Abstract
Now that it is possible to achieve measurement and control fidelities for individual quantum bits (qubits) above the threshold for fault tolerance, attention is moving towards the difficult task of scaling up the number of physical qubits to the large numbers that are needed for faulttolerant quantum computing^{1,2}. In this context, quantumdotbased spin qubits could have substantial advantages over other types of qubit owing to their potential for allelectrical operation and ability to be integrated at high density onto an industrial platform^{3,4,5}. Initialization, readout and single and twoqubit gates have been demonstrated in various quantumdotbased qubit representations^{6,7,8,9}. However, as seen with smallscale demonstrations of quantum computers using other types of qubit^{10,11,12,13}, combining these elements leads to challenges related to qubit crosstalk, state leakage, calibration and control hardware. Here we overcome these challenges by using carefully designed control techniques to demonstrate a programmable twoqubit quantum processor in a silicon device that can perform the Deutsch–Josza algorithm and the Grover search algorithm—canonical examples of quantum algorithms that outperform their classical analogues. We characterize the entanglement in our processor by using quantumstate tomography of Bell states, measuring state fidelities of 85–89 per cent and concurrences of 73–82 per cent. These results pave the way for largerscale quantum computers that use spins confined to quantum dots.
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Acknowledgements
This research was sponsored by the Army Research Office (ARO) under grant numbers W911NF1710274 and W911NF1210607. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the ARO or the US Government. The US Government is authorized to reproduce and distribute reprints for government purposes notwithstanding any copyright notation herein. Development and maintenance of the growth facilities used for fabricating samples is supported by DOE (DEFG0203ER46028). We acknowledge the use of facilities supported by NSF through the University of WisconsinMadison MRSEC (DMR1121288). E.K. was supported by a fellowship from the Nakajima Foundation. We acknowledge financial support from the Marie SkłodowskaCurie actions—Nanoscale solidstate spin systems in emerging quantum technologies—SpinNANO, grant agreement number 676108. We acknowledge discussions with S. Dobrovitski, C. Dickel, A. Rol, J. P. Dehollain, Z. Ramlakhan and members of the Vandersypen group, and technical assistance from R. Schouten, R. Vermeulen, M. Tiggelman, M. Ammerlaan, J. Haanstra, R. Roeleveld and O. Benningshof.
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T.F.W. performed the experiment with help from E.K. and P.S., T.F.W. and S.G.J.P. analysed the data, S.G.J.P. performed the simulations of the algorithms, T.F.W., S.G.J.P., E.K., P.S., M.V., M.F., S.N.C., M.A.E. and L.M.K.V. contributed to the interpretation of the data and commented on the manuscript, D.R.W. fabricated the device, D.E.S. and M.G.L. grew the Si/SiGe heterostructure, T.F.W. wrote the manuscript (S.G.J.P. wrote parts of Methods) and L.M.K.V. conceived and supervised the project.
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The authors declare no competing financial interests.
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Reviewer Information Nature thanks H. Bluhm and the other anonymous reviewer(s) for their contribution to the peer review of this work.
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Extended data figures and tables
Extended Data Figure 1 Schematic of the measurement setup.
The sample was bonded to a printed circuit board (PCB) mounted onto the mixing chamber of a dilution refrigerator. All measurements were performed at the base temperature of the fridge, T_{base} ≈ 20 mK. DC voltages were applied to all of the gate electrodes using roomtemperature digitaltoanalogue converters via filtered lines (not shown). Voltage pulses were applied to plunger gates P1 and P2 using a Tektronix 5014C arbitrary waveform generator (AWG) with 1GHz clock rate. The signals from the AWGs passed through a roomtemperature lowpass filter and attenuators at different stages of the fridge and were added to the DC signals via bias tees mounted on the PCB. Two Keysight E8267D vector microwave sources, MW1 and MW2, were used to apply microwaves (18–20 GHz) to perform EDSR on Q1 and Q2, respectively. The signals passed through roomtemperature DC blocks and custombuilt 15GHz highpass filters and attenuators at different stages of the fridge and were added to the DC signals via bias tees mounted on the PCB. The output of the microwave source (phase, frequency, amplitude and duration) was controlled with I/Q vector modulation. The I/Q signals were generated with another Tektronix 5041C, which was the master device for the entire setup and provided trigger signals for the other devices. In addition to the vector modulation, we used pulse modulation to give an on/off microwave power output ratio of 120 dB. Although I/Q modulation can be used to output multiple frequencies, the bandwidth of the AWG was not sufficient to control both qubits with one microwave source owing to their large separation in frequency (1.3 GHz). The sensor current I was converted to a voltage signal with a custombuilt preamplifier, and an isolation amplifier was used to separate the signal ground, with the measurement equipment grounded to reduce interference. Following this, a 20kHz Bessel lowpass filter was applied to the signal using a SIM965 analogue filter. A fieldprogrammable gate array (FPGA) analysed the voltage signal during the readout and assigned the trace to be spinup if the voltage fell below a certain threshold. The voltage signal could also be measured with a digitizer card in the computer. The shapes of the pulses generated by the AWGs and microwave sources during qubit manipulation, along with the typical timescales, are shown in the lower right. Square pulses were used to perform the CZ gate and as the input for the I/Q modulation to generate microwave pulses. The pulse modulation was turned on 40 ns before turning on the I/Q signal, owing to the time needed for the modulation to switch on.
Extended Data Figure 2 Measurement protocol for two electron spins.
a, Stability diagram of the double quantum dot, showing the positions in gate space used to perform singlequbit gates (red circle) and twoqubit gates (yellow circle). The map shows the numerically obtained differential current dI/dV_{P1} through the quantumdot charge sensor as a function of the voltages on P1 and P2. The white dashed line is the (1, 1)–(0,2) interdot transition line. The white arrow indicates the detuning axis ε used in the experiments. Although the detuning pulse for the twoqubit gate crosses the chargeaddition lines of D1 and D2, the quantum dots remain in the (1, 1) charge state because the pulse time is much shorter than the electron tunnel times to the reservoirs. b, Plot of the voltage pulses applied to plunger gates P1 and P2 and the response of the quantumdot charge sensor over one measurement cycle. D2 is unloaded by pulsing into the (1, 0) charge region for 1.5 ms (purple circle). The electron on D1 is initialized to spindown by pulsing to a spinrelaxation hotspot at the (1, 0) and (0, 1) charge degeneracies (orange circle) for 50 μs (see Extended Data Fig. 5). D2 is loaded with a spindown electron by pulsing to the readout position for 4 ms (blue circle). During manipulation, the voltages on the plunger gates are pulsed to the red circle for singlequbit gates and to the yellow circle for twoqubit gates, where the exchange is about 6 MHz. After manipulation, the spin of the electron on D2 is measured by pulsing to the readout position (blue circle) for 0.7 ms, where the Fermi level of the reservoir is between the spinup and spindown electrochemical potentials of D2. If the electron is spinup then it can tunnel out, after which a spindown electron tunnels back in. These two tunnel events are detected by the quantumdot sensor as a single blip in the current signal. An additional 1.3 ms is spent at the readout position so that D2 is initialized to spindown with high fidelity. Following this, Q1 is measured by first performing a CROT gate at the yellow circle so that , where CROT12 indicates a CROT gate with Q1 as the control and Q2 as the target. A projective measurement of Q1 is then performed by measuring Q2 at the readout position for 0.7 ms (blue circle). Finally, we add a compensation pulse to VP1 and VP2 so that over the measurement cycle V_{DC} = 0 to mitigate charging effects in the bias tees. c, Closeup of the stability diagram in a showing the positions in gate space used for initialization and readout.
Extended Data Figure 3 Singlequbit properties and twoaxis control.
The purple (top) and orange (bottom) data points correspond to measurements performed on Q1 and Q2, respectively, in the (1, 1) regime (red circle in Extended Data Fig. 2). a, Spinup fraction as a function of the microwave frequency of an applied π pulse, showing a resonant frequency of 18.387 GHz (19.670 GHz) for Q1 (Q2). b, The spinrelaxation time is measured by preparing the qubit to spinup and varying the wait time before readout. From the exponential decay in the spinup probability we measure T_{1} > 50 ms (T_{1} = 3.7 ± 0.5 ms) for Q1 (Q2). c, Spinup probability as a function of microwave duration, showing Rabi oscillations of 2.0 MHz for Q1 and Q2. d, The dephasing time is measured by applying a Ramsey pulse sequence and varying the free evolution time τ. Oscillations were added artificially to improve the fit of the decay by making the phase of the last microwave pulse dependent on the free evolution time, ϕ = sin(ωτ), where ω = 4 MHz. By fitting the data with a Gaussian decay, , we extract () for Q1 (Q2). In the measurement for Q1, the first π/2 microwave pulse is a Y gate. The Ramsey measurement was performed over about 20 min with the frequency calibrated every approximately 1 min. e, The coherence time of Q1 (Q2) can be extended to T_{2Hahn} = 19 ± 3 μs (T_{2Hahn} = 7 ± 1 μs) by a Hahn echo sequence. The coherence time is extracted from an exponential fit to the spinup probability as a function of the free evolution time in the Hahn echo sequence. f, Full twoaxis control is demonstrated by applying two π/2 pulses and varying the phase of the second one. All error bars are 1σ from the mean.
Extended Data Figure 4 Randomized benchmarking of singlequbit gates.
Randomized benchmarking of the singlequbit gates for each qubit is performed by applying a randomized sequence of a varying number of Clifford gates m to either the 1〉 or 0〉 state and measuring the final spinup probability or P_{1〉}, respectively. All gates in the Clifford group are decomposed into gates from the set {I, ±X, ±X^{2}, ±Y, ±Y^{2}}. The purple (orange) data points show the difference in the spinup probabilities for Q1 (Q2) as a function of sequence length. For each sequence length m we average over 32 different randomized sequences. From an exponential fit (solid lines) of the data, , we estimate average Cliffordgate fidelities F_{C} = 1 − (1 − p)/2 of 98.8% and 98.0% for Q1 and Q2, respectively. The last three data points from both datasets were omitted from the fits because they deviate from a single exponential^{20}. All error bars are 1σ from the mean.
Extended Data Figure 5 Spinrelaxation hotspots used for highfidelity initialization.
a, Closeup stability diagram of the (1, 0)–(0, 1) charge transition. The white arrow defines the detuning axis between D1 and D2 controlled with P1. b, Schematic of the energylevel diagram as a function of detuning for one electron spin in a double quantum dot. c, Spinrelaxation hotspots are measured by preparing the electron on D1 to spinup using EDSR, applying a voltage pulse along the detuning axis (white arrow in a) for a wait time of 200 ns and performing readout of the electron spin. We observe three dips in the spinup probability, which correspond to spinrelaxation hot spots. The first and third hotspot are due to anticrossings between the (0, ↓) and (↑, 0) states and the (↓, 0) and (0, ↑) states^{24}. The second hotspot occurs at zero detuning. The voltage separation between the first and third hot spot corresponds to the sum of the Zeeman energy of D1 and D2 divided by the gate lever arm α along the detuning axis. Knowing precisely the Zeeman energies from EDSR spectroscopy, we can accurately extract the gate lever arm to be α = 0.09e. d, The spinrelaxation time at zero detuning (orange circle in a) is found to be T_{1} = 220 ns by measuring the exponential decay of the spinup probability as a function of wait time τ at zero detuning. All error bars are 1σ from the mean.
Extended Data Figure 6 Twoqubit CROT gate.
a, Microwave spectroscopy of Q2 close to zero detuning between the (1, 1) and (0, 2) states (yellow dot in Extended Data Fig. 2a) with the exchange coupling on. The blue and red curves show the resonance of Q2 after preparing Q1 into spindown and spinup, respectively. The resonance frequency of Q2 shifts by the exchange coupling, and by applying a π pulse at one of these frequencies we can perform a CROT gate, which is equivalent to a CNOT gate up to a rotation. As discussed in the main text, this CROT gate is used to perform the projective measurement of Q1. All error bars are 1σ from the mean.
Extended Data Figure 7 Measurement of J_{off} using a decoupling sequence.
The exchange coupling J_{off} during singlequbit gates is measured using a twoqubit Hahn echo sequence, which cancels out any unconditional rotations during the free evolution time τ. Fitting the spinup probability as a function of free evolution time τ using the functional form sin(2πJ_{off}τ), we extract J_{off} = 270 kHz. All error bars are 1σ from the mean.
Extended Data Figure 8 Microwave spectroscopy of Q1 and Q2.
a, b, Spectroscopy of Q1 (a) and Q2 (b) versus detuning energy ε after initializing the other qubit to . Towards ε = 0 there are two resonances each for Q1 and Q2, which are separated by the exchange energy J(ε)/h. As discussed, the Zeeman energy E_{Z}(ε) of Q1 and Q2 also depends on detuning because changes to the applied voltages shift the position of the electron in the magneticfield gradient. The four resonance frequencies are fitted (green, blue, red and yellow lines) with f_{jk} = E_{Zj}(ε) + (−1)^{k+1}J(ε), where j denotes the qubit and k denotes the state of the other qubit. The data are fitted well using , , and E_{Z2}(ε) ∝ ε. The fitted Zeeman energies of Q1 and Q2 are shown by the black lines. We observe that the Zeeman energy of Q1 has an exponential dependence towards the (0, 2) charge regime (ε = 0). This observation can be explained by the electron delocalizing from D1 towards D2, which has a much higher Zeeman energy. c, Schematic of the colourcoded transitions that correspond to the resonances in a and b.
Extended Data Figure 9 Decay of the decoupled CZ oscillations.
The normalized spinup probability of Q1 as a function of the total duration time 2τ of the two CZ gates in the DCZ sequence. The data are fitted using a sinusoid, P_{1〉} = 0.5sin(2πJτ) + 0.5, with either a Gaussian (black line; ) or exponential (red line; ) decay. From these fits we find a decay time of T_{2} = 1.6 μs. All error bars are 1σ from the mean.
Extended Data Figure 10 Simulation of the Deutsch–Josza and Grover search algorithms using the DCZ gate.
a, b, Twospin probabilities as a function of the sequence time during the Deutsch–Josza algorithm (a) and the Grover search algorithm (b) for each function, using the decoupled version of the twoqubit CZ gate (the DCZ gate). The solid lines show the outcome of the simulations, which include decoherence due to quasistatic charge noise and nuclearspin noise. All error bars are 1σ from the mean.
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Watson, T., Philips, S., Kawakami, E. et al. A programmable twoqubit quantum processor in silicon. Nature 555, 633–637 (2018). https://doi.org/10.1038/nature25766
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