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A four-qubit germanium quantum processor

Abstract

The prospect of building quantum circuits1,2 using advanced semiconductor manufacturing makes quantum dots an attractive platform for quantum information processing3,4. Extensive studies of various materials have led to demonstrations of two-qubit logic in gallium arsenide5, silicon6,7,8,9,10,11,12 and germanium13. However, interconnecting larger numbers of qubits in semiconductor devices has remained a challenge. Here we demonstrate a four-qubit quantum processor based on hole spins in germanium quantum dots. Furthermore, we define the quantum dots in a two-by-two array and obtain controllable coupling along both directions. Qubit logic is implemented all-electrically and the exchange interaction can be pulsed to freely program one-qubit, two-qubit, three-qubit and four-qubit operations, resulting in a compact and highly connected circuit. We execute a quantum logic circuit that generates a four-qubit Greenberger−Horne−Zeilinger state and we obtain coherent evolution by incorporating dynamical decoupling. These results are a step towards quantum error correction and quantum simulation using quantum dots.

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Fig. 1: Four germanium hole spin qubits.
Fig. 2: Controlled rotations between all nearest-neighbour qubit pairs.
Fig. 3: Resonant one-, two-, three- and four-qubit gates.
Fig. 4: CPHASE gate and dynamical decoupling.
Fig. 5: Coherent generation of a four-qubit GHZ state.

Data availability

All data underlying this study are available from the 4TU ResearchData repository at https://doi.org/10.4121/13663442.

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Acknowledgements

We thank L. M. K. Vandersypen for discussions and S. G. J. Philips for his contributions to software development. M.V. acknowledges support through a Vidi grant, two projectruimte grants, and an NWA grant, all associated with the Netherlands Organization of Scientific Research (NWO).

Author information

Affiliations

Authors

Contributions

N.W.H. performed the experiments and analysed the data. W.I.L.L. fabricated the device. N.W.H. contributed to the design and development. M.R. contributed to theoretical analysis. F.v.R. contributed to the preparation of the experiment. A.S. and G.S. supplied the heterostructures. S.L.d.S. and R.N.S. contributed to software and hardware development. N.W.H. and M.V. wrote the manuscript with input from all other authors. M.V. supervised the project.

Corresponding authors

Correspondence to Nico W. Hendrickx or Menno Veldhorst.

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Competing interests

The authors declare no competing interests.

Additional information

Peer review information Nature thanks Hendrik Bluhm, Michel Pioro-Ladrière, David Reilly and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

Extended data

is available for this paper at https://doi.org/10.1038/s41586-021-03332-6.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Extended data figures and tables

Extended Data Fig. 1 Schematic of the measurement set-up.

The sample is bonded to a PCB mounted on the cold finger of a dilution refrigerator. Direct-current (dc) voltages (orange) are applied to the gates using galvanically isolated digital-to-analogue converters (DACs) through lines filtered at the mixing chamber stage of the refrigerator. The DACs are custom built and part of a custom serial peripheral interface (SPI) measurement rack. Voltage pulses (red) are applied to the same gates using Keysight M3202A AWGs. The Keysight AWGs and digitizer are part of a Keysight M9019A peripheral component interconnect express extensions for instrumentation (PXIe) chassis. The coaxial cables (coax) pass through a ferrite common mode choke and are attenuated at different stages of the fridge and combined with the dc signal using on-PCB bias tees. The microwave excitation used for spin resonance is generated by three separate vector sources and is combined with the AWG signal on plungers P2, P3 and P4 through room-temperature diplexers with pass bands dc-400 MHz (LF) and 1.5–10 GHz (RF). The output of the vector sources is modulated by quadrature modulation using signals generated on the AWGs (dark blue). By applying sine waves with a phase difference of π/2 to the in-phase (I) and quadrature (Q) inputs of the vector source, a single-sideband signal with controllable amplitude, frequency, phase and duration is acquired. Additionally, we apply a pulse modulation (PM) envelope around the microwave pulses to improve the microwave suppression to a total of −120 dB. Using an in-house built reflectometry set-up, we generate two microwave tones resonant with the tank circuit on the PCB (light blue). The radio-frequency sources (rf src) are pulse modulated by a trigger (trig) generated on the AWG and 50-Ω split (split) across all sources, to mute them during qubit manipulation. The radio-frequency signals are combined (rf cmb), filtered and attenuated before reaching a directional coupler at the mixing chamber stage. Here, the signal propagates through two on-PCB bias-tees to the two NbTiN resonators, which form the tank circuits. The reflected signal is then split off and amplified by 35 dB at the 4 K stage of the fridge using a CITLF3 cryogenic amplifier. The radio-frequency signal is amplified (rf amp) again at room temperature (RT) after which it is demodulated separately at both of the carrier frequencies (ref) to retrieve the charge sensor signal. After filtering the signal, it is recorded and temporally averaged \(\bar{V}(t)\) using the field-programmable gate array (FPGA) in the digitizer (digi) to reduce the amount of data transferred to the computer.

Extended Data Fig. 2 Screening of qubit devices.

a, The qubit devices undergo a visual screening as well as a transport screening at a temperature of T = 4.2 K. Out of the full batch of 16 nominally identical devices, 15 passed visual inspection. Seven of these devices were tested at T = 4.2 K and two devices were found to pass all testing, of which one (the device presented in this work) was mounted in a dilution refrigerator. b, Transport current I at a temperature of 4.2 K for the device presented in this work. Three different channels are turned on by sweeping all gates down to Vgate = −1,500 mV (black), thereby accumulating charge in the undoped strained Ge quantum well. Next, the effect of the individual gates is tested by sweeping them up and down to Vgate = 0 V (coloured lines). Both the north and south single hole transistor (SHT) sensors are tested, as well as a transport channel through the full device. All channels turn on and the gates affect the transport current as expected from the device layout.

Extended Data Fig. 3 Readout characteristics.

a, b, We measure the difference in charge sensor signal between the blocked and non-blocked states as a function of the measurement time at the readout point. An exponential decay can be observed related to the tunnelling time Tin of Q2 (or Q4) to the reservoir for the Q1Q2 (a) and Q3Q4 (b) readout system, respectively. cf, We vary the ramp time tramp between the manipulation phase and the readout phase and measure the blocked state probability Pblocked of the four different two-qubit basis states by applying preparation π pulses to the relevant qubits, both for the Q1Q2 readout system (a, b) and the Q3Q4 readout system (c, d). By increasing the interdot coupling during the readout and elongating the ramp between the manipulation and readout point, we can switch between a parity readout (a, c) and a single-state readout (b, d). The dashed line corresponds to the optimized readout ramp time used for our measurements.

Extended Data Fig. 4 Randomized benchmarking of the Clifford group and spin relaxation times of the different qubits.

a, We quantify the quality of the single-qubit gates by performing randomized benchmarking of the single-qubit Clifford group43. The decay curve of the qubit state is measured as a function of the number of Clifford gates applied. Each data point consists of 1,000 single shots for 30 different randomly selected Clifford sequences of length NCliffords. The decay is fitted to Pup = aexp(−(NCliffordsm)) + y0, with a the initial spin-up probability, m the decay parameter, and y0 an offset. F = 1 – m/(2 × 1.875) is extracted based on the average single qubit gate length of 1/1.875 Clifford gates. Uncertainty margins (in parentheses) correspond to 1σ. b, The spin relaxation time T1 is measured at the manipulation point by applying a πX-pulse separated by a waiting time twait from the readout phase. By fitting the normalized spin-up fraction to P = exp(−twait/T1), we find spin relaxation times of \({T}_{1}^{{\rm{Q}}1}\)  = 0.84(6) ms, \({T}_{1}^{{\rm{Q}}2}\) = 7.6(5) ms, \({T}_{1}^{{\rm{Q}}3}\)  = 16.1(8) ms, and \({T}_{1}^{{\rm{Q}}4}\)  = 11.5(5) ms. Uncertainty margins (in parentheses) correspond to 1σ.

Extended Data Fig. 5 Ramsey, Hahn echo and CPMG measurements on the different qubits.

a, The phase coherence time \({T}_{2}^{* }\) is measured using a Ramsey sequence consisting of two X(π/2)-pulses separated by a waiting time τ as illustrated in the schematic at the top. By fitting the data to \(P=\,\cos (2{\rm{\pi }}\Delta f\tau +{\varphi }_{0})\exp (-{(\tau /{T}_{2}^{* })}^{\alpha })\), with Δf the frequency detuning, ϕ0 a phase offset and α the power of the decay, we find spin dephasing times of \({T}_{2,{\rm{Q}}1}^{* }\) = 201 ns, \({T}_{2,{\rm{Q}}2}^{* }\) = 146 ns, \({T}_{2,{\rm{Q}}3}^{* }\) = 445 ns, and \({T}_{2,{\rm{Q}}4}^{* }\) = 150 ns for Q1 to Q4, respectively. b, Using an additional X(π)-pulse, low-frequency fluctuations of the qubit resonance frequency can be echoed out, allowing us to probe the Hahn-echo decay time \({T}_{2}^{{\rm{Hahn}}}\). Fitting the data to \(P=\exp (-{(\tau /{T}_{2}^{{\rm{Hahn}}})}^{\alpha })\), we find Hahn echo times of \({T}_{2,{\rm{Q}}1}^{{\rm{Hahn}}}\) = 4.3 μs, \({T}_{2,{\rm{Q}}2}^{{\rm{Hahn}}}\) = 5.5 μs, \({T}_{2,{\rm{Q}}3}^{{\rm{Hahn}}}\) = 3.8 μs, and \({T}_{2,{\rm{Q}}4}^{{\rm{Hahn}}}\) = 2.9 μs. c, Using a CPMG sequence of repeated Y(π) pulses, we can increase the echo bandwidth and extend the phase coherence to over \({T}_{2,{\rm{Q}}1}^{{\rm{CPMG}}}\) >100 μs. The phase coherence can be observed to increase with the amount of refocusing pulses Nπ (left), with exemplary decay traces for Q1 plotted in the right panel.

Extended Data Fig. 6 Noise spectroscopy using Ramsey and CPMG measurements.

We measure the effective noise spectrum acting on the qubit, both tracing the resonance frequency using repeated Ramsey measurements20 (in blue), as well as by using the filter function of a dynamical decoupling measurement18,52 (in red). Dashed blue and red lines are fits to the Ramsey and CPMG data respectively. The black line is a fit to the combined dataset, where the weight of both sets is normalized to the amount of data points. The effective noise can be observed to increase towards low frequencies, consistent with the upwards trend of \({T}_{2}^{{\rm{CPMG}}}\) observed in Extended Data Fig. 3c. The effective charge noise measured in this heterostructure is Scn(f) = 6 μV \(\sqrt{{\rm{Hz}}}\)–1 at 1 Hz (ref. 27). Combining this with a typical resonance frequency slope of df/dV = 5 MHz mV–1 (ref. 39) results in an effective resonance-frequency noise power of S(f) = 9 × 108 Hz2 Hz–1, comparable to what is observed experimentally, suggesting that coherence is limited by charge noise in our system. The effect of charge noise could be mitigated by careful optimization of the electric field environment53 or moving to a multi-hole charge occupancy, screening the influence of charge impurities54, potentially enabling even higher-fidelity operations. Alternatively, noise could originate in the nuclear spin bath present in natural germanium, which could be overcome by isotopically enriching the material.

Extended Data Fig. 7 Driving of all resonance lines of the coupled three- and four-qubit system.

a, Both the coupling between Q2 and Q1 as well as Q2 and Q3 are enabled, using the respective virtual barrier gates. This splits the resonance line into four, as shown in Fig. 3. Driving each of the separate lines results in the conditional rotations of Q2 depending on the states of Q1 and Q3. We measure the spin-up probability Pup after driving each of the four resonance lines for time tp, for all four permutations of the Q1 and Q3 basis states as initial state, following the colour scheme of Fig. 3. The driving power is adjusted for each of the transitions to synchronize the π-rotation times, with \({a}_{{f}_{1}}\) = 330 mV, \({a}_{{f}_{2}}\) =500 mV, \({a}_{{f}_{3}}\) = 280 mV, and \({a}_{{f}_{4}}\) = 400 mV, for f1f4 from low to high. b, Similarly, by additionally opening up the coupling between Q3 and Q4 as well, the resonance line splits into eight and we can drive all separate lines individually. The eight lines are driven using the same microwave power in this figure and a strong difference in rotation frequencies can be observed for the different transitions f1f8 from low to high. This also results in a small off-resonant driving effect for some of the lines.

Extended Data Fig. 8 Tuning of the CZ-gates.

a, b, The CZ gates between all four qubit pairs are tuned using a Ramsey sequence (analogous to Fig. 4), where the spin-up probability is measured as a function of the phase θ of the final π/2 pulse as well as the depth of the exchange pulse VvBmn, with m and n the relevant qubits (a). We choose to tune the height of the voltage pulse rather than its length, owing to the limited temporal resolution of the exchange pulses (1 ns). The acquired phase θ0 is obtained by fitting each line to P = Acos(θ + θ0) + y0, with A the visibility and y0 an offset. A CZ gate is achieved when the difference in acquired phase is exactly Δθ = π, for the situation where the control qubit is |↓ (blue) compared to |↑ (orange). The barrier gate voltage at which this occurs is obtained from the intersection of two locally linear fits to the extracted acquired phase (b). c, The CZ gates between all four qubit pairs probed by a Ramsey sequence using the inverse qubits as target and control as compared to the data in Fig. 4b. The target qubit is marked in boldface.

Extended Data Fig. 9 Time evolution of the four-qubit GHZ state.

a, Circuit diagram of the experiments performed in b, c. We first apply a preparation pulse to Q3 and then generate a four-qubit GHZ state analogously to Fig. 5. Next we let the entangled system evolve for time twait, then apply an optional Y2 decoupling pulse and finally disentangle the GHZ state again. b, c, We vary both the waiting time twait and preparation time tprep and plot the non-blocked state fraction for Q3Q4 in the case without (b) and with (c) a decoupling pulse. It can be clearly observed that without the echo pulse, the system has fully decohered at the end of the algorithm. However, by applying the decoupling pulse, the coherence of the entangled system can be maintained for a prolonged timescale, with a characteristic decay time of τ = 390 ns.

Extended Data Fig. 10 Dephasing of the four-qubit GHZ state.

We model the quantum circuit performed in Fig. 5 and account for qubit decoherence by applying a depolarizing channel Λλ(ρ) = λρ + [(1 − λ)/d]\({\mathbb{1}}\), with ρ the density matrix, λ the depolarization parameter, d the dimension of the Hilbert space, and \({\mathbb{1}}\) the identity operator. We plot the expected measurement outcomes for qubit pairs Q1Q2 (blue) and Q3Q4 (orange). The top row corresponds to the case of perfect coherence in each panel. In the centre row, we fit the depolarization parameter to the measurement data. The finite rotations visible in panels IV and V in Fig. 5b can be reproduced by including gate or readout errors in the model. Finally, the bottom row corresponds to a full depolarization of the state. If the qubit system is completely dephased at any point in time, no recovery of the signal can be observed in panel IX.

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Hendrickx, N.W., Lawrie, W.I.L., Russ, M. et al. A four-qubit germanium quantum processor. Nature 591, 580–585 (2021). https://doi.org/10.1038/s41586-021-03332-6

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