shorter than the rise and fall times combined, and therefore, the pulses can be regarded as triangular. This pulse width is also small compared with T_{2}* away from the balance point, which was estimated to be ∼300 ps by studying the inhomogeneous broadening of the photonassisted tunnelling line width^{5}. We initialize the lower qubit in state 0> and sweep the lower pulse amplitude. The lower qubit is driven to pass through its balance point if the pulse amplitude is larger than its detuning, that is, from working point R_{a} to cross S_{a}. This adiabatic passage through the balance point induces the Landau–Zener–Stuckelberg effect, corresponding to rotation around both the x and z axis in the Bloch sphere^{6}.
Immediately following the lower pulse, a rectangular pulse is applied to the upper qubit and induces Larmor oscillations in the upper qubit, that is, from working point R_{a} to cross R_{b}. Here we demonstrate that the Larmor oscillations of the upper qubit are controlled by the lower qubit’s phase accumulation caused by Landau–Zener–Stuckelberg interference. First, let us suppose that the triangular pulse can independently drive the lower qubit from the 0> state into the U(β,ψ) 0>+V(β,ψ) 1> state, where U^{2}(β,ψ)+V^{2}(β,ψ)=1, and that the rectangular pulse can independently drive the upper qubit from the 0> state into the cosα 0>+sinα 1> state if the two qubits are completely uncorrelated.
In reality, the two qubits are coupled, and the CNOT gate logic ensures that the upper triangular pulse can only cause the upper qubit to rotate when the lower qubit is in the 0> state. Consequently, after both the lower and upper pulses, the final entangled twoqubit state will be as follows: U(β,ψ) cosα 00>+U(β,ψ) sinα 10>+V(β,ψ) 01>. The probability of finding the upper qubit in state 0> is P_{U}^{0}=1U^{2}(β,ψ) sin^{2}α, and the probability of finding the lower qubit in state 0> is P_{L}^{0}=U^{2}(β,ψ). U^{2}(β,ψ) oscillates with both the amplitude (through β) and phase (through ψ) of the lower qubit’s state. However, the oscillation of the phase is much faster than that of the amplitude. Therefore, in the time window of our experiment, we predominantly observe periodic oscillations with the phase. Thus, P_{U}^{0} will exhibit cosine oscillations with both W_{1} (through α) and A_{2} (mainly through ψ), and P_{L}^{0} will oscillate only with A_{2}(predominantly through ψ). Again, we note that the dependence of P_{U}^{0} on A_{2} is out of phase by π compared with that of P_{L}^{0}. In Fig. 4b,c, we present the simulated responses of P_{U}^{0} to A_{2}, respectively.
This interpretation explains the data depicted in Fig. 4d,e, where we present the oscillations of the upper qubit with W_{1} and A_{2}. As expected, the upper qubit exhibits not only Larmor oscillations with respect to W_{1} through angleα but also oscillations with A_{2} through the phase ψ. Its dependence on ψ is also out of phase by ∼π as compared with that of the lower qubit. This indicates that the phase of the lower qubit’s state can be used to control the state of the upper qubit. Our CNOT gate is thus proven to operate on quantum states of the qubits. In Supplementary Fig. 3 and Supplementary Note 3, we demonstrate that we can even rotate the phase and amplitude of both the upper and lower qubits while preserving the CNOT gate quantum logic.
We also need to point out that further experiment is still needed to explicitly quantify the degree of entanglement between the two qubits during the operation process to realize quantum twoqubit logic gates.
Discussion
In summary, a string of technical accomplishments, including the achievement of a strong interqubit coupling and the synchronization of multiple ultrafastpulses, enabled us to demonstrate universal twoqubit operations in an all electrically controlled semiconductor charge system. CNOT gate truth table shows high fidelities comparable to that of electron spin twoqubit gates. However, a quantum process tomography measurement^{22} is required to definitively compare the qubit metrics of the two systems. At current stage, trading shorter dephasing time for faster qubit operation time, charge qubits can perform well in the twoqubit level. Optimistically, we argue that with a better control of pulse shape and a better design of the dispersion relations, which are completely deterministic, the semiconductor charge qubits may become a force to contend with in the scalable quantum computation arena.
Methods
Devices
The twoDQD device was defined via electron beam lithography on a molecular beam epitaxially grown GaAs/AlGaAs heterostructure. A twodimensional electron gas is present 95 nm below the surface. The twodimensional electron gas has a density of 3.2 × 10^{11} cm^{−2} and a mobility of 1.5 × 10^{5} cm^{−2}V^{−1}s^{−1}. Figure 1a presents a scanning electron micrograph of the surface gates. Five upper gates U_{1}–U_{5} and two horizontal gates H_{1} and H_{2} form the upper DQD. Five lower gates L_{1}–L_{5} and two horizontal gates H_{1} and H_{2} form the lower DQD. The horizontal gates H_{1} and H_{2} also tune the capacitive coupling strength between the upper and lower DQDs. Direct electron tunnelling is suppressed between the two DQDs by ensuring adequate negative bias voltages on gates H_{1} and H_{2}. The four gates Q_{1}–Q_{4} define QPCs for the monitoring of the charge status on each DQD.
Measurements
The experiments are performed in an Oxford Triton dilution refrigerator with a base temperature of 10 mK. Two Agilent 81134A pulse generators, which have a rise time of 65 ps and a time resolution of 1 ps, are used to deliver fast pulse trains through semirigid coaxial transmission lines to the device. Standard lockin modulation and detection techniques are used for the chargesensing readout. Through electronic transport and photonassisted tunnelling measurements, wherein the electron energy can be read directly from the sourcedrain bias voltage and the photon frequency, we conclude that the energy–voltage lever arm is ∼100 μeV per mV for the barrier gates (U_{1}, U_{5}, L_{1} and L_{5}) and 30 μeV per mV for the plunger gates (U_{2}, U_{4}, L_{2} and L_{4}).
Additional information
How to cite this article: Li, H. O. et al. Conditional rotation of two strongly coupled semiconductor charge qubits. Nat. Commun. 6:7681 doi: 10.1038/ncomms8681 (2015).
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Acknowledgements
This work was supported by the National Fundamental Research Program (grant no. 2011CBA00200), the National Natural Science Foundation (grant nos. 11222438, 11174267, 61306150, 11304301 and 91121014) and the Chinese Academy of Sciences.
Author information
Author notes
 HaiOu Li
 , Gang Cao
 & GuoDong Yu
These authors contributed equally to this work
Affiliations
Key Laboratory of Quantum Information, CAS, University of Science and Technology of China, Hefei, Anhui 230026, China.
 HaiOu Li
 , Gang Cao
 , GuangCan Guo
 & GuoPing Guo
Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China.
 HaiOu Li
 , Gang Cao
 , GuoDong Yu
 , Ming Xiao
 , GuangCan Guo
 & GuoPing Guo
Department of Optics and Optical Engineering, University of Science and Technology of China, Hefei, Anhui 230026, China.
 GuoDong Yu
 & Ming Xiao
Department of Physics and Astronomy, University of California, Los Angeles, California 90095, USA.
 HongWen Jiang
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Contributions
H.O.L. and G.P.G. performed the measurements. G.C., G.C.G. and H.W.J. provided theoretical support and analysed the data. G.D.Y. and M.X. fabricated the samples. M.X. and G.P.G. supervised the project. All authors contributed to write the paper.
Competing interests
The authors declare no competing financial interests.
Corresponding authors
Correspondence to Ming Xiao or GuoPing Guo.
Supplementary information
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Supplementary Information
Supplementary Figures 14, Supplementary Notes 14 and Supplementary References
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