Abstract
We report the simultaneous operation and twoqubitcoupling measurement of a pair of twoelectron spin qubits, actively decoupled from quasistatic nuclear noise in a GaAs quadruple quantum dot array. Coherent Rabi oscillations of both qubits (decay time ≈2 μs; frequency few MHz) are achieved by continuously tuning their drive frequency using rapidly converging realtime Hamiltonian estimators. We observe strong twoqubit capacitive interaction (>190 MHz), combined with detuning pulses, inducing a stateconditional frequency shift. The twoqubit capacitive interaction is beyond the bilinear regime, consistent with recent theoretical predictions. We observe a high ratio (>16) between coherence and conditional phaseflip time, which supports the possibility of generating highfidelity and fast quantum entanglement between encoded spin qubits using a simple capacitive interaction.
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Introduction
Spins in semiconductor quantum dot (QD) nanostructures offer a promising platform for realizing scalable quantum informationprocessing units with highfidelity universal quantum control^{1,2,3}. Recent progress in III–V and IV semiconducting materials demonstrated several achievements including the demonstration of single and twoqubit gate fidelities exceeding 99% in ^{Nat}Si and ^{28}Si (ref. ^{2,3}), simultaneous qubit operations in GaAs with a coherence time over 2 µs (refs. ^{4,5}), a few qubit entanglements in Ge and ^{28}Si (refs. ^{6,7,8}), hightemperature operations of spin qubits^{9,10,11}, and longrange coupling of spin qubits using superconducting cavity structures^{12,13}. The field is currently moving towards the highfidelity control of multiple qubits and the generation of controlled entanglement.
However, lowfrequency noise, including quasistatic nuclear fluctuation and slow charge noise, is one of the main factors reducing coherence times below the intrinsic limit of a given host material^{14,15,16,17,18,19,20}. For example, the spin coherence time is often affected by charge noise coupled through the inhomogeneous magnetic field generated by micromagnets^{18,19}. In addition, exchange or capacitivecouplingbased twoqubit control is inherently susceptible to charge noise^{14,16,20,21,22}. Thus, eliminating or mitigating slow magnetic and electric noises is an important task in semiconductor QD platforms.
Realtime Hamiltonian parameter estimation and measurementbased feedback^{4,23,24} are two complementary techniques to coherent quantum feedback^{25} capable of error mitigation, which is compatible —albeit sequentially—with general qubit controls. Previously, the Hamiltonian parameter estimation applied to GaAs has shown that the effect of quasistatic nuclear spin fluctuations can be strongly suppressed for singlespin^{23} and singlet–triplet (ST_{0}) qubits^{4}. Nevertheless, extending the technique to a multiqubit system is desirable. While this has not been demonstrated to date, the simultaneous Hamiltonian estimation is also crucial for the accurate measurement of interqubit coupling strength. This is particularly important in the case of GaAs, for which the application of realtime calibrated singlequbit rotations on each qubit is a prerequisite.
In this study, we demonstrate the simultaneous drive of a pair of ST_{0} qubits in GaAs. The quasistatic nuclear noise for each qubit is actively decoupled using a Bayesian inferencebased realtime Hamiltonian estimation circuit. We show highquality Rabi oscillations for both qubits with an oscillation quality factor above 10. We further exploit this result to demonstrate the measurement of the electrostatic coupling of two ST_{0} qubits, which grows beyond an empirically assumed bilinear form^{26,27} for large intraqubit exchange energies. Combining this with the spinecho sequence, we assess the potential to generate a highfidelity and fast conditional phase gate using capacitive interactions in a linear QD array.
Results
Simultaneous Hamiltonian parameter estimation
Figure 1a shows a quadruple QD device on top of a GaAs/AlGaAs heterostructure, hosting a pair of ST_{0} qubits with singlet \(\left S \right\rangle\) and tripletzero \(\left {T_0} \right\rangle\) basis states (See methods section and ref. ^{28} for details of the material structure and device fabrication). Highfrequency and synchronous voltage pulses, combined with DC voltages through biastees, were input to gates V_{1}–V_{6}. Fast dual RF reflectometry was performed by injecting a carrier signal having a frequency \(\approx\)125 MHz (153 MHz) and power of –100 dBm at the Ohmic contacts of the left (right) RF singleelectron transistors (see Fig. 1a). The reflected power was monitored through a frequencymultiplexed homodyne detection. The device was operated in a dilution refrigerator with a base temperature\(\approx \,7{{{\mathrm{ }}}}mK\), where an external magnetic field (0.7 T) was applied in the direction shown in Fig. 1a. The measured electron temperature is \(\approx\)72 mK (ref. ^{28}).
The Hamiltonian of the twoqubit system is given (up to a constant term) by^{26}
where \(J_{{{\mathrm{L}}}}\left( {\varepsilon _{{{\mathrm{L}}}}} \right)\left( {J_{{{\mathrm{R}}}}\left( {\varepsilon _{{{\mathrm{R}}}}} \right)} \right)\) is the exchange splitting between states \(\left S \right\rangle\) and \(\left {T_0} \right\rangle\), controlled by potential detuning \(\varepsilon _{{{\mathrm{L}}}}\)\(\left( {\varepsilon _{{{\mathrm{R}}}}} \right)\) of the left (right) qubit (Q_{L} and Q_{R}). \(\sigma _{i = {{{\mathrm{xL}}}},z{{{\mathrm{L}}}}}\)\(\left( {\sigma _{i = {{{\mathrm{xR}}}},z{{{\mathrm{R}}}}}} \right)\) is the Pauli matrix for Q_{L} (Q_{R}), I is the identity matrix, and ΔB_{zL} (ΔB_{zR}) is the magnetic field difference between the constituent QDs of each qubit, set by the local hyperfine interaction with the host Ga and As nuclei. Here, we adopted \(g^\ast \mu _B/h\) = 1 for units, where g* ≈ −0.44 is the effective gyromagnetic ratio in GaAs, \(\mu _B\) is the Bohr magneton, and h is Planck’s constant.
For the realtime Hamiltonian estimation of the quasistatic fluctuations of ΔB_{zL} and ΔB_{zR}, owing to statistical fluctuations of the nuclei, we extended the methodology developed in refs. ^{4,23,28} for twoqubit systems. Briefly, we initialized and measured the (2,0)_{L} ((0,4)_{R}) charge configuration for Q_{L} (Q_{R}), respectively, via fast energy selective tunneling (EST)based singleshot readout and adaptive initialization^{29,30,31} (see also Supplementary Note 1). In contrast to previous studies, in which sequential measurements were performed using the Pauli spin blockade (PSB)based readout^{26,32}, here we simultaneously measured both qubits to minimize the estimation latency. During the estimation sequence, when qubits are in the evolution step, \(J_{{{\mathrm{L}}}}\left( {\varepsilon _{{{\mathrm{L}}}}} \right)\), \(J_{{{\mathrm{R}}}}\left( {\varepsilon _{{{\mathrm{R}}}}} \right)\), and \(J_{{{{\mathrm{RL}}}}}\left( {\varepsilon _{{{\mathrm{L}}}},\varepsilon _{{{\mathrm{R}}}}} \right)\) are abruptly turned off. Each qubit evolves around the x axis of the respective Bloch sphere during the evolution time t_{k} = 1.67 k ns at the kth estimation trial (Fig. 1b). For each simultaneous singleshot measurement, the Bayesian inference is simultaneously performed according to the following rule (up to a normalization constant)^{4}
where N is the number of singleshot measurements per Hamiltonian estimation, \(P_0\left( {{\Delta}B_{{{{\mathrm{z}}}}i}} \right)\) is the uniform initial distribution, \(r_{ki}\) = 1(−1) for \(m_{ki} = \left S \right\rangle \left( {\left {T_0} \right\rangle } \right)\), and \(\alpha = 0.1\left( {\beta = 0.8} \right)\) is the parameter determined by the axis of rotation on the Bloch sphere (oscillation visibility). After the Nth round, the value of ΔB_{zi} is estimated, where the posterior distribution \(P\left( {{\Delta}B_{{{{\mathrm{z}}}}i}\left {m_{{{{\mathrm{N}}}}i},} \right.m_{({{{\mathrm{N}}}}  1)i}, \ldots m_{1i}} \right)\)reaches its maximum. The Bayesian circuit was implemented using a commercial fieldprogrammable gate array (FPGA)^{4} (see Supplementary Note 2).
Figure 1c shows a typical time trace of the estimated ΔB_{zL} and ΔB_{zR}. The time resolution of these estimations is ~1.8 ms, which consists of (N = 70) × 26 μs, where a single Bayesian update takes 16 μs and 10 μs for a singleshot measurement and calculation according to Eq. (2), respectively. We found that both ΔB_{z}s exhibit nonzero average values, likely arising from unintentional nuclear polarization, as reported previously for similarly prepared GaAs devices^{17,33}. When ensembleaveraged, this fluctuation leads to a nuclear fluctuationlimited coherence time T_{2}* of the order of 20 ns. Moreover, the difference between the mean values of ΔB_{zL} and ΔB_{zR} is at least twice the standard deviation. While the microscopic origin of this phenomenon is not well known yet, we used this difference to set the range of qubit frequencies to 25 < ΔB_{zL} < 50 (100 < ΔB_{zR} < 160) MHz for simultaneous qubit drive and active frequency feedback.
As shown in the inset of Fig. 2a, we concatenate the Bayesian estimators, acting as probe and operation steps, to control the two qubits in the frequency feedback mode. The controller triggers the operation step when the estimated ΔB_{z} in the probe step is in the range described above; otherwise, the cases are discarded. However, the preset range of allowed ΔB_{z} covers almost the entire distribution (Fig. 1c), so heralding does not significantly increase the total experimental time. When triggered, each qubit is first adiabatically initialized near the x axis on the Bloch sphere. The controller also adjusts the RF drive frequency using the probed ΔB_{z} for each qubit. Then, an RF pulse, with a varied pulse length, is applied to gate V_{2} (V_{5}) to resonantly modulate J_{L} (J_{R}) and induce Rabi oscillation. In all experiments, ~50 shots of operations were performed after one probe step.
The main panel of Fig. 2a shows the coherent Rabi oscillation for each qubit measured as a function of the RF pulse duration t_{RF} and the controlled detuning δf with respect to the actively adjusted resonant frequency. The chevron pattern of Q_{R} shows a resonance frequency shift, most likely caused by the AC stark effect^{34}. Figure 2b compares the Rabi oscillations for individual and simultaneous qubit operations under resonant conditions. For the former (latter), both the probe and RF pulses were applied to only one qubit (simultaneously on both qubits). For individual operations, Q_{L} (Q_{R}) shows a Rabi decay time T_{Rabi} = 1.75 μs (1.88 μs) at the Rabi frequency f_{Rabi} = 3.09 MHz (5.69 MHz) and oscillation visibility of 90.8% (93.6%), yielding the oscillation quality factor \(Q = f_{{{{\mathrm{Rabi}}}}}T_{{{{\mathrm{Rabi}}}}} \approx 5\;\left( {11} \right)\). For simultaneous operations, corresponding results are 1.68 μs (1.59 μs), 3.12 MHz (5.68 MHz), and 88.4% (88.9%).
The comparison reveals that the Rabi frequency remains virtually unchanged regardless of the operation scheme, indicating a negligible RF crosstalk between the two qubits. This is expected from the Rabi chevron pattern for each qubit (Fig. 2a) because the separation between the set target frequency range (50 MHz) is larger than the width of each chevron pattern (~10 MHz). In contrast, simultaneous qubit operation generally reduces T_{Rabi} and oscillation visibility. The time required for the Bayesian calculation depends on the type of operation mode. It takes 10 μs for the singlequbit probefeedback mode and twoqubit probeonly mode. However, it takes 50 μs for the twoqubit probefeedback mode, leading to ~70 shots × 65 μs = 4.6 ms of latency per Hamiltonian estimation. This is because of the limited resources available on the FPGA setup for parallel computations. Thus, we ascribe the reduced T_{Rabi} to the increased latency for simultaneous Bayesian estimation and frequency feedback, during which the uncertainty of the estimation increases owing to the diffusion of ΔB_{z}. In addition, reduced oscillation visibility is induced by the finite readout crosstalk between two qubits (see Supplementary Note 3).
Furthermore, we performed a simultaneous Ramsey interference of two qubits by applying calibrated π/2 pulses separated by t_{W} and by varying δf, as shown in Fig. 2c. We then extract \(T_2^\ast \approx 151\left( {183} \right){{{\mathrm{ns}}}}\) for Q_{L} (Q_{R}) (inset of Fig. 3c), by fitting the Ramsey amplitude decay to a Gaussian function at each resonant frequency^{20}. There is room for further improvements in T_{2}* by increasing computational resources of the FPGA setup and removing the readout crosstalk, using newly developed machine learning techniques^{35}, for example. Nevertheless, we keep these tasks for future work.
Capacitive coupling between two ST_{0} qubits
We now discuss the twoqubit capacitivecoupling measurements using the dual Hamiltonian estimation circuit discussed in the previous section. Specifically, throughout the experiment, a resonant RF pulse is applied to the control qubit to observe the statedependent frequency shift of the target qubit, whose frequency is estimated by the Hamiltonian estimator. In addition, using the simultaneous Hamiltonian estimation circuit, the control qubit is operated when the separation between the qubit frequency is larger than 50 MHz to prevent the unwanted flip of the target qubit by the RF crosstalk.
The capacitive coupling between singlet–triplet qubits originates from the different electric dipole moments of states \(\left S \right\rangle\) and \(\left {T_0} \right\rangle\) (ref. ^{36}). It has been considered to be a simple method to generate leakagefree twoqubit gates^{22,26,27,37} unlike the interqubit exchange couplingbased method, in which the interqubit magnetic field difference should be sizable to prevent leakage of the qubits outside the computational space^{22}. Nevertheless, the weak coupling dependent on the intraqubit exchange energies constitutes the main disadvantage of the capacitivecoupling method. For example, the pioneering demonstration of entanglement in GaAs^{26} used a coupling strength on the order of a few MHz, whereas individual exchange energies were approximately 300 MHz. Moreover, it has been assumed that capacitive coupling follows a bilinear form J_{RL} ∝ J_{L}J_{R}. In this bilinear form, the entanglement fidelity is expected to remain constant since the fidelity is limited by the dephasing of an individual qubit in \(J_{{{{\mathrm{RL}}}}} < < J_{{{{\mathrm{L}}}}({{{\mathrm{R}}}})}\), giving a constant quality factor for \(T_{2{{{\mathrm{L}}}}\left( {{{\mathrm{R}}}} \right)}^\ast \propto \left( {J_{{{{\mathrm{L}}}}\left( {{{\mathrm{R}}}} \right)}} \right)^{  1}\) (ref. ^{26}). This constant entanglement fidelity is experimentally confirmed in previous research, indicating that the bilinear form seems to hold, at least for the experiment in which the interqubit distance is larger than the distance between dots within a qubit^{26,36}.
The validity of the previously assumed scaling of J_{RL} was experimentally tested in a regularly and compactly spaced linear QD array. Motivated by theoretical works showing that J_{RL} can actually be a stronger function of J_{L} and J_{R} (ref. ^{37}), we measured J_{RL} by performing statedependent exchange oscillation in combination with the dual Hamiltonian estimator. Figure 3a shows the pulse sequence for the target and control qubits. After the probe step, the control qubit is initialized to the x axis of a Bloch sphere, followed by an optional π pulse. J_{RL} is then adiabatically switched on by slowly adjusting the detuning of the control qubit while the target qubit is initialized to the x axis of a Bloch sphere. The exchange oscillation of the target qubit is then performed by diabatically changing the detuning of the target qubit for a time t_{exch} to induce exchange oscillations, whose frequency depends on the control qubit state as \(\scriptstyle f^{{{\mathrm{T}}}} = \sqrt {(J^{{{\mathrm{T}}}}  J_{{{{\mathrm{RL}}}}}r_{{{\mathrm{C}}}})^2 + {\Delta}B_{{{\mathrm{z}}}}^2}\) according to Eq. (1), where \(J^{{{\mathrm{T}}}}\) is the intraqubit exchange energy of the target qubit and r_{C} = 0 (1) when the state of the control qubit is \(\left S \right\rangle \left( {\left {T_0} \right\rangle } \right)\).
Figure 3b (Fig. 3c) shows the resultant stateconditional frequency shift of Q_{L} (Q_{R}) as a function of t_{exch}, with J_{R(L)} ~3.61 (4.13) GHz. The precession frequency of the qubit is lower when the control qubit is in the \(\left S \right\rangle\) state for both cases, which is consistent with the charge configuration of the QD array^{38}. The observed frequency shifts of 34.9 (40.6) MHz for Q_{L} (Q_{R}) is a direct measure of J_{RL}, which is significantly larger than the value reported in (ref. ^{38}). As predicted in recent theoretical works, we hypothesize that the different relative orientations and the shorter distance between the qubits are related to this enhancement^{39}. In addition, we observe the beating of the target qubit oscillation when the control qubit is prepared as a superposition of \(\left S \right\rangle\) and \(\left {T_0} \right\rangle\) (see Supplementary Note 4).
We also measured T_{2}* and spinecho coherence time, T_{echo}, for each qubit to quantify the quality factor of the conditional phaseflip operation. Figure 4a, b shows T_{2}* and T_{echo} for each qubit, where T_{2}* is extracted from the exponentially decaying exchange oscillation and T_{echo} is measured by fitting the data to the echo envelope using calibrated π/2 and π pulses. Along with the form \(J \propto \exp \left( {  \varepsilon } \right)\) (inset of Fig. 4a, b), we observe charge noiselimited coherence time, where T_{2}* and T_{echo} are close to the form (dJ/dε)^{−1} (ref. ^{20}). This essentially explains why previously demonstrated entanglement quality showed no improvement when increasing J_{L} and J_{R} if J_{RL} \(\propto\) J_{L}J_{R} (ref. ^{26}).
Next, we perform the experiment in Fig. 3 with varying J_{L} and J_{R} near \(J_R = J_L\) to investigate the superlinearity of J_{RL} when both qubits show reasonable coherence. Figure 4c shows the nonlinear behavior of J_{RL} as J_{L} and J_{R} increase, manifesting the deviation from the bilinear proportionality^{26,27} in our device. Here, the error bar of the estimated J_{RL} is determined by the fitting uncertainty limited by the sampling rate of the arbitrary waveform generator (see Supplementary Note 5). By fitting the measured J_{RL} to (J_{L}J_{R})^{a}, the agreement with experimental data is found for \(a = 2.14\), which is close to the theoretically expected form J_{RL}~(J_{L}J_{R})^{2} using the effective Hamiltonian obtained from a Hund–Mulliken model independent of the details of the confinement potential in the regime where intraqubit tunnel coupling overwhelms the intraqubit exchange energy^{37} (see Supplementary Note 6). Moreover, we estimated the dipolar energy D ≈ 46 GHz, the order of which is consistent with the recent experimental work using a similar interdot spacing^{40}. With this superlinear proportionality, we observe J_{RL} > 190 MHz when J_{L} and J_{R} \(\approx\) 900 MHz, showing that more than 20% of the stateconditional qubit frequency shift can be obtained in a closely spaced QD array.
Discussion
The nonlinear J_{RL}(J_{L}, J_{R}) form implies that the twoqubit gate quality should increase at larger J_{L} and J_{R}. We calculated \(Q_{T_2 \ast \left( {{{{\mathrm{echo}}}}} \right)} \equiv 2JT_{{{\mathrm{2}}}} \ast \left( {T_{{{{\mathrm{echo}}}}}} \right)\), which quantifies the number of conditional phase flips within T_{2}*(T_{echo}), as shown in Fig. 4d. We observed Q_{echo} as high as ~16 (~7) for Q_{L} (Q_{R}), predicting that the fidelity of a conditional phaseflip operation of Q_{L} (Q_{R}) with Q_{R} (Q_{L}) in the σ_{z} eigenstate reaches as high as \(e^{  1/Q_{{{{\mathrm{echo}}}}}} = 94.0{{{\mathrm{ }}}}(86.7){{{\mathrm{ \% }}}}\) and also monotonically increases as a function of J_{L} and J_{R}. In addition, the simulation based on the measured values predicts that the maximum attainable Bell state fidelity F_{Bell} reaches \(\approx\)95% and increases at larger J_{L} and J_{R}, where the Bell state is prepared by the echolike pulse implemented in ref. ^{26} in which F_{Bell} maximizes at \(\approx\)72% with \(J_{{{{\mathrm{RL}}}}} \approx 1\;{{{\mathrm{MHz}}}}\)(see Supplementary Note 7).
Previously, a Bell state fidelity with a capacitive coupling has enhanced to \(\approx\)93% with simultaneous rotary echo and rapid dynamic nuclear polarization (DNP), which enabled an approximately tenfold increase of a coherence time with ΔB_{z} = 900 MHz, but J_{RL} on the order of a few MHz was rather exploited^{27}. Thus we expect that F_{Bell} could be enhanced more by applying simultaneous rotary echo to the closely spaced QD array at a large J_{RL}, although the application is not currently viable in our device due to the insufficient DNP rate. In addition, the minimum time step and pulse rise times are currently limited by the sample rate of the waveform generator (2.4 Gsa/s), which also prevents performing full twoqubit gate operations and entanglement demonstration. Therefore further optimization with a faster signal source is still required. Note also that the pulse sequence in Fig. 3a is proper only for a twoqubit interaction measurement since the control qubit is likely to decohere while adiabatically turning on the interaction. Thus, a different qubit driving strategy (for example, using a nonadiabatic pulse) should be devised for entanglement demonstration, which will be considered in future work. Nonetheless, as our Hamiltonian estimation technique and readout method are compatible with large ΔB_{z}, we anticipate that performing a full twoqubit experiment in a regularly and closely spaced linear QD array, with increased ΔB_{z} by micromagnets or dynamic nuclear polarization, may show an even higher twoqubit gate fidelity that is also fast, exploiting large J_{RL}.
In conclusion, we demonstrated the simultaneous Hamiltonian parameter estimation and active suppression of the quasistatic noise of two ST_{0} qubits in a GaAs quadruple QD array. Using fast qubit calibration routines, we also showed that both the magnitude and scaling of the capacitive coupling in a closely spaced QD array can be stronger than the previously measured bilinear form, leading to a stateconditional frequency shift of over 20 % and a quality factor of conditional phase flip of over 16. Our measurement confirms recent theoretical calculations and supports the possibility of realizing a highfidelity and fast entanglement of encoded spin qubits in both GaAs and Si using a simple capacitive interaction.
Methods
Device fabrication
The quadruple QD device shown in Fig. 1a was fabricated on a GaAs/AlGaAs heterostructure where twodimensional electronic gas (2DEG) is located 70 nm below the surface. Mesa was defined by a wet etching technique to eliminate 2DEG outside the region of interest to suppress unwanted leakage. Five ohmic contacts were formed by metal diffusion with thermal annealing. Nanogates were fabricated by ebeam lithography and metal evaporation.
Measurement
The device was placed on the 7 mK plate in a commercial dilution refrigerator (Oxford Instruments, Triton500). The batteryoperated voltage sources (Stanford Research Systems, SIM928) supplied by stable DC voltages rapid voltage pulses generated by the arbitrary waveform generator with the maximum sampling rate of 2.4 Gsa/s (Zurich Instruments, HDAWG) were applied to metallic gates through onboard biastees. A detailed description of the experimental setup and FPGA implementation can be found in Supplementary Note 2.
Data availability
The data that support the findings of this study are available from the corresponding author upon request.
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Acknowledgements
This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean Government (MSIT) (No. 2018R1A2A3075438, No. 2019M3E4A1080144, No. 2019M3E4A1080145, and No. 2019R1A5A1027055), Korea Basic Science Institute (National Research Facilities and Equipment Center) grant funded by the Ministry of Education (No. 2021R1A6C101B418), and CreativePioneering Researchers Program through Seoul National University (SNU). The cryogenic measurement used equipment supported by the Samsung Science and Technology Foundation under Project Number SSTFBA1502–03.
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D.K. and J.Y. conceived the project. J.Y. performed the measurements and analyzed the data. J.K. and H.J. fabricated the device. J.P., H.J., W.J., Y.S., M.C., and H.S. built the experimental setup and configured the measurement software. V.U. synthesized and provided the GaAs heterostructure. All the authors contributed to the preparation of the manuscript.
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Yun, J., Park, J., Jang, H. et al. Probing twoqubit capacitive interactions beyond bilinear regime using dual Hamiltonian parameter estimations. npj Quantum Inf 9, 30 (2023). https://doi.org/10.1038/s41534023006994
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DOI: https://doi.org/10.1038/s41534023006994
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