A LD qubit and a ST qubit formed in a triple quantum dot (TQD)

A hybrid system comprising a LD qubit and a ST qubit is implemented in a linearly-coupled gate-defined TQD shown in Fig. 1a. The LD qubit (Q LD ) is formed in the left dot while the ST qubit (Q ST ) is hosted in the other two dots. We place a micro-magnet near the TQD to coherently and resonantly control Q LD via electric dipole spin resonance (EDSR)20,21,22,23,26. At the same time it makes the Zeeman energy difference between the center and right dots, \({\mathrm{\Delta }}E_{\mathrm{Z}}^{{\mathrm{ST}}}\), much larger than their exchange coupling JST, such that the eigenstates of Q ST become |↑↓〉 and |↓↑〉 rather than singlet |S〉 and triplet |T〉. We apply an external in-plane magnetic field B ext = 3.166 T to split the Q LD states by the Zeeman energy E Z as well as to separate polarized triplet states |↑↑〉 and |↓↓〉 from the Q ST computational states. The experiment is conducted in a dilution refrigerator with an electron temperature of approximately 120 mK. The qubits are manipulated in the (N L , N C , N R ) = (1,1,1) charge state while the (1,0,1) and (1,0,2) charge states are also used for initialization and readout (see Fig. 1b). Here, N L(C,R) denotes the number of electrons inside the left (center, right) dot.

Fig. 1 Hybrid system of a LD qubit and a ST qubit realized in a TQD. a False color scanning electron microscope image of a device identical to the one used in this study. The TQD is defined in a two-dimensional electron gas at the GaAs/AlGaAs heterointerface 100 nm below the surface. The upper single electron transistor is used for radiofrequency-detected charge sensing24,25. A MW with a frequency of 17.26 GHz is applied to the S gate to drive EDSR. b Stability diagram of the TQD obtained by differentiating the charge sensing signal V rf . c Hybrid system of a LD qubit and a ST qubit coupled by the exchange coupling JQQ. d Rabi oscillation of Q LD (rotation around x-axis) driven by EDSR with JQQ ~ 0 at point RL in Fig. 1b. The data is fitted to oscillations with a Gaussian decay of \(T_2^{{\mathrm{Rabi}}}\) = 199 ns. e Pulse sequence used to produce Fig. 1d showing gate voltages V PL and V PR applied to the PL and PR gates and a MW burst V MW . f Precession of Q ST (rotation around z-axis) with a frequency of fST = 280 MHz due to \({\mathrm{\Delta }}E_{\mathrm{Z}}^{{\mathrm{ST}}}\), taken at point E marked by the white circle in (1,1,1) in Fig. 1b, where JQQ and JST ~ 0. The data follow the Gaussian decay with a decay time of 207 ns (see Supplementary Fig. 2a) induced by the nuclear field fluctuations29. g Pulse sequence used to produce Fig. 1f Full size image

We first independently measure the coherent time evolution of each qubit to calibrate the initialization, control, and readout. We quench the inter-qubit exchange coupling by largely detuning the energies of the (1,1,1) and (2,0,1) charge states. For Q LD , we observe Rabi oscillations4 with a frequency f Rabi of up to 10 MHz (Fig. 1d) as a function of the microwave (MW) burst time t MW , using the pulse sequence in Fig. 1e. For Q ST , we observe the precession between |S〉 and |T〉 (ST precession) (Fig. 1f) as a function of the evolution time t e , using the pulse sequence in Fig. 1g (see Supplementary Note 2 for full control of Q ST ). We use a metastable state to measure Q ST with high fidelity13 (projecting to |S〉 or |T〉) in the presence of large \({\mathrm{\Delta }}E_{\mathrm{Z}}^{{\mathrm{ST}}}\) with which the lifetime of |T〉 is short27.

Calibration of the two-qubit coupling

The two qubits are interfaced by exchange coupling JQQ between the left and center dots as illustrated in Fig. 1c. We operate the two-qubit system under the conditions of \(E_{\mathrm{Z}} \gg {\mathrm{\Delta }}E_{\mathrm{Z}}^{{\mathrm{ST}}},{\mathrm{\Delta }}E_{\mathrm{Z}}^{{\mathrm{QQ}}} \gg J^{{\mathrm{QQ}}} \gg J^{{\mathrm{ST}}}\) where \({\mathrm{\Delta }}E_{\mathrm{Z}}^{{\mathrm{QQ}}}\) is the Zeeman energy difference between the left and center dots. Then, the Hamiltonian of the system is

$${\cal H} = - E_{\mathrm{Z}}\hat \sigma _z^{{\mathrm{LD}}}/2 - {\mathrm{\Delta }}E_{\mathrm{Z}}^{{\mathrm{ST}}}\hat \sigma _z^{{\mathrm{ST}}}/2 + J^{{\mathrm{QQ}}}(\hat \sigma _z^{{\mathrm{LD}}}\hat \sigma _z^{{\mathrm{ST}}} - 1)/4$$ (1)

where \(\hat \sigma _z^{{\mathrm{LD}}}\) and \(\hat \sigma _z^{{\mathrm{ST}}}\) are the Pauli z-operators of Q LD and Q ST , respectively18 (Supplementary Note 3). The last term in Eq. (1) reflects the effect of the inter-qubit coupling JQQ: for states in which the spins in the left and center dots are antiparallel, the energy decreases by JQQ/2 (see Fig. 2a). In the present work, we choose to operate Q LD as a control qubit and Q ST as a target, although these are exchangeable. With this interpretation, the ST precession frequency fST depends on the state of Q LD, \(f_{\sigma _z^{{\mathrm{LD}}}}^{{\mathrm{ST}}} = \left( {{\mathrm{\Delta }}E_{\mathrm{Z}}^{{\mathrm{ST}}} - \sigma _z^{{\mathrm{LD}}}J^{{\mathrm{QQ}}}/2} \right)/h\). Here \(\sigma _z^{{\mathrm{LD}}}\) represents |↑〉 or |↓〉 and +1 or −1 interchangeably. This means that while JQQ is turned on for the interaction time t int , Q ST accumulates the controlled-phase ϕ C = 2πJQQt int /h, which provides the CPHASE gate (up to single-qubit phase gates; see Supplementary Note 7) in t int = h/2JQQ. An important feature of this two-qubit gate is that it is intrinsically fast, scaling with JQQ/h which can be tuned up to ~100 MHz, and is limited only by the requirement \(J^{{\mathrm{QQ}}}/h \ll {\mathrm{\Delta }}E_{\mathrm{Z}}^{{\mathrm{QQ}}}/h \sim 500\,{\mathrm{MHz}}\) in our device.

Fig. 2 ST qubit frequency controlled by the LD qubit. a Energy diagram of the two-qubit states for \(E_{\mathrm{Z}} \gg {\mathrm{\Delta }}E_{\mathrm{Z}}^{{\mathrm{ST}}},{\mathrm{\Delta }}E_{\mathrm{Z}}^{{\mathrm{QQ}}} \gg J^{{\mathrm{QQ}}}\) (JST = 0). The ST qubit frequency is equal to \({\mathrm{\Delta }}E_{\mathrm{Z}}^{{\mathrm{ST}}}\) for JQQ = 0, and shifts by ±JQQ/2 depending on the Q LD state for finite JQQ. b The quantum circuit for demonstrating the phase control of Q ST depending on Q LD . After preparing an arbitrary state of Q LD (stages A and B), we run modified stages from D to H (shown in the upper panel) 100 times with t int values ranging from 0.83 to 83 ns to observe the time evolution of Q ST without reinitializing or measuring Q LD . Stages A, B and C take 202 μs in total and the part from D to H is 7 μs long. c FFT spectra of fST with different interaction points shown by the white corresponding symbol in Fig. 1b (traces offset for clarity). In addition to the frequency splitting due to JQQ, the center frequency of the two peaks shifts because \({\mathrm{\Delta }}E_{\mathrm{Z}}^{{\mathrm{ST}}}\) is also dependent on the interaction point (Methods). d Interaction point dependence of the ST qubit frequency splitting, i.e. the two-qubit coupling strength JQQ/h, fitted with the black model curve (see Supplementary Note 4 for the data extraction and fitting). e ST precession for the Q LD input state |↑〉 fitted with the Gaussian-decaying oscillations with a decay time of 72 ns. f ST precession for the Q LD input state |↓〉 with a fitting curve. The decay time is 75 ns. The total data acquisition time for e and f is 451 ms Full size image

Before testing the two-qubit gate operations, we calibrate the inter-qubit coupling strength JQQ, and its tunability by gate voltages. The inter-qubit coupling in pulse stage F (Fig. 2b) is controlled by the detuning energy between (2,0,1) and (1,1,1) charge states (one of the points denoted E in Fig. 1b). To prevent leakage from the Q ST computational states, we switch JQQ on and off adiabatically with respect to \({\mathrm{\Delta }}E_{\mathrm{Z}}^{{\mathrm{QQ}}}\) by inserting voltage ramps to stage F with a total ramp time of t ramp = 24 ns (Fig. 2b)28. The coherent precession of Q ST is measured by repeating the pulse stages from D to H without initializing, controlling and measuring Q LD , which makes Q LD a random mixture of |↑〉 and |↓〉. Figure 2c shows the FFT spectra of the precession measured for various interaction points indicated in Fig. 1b. As we bring the interaction point closer to the boundary of (1,1,1) and (2,0,1), JQQ becomes larger and we start to see splitting of the spectral peaks into two. The separation of the two peaks is given by JQQ/h which can be controlled by the gate voltage as shown in Fig. 2d.

We now demonstrate the controllability of the ST precession frequency by the input state of Q LD , the essence of a CPHASE gate. We use the quantum circuit shown in Fig. 2b, which combines the pulse sequences for independent characterization of Q LD and Q ST . Here we choose the interaction point such that JQQ/h = 90 MHz. By using either |↑〉 or |↓〉 as the Q LD initial state (the latter prepared by an EDSR π pulse), we observe the ST precessions as shown in Fig. 2e, f. The data fit well to Gaussian-decaying oscillations giving \(f_{| \uparrow \rangle}^{{\mathrm{ST}}} = 434 \pm 0.5\,{\mathrm{MHz}}\) and \(f_{| \downarrow \rangle}^{{\mathrm{ST}}} = 524 \pm 0.4\,{\mathrm{MHz}}\) [These are consistent with the values determined by Bayesian estimation discussed in Methods]. This demonstrates the control of the precession rate of Q ST by JQQ/h depending on the state of Q LD .

Demonstration of a CPHASE gate

To characterize the controlled-phase accumulated during the pulse stage F, we separate the phase of Q ST into controlled and single-qubit contributions as \(\phi _{\sigma _z^{{\mathrm{LD}}}} = - \pi \sigma _z^{{\mathrm{LD}}}J^{{\mathrm{QQ}}}\left( {t_{{\mathrm{int}}} + t_0} \right)/h\) and \(\phi ^{{\mathrm{ST}}} = 2\pi {\mathrm{\Delta }}E_{\mathrm{Z}}^{{\mathrm{ST}}}( {t_{{\mathrm{int}}} + t_{{\mathrm{ramp}}}} )/h + \phi _0\), respectively. Here t 0 (≪t ramp ) represents the effective time for switching on and off JQQ (Supplementary Note 5). A phase offset ϕ 0 denotes the correction accounting for nonuniform \({\mathrm{\Delta }}E_{\mathrm{Z}}^{{\mathrm{ST}}}\) during the ramp (Supplementary Note 5). Then the probability of finding the final state of Q ST in singlet is modeled as

$$P_{{\mathrm{S}},{\mathrm{model}}} = a{\mathrm{cos}}\left( {\phi _{\sigma _z^{{\mathrm{LD}}}} + \phi ^{{\mathrm{ST}}}} \right)\exp \left( { - (t_{{\mathrm{int}}}/T_2^ \ast )} \right)^2 + b$$ (2)

where a, b and \(T_2^ \ast\) represent the values of amplitude, mean and the dephasing time of the ST precession, respectively. We use maximum likelihood estimation (MLE) combined with Bayesian estimation29,30 to fit all variables in Eq. 2, that are \(a,b,t_0,J^{{\mathrm{QQ}}},T_2^ \ast ,\phi _0\), and \({\mathrm{\Delta }}E_{\mathrm{Z}}^{{\mathrm{ST}}}\), from the data (Methods). This allows us to extract the t int dependence of \(\phi _{\sigma _z^{{\mathrm{LD}}}}\) (Fig. 3a) (Methods) and consequently ϕ C = ϕ |↓〉 − ϕ |↑〉 (Fig. 3b). It evolves with t int in the frequency of JQQ/h = 90 MHz, indicating that the CPHASE gate time can be as short as h/2JQQ = 5.5 ns (up to single-qubit phase). On the other hand, \(T_2^ \ast\) obtained in the MLE is 211 ns, much longer than what is observed in Fig. 2e, f because the shorter data acquisition time used here cuts off the low-frequency component of the noise spectrum29. We note that this \(T_2^ \ast\) is that for the two-qubit gate while JQQ is turned on8, and therefore it is likely to be dominated by charge noise rather than the nuclear field fluctuation (Supplementary Note 6). The ratio \(2J^{{\mathrm{QQ}}}T_2^ \ast /h\) suggests that 38 CPHASE operations would be possible within the two-qubit dephasing time. We anticipate that this ratio can be further enhanced by adopting approaches used to reduce the sensitivity to charge noise in exchange gates such as symmetric operation31,32 and operation in an enhanced field gradient33.

Fig. 3 Controlled-phase evolution. a Interaction time t int dependence of \(\phi _{\sigma _z^{{\mathrm{LD}}}}\) controlled by Q LD . The blue and red data are for Q LD = |↑〉 and |↓〉, respectively. The solid curves are sin(πJQQ(t int + t 0 )/h) (red) and sin(−πJQQ(t int + t 0 )/h) (blue) where the values of JQQ and t 0 are obtained in the MLE. The curves are consistent with the data as expected. b Controlled-phase ϕ C = ϕ |↓〉 − ϕ |↑〉 extracted from Fig. 3a. Including the initial phase accumulated during gate voltage ramps at stage F, ϕ C reaches π first at t int = 4.0 ns and increases by π in every 5.5 ns afterwards Full size image

Finally we show that the CPHASE gate operates correctly for arbitrary Q LD input states. We implement the circuit shown in Fig. 4a in which t int is fixed to yield ϕ C = π, while a coherent initial Q LD state with an arbitrary \(\sigma _z^{{\mathrm{LD}}}\) is prepared by EDSR. We extract the averaged \(\phi _{\sigma _z^{{\mathrm{LD}}}}\), \(\left\langle \phi _{\sigma _z^{{\mathrm{LD}}}} \right\rangle\) by Bayesian estimation29,30, which shows an oscillation as a function of t MW in agreement with the Rabi oscillation measured independently by reading out Q LD at stage C as shown in Fig. 4b (see Methods for the estimation procedure and the origin of the low visibility, i.e., \({\mathrm{max}}| {\langle {\phi _{\sigma _z^{{\mathrm{LD}}}}} \rangle } | < \pi /2\)). These results clearly demonstrate the CPHASE gate functioning for an arbitrary Q LD input state.