A four-qubit germanium quantum processor
N.W. Hendrickx, W.I.L. Lawrie, M. Russ, F. van Riggelen, S.L. de Snoo, R.N. Schouten, A. Sammak, G. Scappucci, M. Veldhorst
AA four-qubit germanium quantum processor
N.W. Hendrickx, ∗ W.I.L. Lawrie, M. Russ, F. van Riggelen, S.L. de Snoo, R.N. Schouten, A. Sammak, G. Scappucci, and M. Veldhorst † QuTech and Kavli Institute of Nanoscience, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands QuTech and Netherlands Organisation for Applied Scientific Research (TNO), Stieltjesweg 1, 2628 CK Delft, The Netherlands
The prospect of building quantum circuits using advanced semiconductor manufacturing positionsquantum dots as an attractive platform for quantum information processing. Extensive studieson various materials have led to demonstrations of two-qubit logic in gallium arsenide, silicon,and germanium. However, interconnecting larger numbers of qubits in semiconductor devices hasremained an outstanding challenge. Here, we demonstrate a four-qubit quantum processor basedon 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 implementedall-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 high-connectivity circuit. Weexecute a quantum logic circuit that generates a four-qubit Greenberger-Horne-Zeilinger state andwe obtain coherent evolution by incorporating dynamical decoupling. These results are an importantstep towards quantum error correction and quantum simulation with quantum dots.
Fault-tolerant quantum computers utilizing quantumerror correction [1] to solve relevant problems [2] will relyon the integration of millions of qubits. Solid-state imple-mentations of physical qubits have intrinsic advantagesto accomplish this formidable challenge and remarkableprogress has been made using qubits based on supercon-ducting circuits [3]. While the development of quantumdot qubits has been at a more fundamental stage, theirresemblance to the transistors that constitute the build-ing block of virtually all our electronic hardware promisesexcellent scalability to realize large-scale quantum cir-cuits [4, 5]. Fundamental concepts for quantum infor-mation, such as the coherent rotation of individual spins[6] and the coherent coupling of spins residing in neigh-boring quantum dots [7], were first implemented in gal-lium arsenide heterostructures. The low disorder in thequantum well allowed the construction of larger arraysof quantum dots and to realize two-qubit logic using twosinglet-triplet qubits [8]. However, spin qubits in groupIII-V semiconductors suffer from hyperfine interactionswith nuclear spins that severely limit their quantum co-herence. Group IV materials naturally contain higherconcentrations of isotopes with a net-zero nuclear spinand can furthermore be isotopically enriched [9] to con-tain only these isotopes. In silicon electron spin qubits,quantum coherence can therefore be sustained for a longtime [10, 11] and single qubit logic can be implementedwith fidelities exceeding 99.9 % [12, 13]. By exploitingthe exchange interaction between two spin qubits in ad-joining quantum dots or closely separated donor spins,two-qubit logic could be demonstrated [14–20]. Silicon,however, suffers from a large effective mass and valley de-generacy [21], which has hampered progress beyond two-qubit demonstrations.Holes in germanium are emerging as a promising alter-native [22] that combine favorable properties such as zeronuclear spin isotopes for long quantum coherence [23],low effective mass and absence of valley states [24] forrelaxed requirements on device design, low charge noise for a quiet qubit environment [25], and low disorder forreproducible quantum dots [26, 27]. In addition, strainedgermanium quantum wells defined on silicon substratesare compatible with semiconductor manufacturing [28].Furthermore, hole states can exhibit strong spin-orbitcoupling that allows for all-electric operation [29–31] andthat removes the need for microscopic components suchas microwave striplines or nanomagnets, which is par-ticularly beneficial for the fabrication and operation oftwo-dimensional qubit arrays. The realization of strainedgermanium quantum wells in undoped heterostructures[32] has led to remarkable progress. In two year’s time,germanium has progressed from the formation of stablequantum dots and quantum dot arrays [26, 27, 33], todemonstrations of single qubit logic [34], long spin life-times [35], and the realization of fast two-qubit logic ingermanium double quantum dots [31].Here, we advance semiconductor quantum dots beyondtwo-qubit realizations and execute a four-qubit quan-tum circuit using a two-dimensional array of quantumdots. We achieve this by defining the four-qubit sys-tem on the spin states of holes in gate-defined germa-nium quantum dots. Fig. 1A shows a scanning-electron-microscopy (SEM) image of the germanium quantum pro-cessor. The quantum dots are defined in a strained ger-manium quantum well on a silicon substrate (Fig. 1B)[25] using a double layer of electrostatic gates and con-tacted by aluminum ohmic contacts. A negative poten-tial on plunger gates P1-P4 accumulates a hole quantumdot underneath that serves as qubit Q1-Q4, which can becoupled to neighboring quantum dots through dedicatedbarrier gates. In addition, two quantum dots are placedto the side of the two-by-two array, and the total sys-tem comprises six quantum dots. Via an external tankcircuit, we configure these additional two quantum dotsas radio frequency (rf) charge sensors for rapid chargedetection. Using the combined signal of both charge sen-sors [33], we measure the four quantum dot stability dia-gram as shown in Fig. 1C. Making use of two virtual gate a r X i v : . [ c ond - m a t . m e s - h a ll ] S e p (0,0,0,0) -10 0 10(P -P +0.75P -0.75P ) (mV)-20-1001020 ( P + P + P + P ) ( m V ) GeSiGeSiGe P P p (ns) 0.20.40.60.8 P p (ns)00.20.40.60.8 P -10 -5 0 5 (mV)-10-505 U ( m V ) visibility 0 0.4 0.8 ( , , , )( , , , ) (1,1) P (1,1) AP (0,1) (0,2) S (0,2) T slowfast (1,2,1,1)(0,1,1,1) BA C E FD (0,1,0,1)(1,2,1,2)(2,1,2,1)(1,1,1,1)(1,0,1,0)(0,0,0,0) Q1 Q2Q3Q4P1S1 S2P3P4 P2 T in Figure 1.
Four germanium hole spin qubits. ( A ) Scanning electron microscope image of the four quantum dot device.We define qubits underneath the four plunger gates indicated by P1-P4. The qubits can be measured using the two chargesensors S1 and S2. The scale bar corresponds to 100 nm. ( B ) Schematic drawing of the Ge/SiGe heterostructure. Startingfrom a silicon wafer, a germanium quantum well is grown in between two Si . Ge . layers at a depth of 55 nm from thesemiconductor/dielectric interface. ( C ) Four quantum dot charge stability diagram as a function of two virtual gates. At thevertical and diagonal bright lines a hole can tunnel between two quantum dots or a quantum dot and its reservoir respectively. Asa result of the virtual axes, the addition lines of the different quantum dots have different slopes, allowing for an easy distinctionof the different charge occupations indicated in the white boxes as (Q1, Q2, Q3, Q4). ( D ) Energy diagram illustrating thelatched Pauli spin blockade readout. When pulsing from the (1,1) charge state to the (0,2) charge state, only the polarizedtriplet states allow the holes to move into the same quantum dot, leaving an (0,2) charge state (green). Interdot tunneling isblocked for the two antiparallel spin states and as a result the hole on the first quantum dot will subsequently tunnel to thereservoir leaving an (0,1) charge state (red), locking the different spin states into different charge states. ( E ) Readout visibilityas defined by the difference in readout between either applying no rotation and a π -rotation to Q2. The readout point is movedaround the (1,1)-(0,2) anticrossing of the Q1Q2 system and a clear readout window can be observed bounded by the different(extended) reservoir transition lines indicated by the dotted lines. ( F ) The qubits can be rotated by applying a microwave toneresonant with the Zeeman splitting of the qubit. Coherent Rabi rotations can be observed as a function of the microwave pulselength t p for all qubits Q1-Q4. axes, we arrange the reservoir addition lines of the fourquantum dots to have different relative slopes of approx-imately − , +1 , − . , . mV/mV for Q1, Q2, Q3, andQ4 respectively. Well defined charge regions (indicatedas (Q1,Q2,Q3,Q4) in the white boxes) are observed, withvertical anticrossings marking the different interdot tran-sitions. The high level of symmetry in the plot is a signof comparable gate lever arms and quantum dot chargingenergies, confirming the uniformity in this platform andsimplifying the operation of quantum dot arrays.For the qubit readout we make use of Pauli-spin block-ade to convert the spin states into a charge signal thatcan be detected by the sensors. In germanium, however,the spin-orbit coupling can significantly lower the spinlifetime during the readout process, in particular whenthe spin-orbit field is perpendicular to the external mag-netic field, reducing the readout fidelity [34, 36]. Here, weovercome this effect by making use of a latched readoutprocess [37]. During the readout process, as illustratedin Fig. 1D, a hole can tunnel spin-selectively to the reser-voir as a result of different tunnel rates of both quan-tum dots to the reservoir. After this process, the systemis locked in this charge state for the slow reservoir tun-nel time T in . We achieve this effect by pulsing into thearea in the (0,2) charge region bounded by the extended (1,1)-(0,1) (fast) and the extended (1,1)-(1,2) (slow) tran-sitions (dotted lines in Fig. 1E). When the interdot tun-neling into the (0,2) charge state is blocked, the hole inthe first quantum dot will quickly tunnel into the reser-voir. This locks the spin state in the metastable (0,1)charge state, with the decay to the (0,2) ground stategoverned by the slow tunnel rate T in between the secondquantum dot and the reservoir. The high level of controlin germanium allows the tuning of T in to arbitrarily longtime scales by changing the potential applied to the corre-sponding reservoir barrier gates. We set T in, Q2 = 200 µ sand T in, Q4 = 2 . ms (Fig. S2), both significantly longerthan the signal integration time T int = 10 µ s. We operatein a parity readout mode where we observe both antipar-allel spin states to be blocked (Fig. S3AB). We speculatethis is caused by the strong spin-orbit coupling mixingthe parallel (1,1) states with the (0,2) state, and caus-ing strong relaxation of the upper parallel spin state. Byboth increasing the interdot coupling and elongating theramp between the manipulation and readout point, wecan transition into a state selective readout where onlythe |↓↑(cid:105) state results in spin blockade (Fig. S3CD), witha slightly reduced readout visibility.In our experiments, we configure the system such thatthe spin-orbit field is oriented along the direction of the c on t r o l (r ad ) Q1 (rad)023 c on t r o l (r ad ) Q1 (rad)023 c on t r o l (r ad ) c on t r o l (r ad ) Q2 (rad) 0 2 3 Q2 (rad) 0 2 3 Q3 (rad) 0 2 3 Q3 (rad) 0 2 3 Q4 (rad) 0 2 3 Q4 (rad) P blocked ϕθ f low f high f low f high Figure 2.
Controlled rotations between all nearest-neighbor qubit pairs.
By selectively enabling the exchange inter-action between each pair of qubits, we can implement two-qubit controlled rotations (CROTs). The pulse sequence consists ofa single preparation gate with length θ on the control qubit (labeled green), followed by a controlled rotation on one of theresonance lines of the target qubit (labelled in red). Both qubit pairs Q1Q2 and Q3Q4 are read out in single-shot mode andthe position of the eye on top of each column indicates the respective readout pair. Each of the four main columns correspondsto conditional rotations on a different qubit as indicated by the red dot. Rows one and two show the results for the horizontalinteraction (dark green), while rows three and four show the two-qubit interaction for the vertical direction (light green) withrespect to the external magnetic field, as indicated in the top left. Rows one and three correspond to driving the lower frequency f low conditional resonance line, while rows two and four show driving of the other resonance line f high . external magnetic field B = 1 . T. This minimizes re-laxation and we project all qubit measurements onto thisreadout direction, thus reading out qubit pairs Q1Q2 andQ3Q4. Each charge sensor can detect transitions in bothqubit pairs, but is mostly sensitive to their respectivenearby quantum dots. We maximize the readout visibil-ity as defined by the difference between the readout ofa spin-up and spin-down state by scanning the readoutlevel around the relevant anticrossing. This is illustratedfor the Q1Q2 pair in Fig. 1E, where a clear readout win-dow with maximum visibility can be observed boundedbetween the (extended) reservoir transitions of the twoquantum dots.Coherent rotations can be implemented by applyingelectric microwave signals to the plunger gates that definethe qubits, exploiting the spin-orbit coupling for fast driv-ing [30, 38]. We initialize the system in the |↓↓↓↓(cid:105) stateby sequentially pulsing both the Q1Q2 and Q3Q4 doublequantum dot systems from their respective (0 , S statesadiabatically into their (1 , T − states. We then performthe qubit manipulations, after which we perform the spinreadout as described above. We observe qubit resonancesat f Q1 = 2 . GHz, f Q2 = 3 . GHz, f Q3 = 3 . GHz,and f Q4 = 3 . GHz, corresponding to effective g -factors of g Q1 = 0 . , g Q2 = 0 . , g Q3 = 0 . , and g Q4 = 0 . .We note that these g -factors can be electrically modu-lated using nearby gates as a means to ensure individualqubit addressability. Fig. 1F shows the single-shot spin-up probability P ↑ for each of the four qubits after apply-ing an on-resonant microwave burst with increasing timeduration t p , resulting in coherent Rabi oscillations.To quantify the quality of the single qubit gates, weperform benchmarking of the Clifford group [39] (Fig. S4)and find single qubit gate fidelities exceeding 99 % for allqubits. The fidelity of Q3 is even above 99.9 %, therebycomparing to benchmarks for quantum dot qubits in iso-topically purified silicon [12, 13]. We find spin lifetimesbetween T = 1 − ms (Fig. S5), comparable to val-ues reported before for holes in planar germanium [35].Furthermore, we observe T ∗ to be between 150-400 nsfor the different qubits (Fig. S6A), but are able to ex-tend phase coherence up to T CPMG = 100 µ s by per-forming Carr-Purcell-Meiboom-Gill (CPMG) refocusingpulses (Fig. S6C), more than two orders of magnitudelarger than previously reported for hole quantum dotqubits [29–31]. This indicates the qubit phase coher-ence is mostly limited by low-frequency noise, which isconfirmed by the predominantly /f α noise spectrum we P up P up P up P up P up P up P up P up P up f q (GHz)f q (GHz)f q (GHz)f q (GHz) P up CBAD FE GHI JKL t p (ns) ↓↓↓↑↑↑↓↑↑↑↓↑↓↓↑↑↑↓↓↑↓↑↓↓
1Q ( C ) 2Q ( D-E ) 3Q (
F-I ) 4Q ( J ) f r equen cy RRR RRR π,f q π,f q π,f q π,f q Figure 3.
Resonant one, two, three, and four-qubit gates. ( A ) Circuit diagram of the experiment performed in panels C-L . All eight permutations of the three control qubit eigenstates are prepared, with R being either no pulse or a π -pulse onthe respective qubit. Next, the resonance frequency of the target qubit is probed using a π -rotation with varying frequency f q . Finally, the prepared qubits are projected back and the target qubit state is measured. By changing the different interdotcouplings J , we can switch between resonant single, two, three, and four-qubit gates as indicated in the dashed boxes. ( B )Turning on the exchange interaction between the different qubit pairs splits the resonance frequency in two, four, and eightfor 1, 2 and 3 enabled pairs respectively. The colors of the line segments correspond to the colors in panels C-L . ( C ) Byturning all exchange interactions off, the qubit resonance frequency of Q2 is independent of the prepared state of the otherthree qubits, resulting in an effective single-qubit rotation. ( D-E ) By turning on a single exchange interaction J ( D ) or J ( E ), the resonance line splits in two. ( F-I ), Turning on both exchange interactions to the neighboring quantum dots results inthe resonance line splitting in four, for Q2 ( F ), Q1 ( G ), Q3 ( H ), Q4 ( I ) respectively. ( J ) Turning on the exchange interactionsbetween three pairs of quantum dots J , J , J splits the resonance line in eight. ( K-L ) Resonant driving of the three-qubitgate ( K ) and the four-qubit gate ( L ) with Q2 being the target qubit, shows Rabi driving as a function of pulse length t p ,demonstrating the coherent evolution of the operation. observe by Ramsey and dynamical decoupling noise spec-troscopy (Fig. S7). This noise could originate in the nu-clear spin bath present in germanium, which could bemitigated by isotopic enrichment. Alternatively, it couldbe caused by charge noise acting on the spin state throughthe spin-orbit coupling and it is predicted that the sen-sitivity to this type of noise could be mitigated by care-ful optimization of the electric field environment [40] ormoving to a multi-hole charge occupancy, screening theinfluence of charge impurities [41], potentially enablingeven higher fidelity operations.Universal quantum logic can be accomplished by com-bining the single qubit rotations with a two-qubit en-tangling gate. We implement this using a conditionalrotation (CROT) gate, where the resonance frequencyof the target qubit depends on the state of the controlqubit, mediated by the exchange interaction J betweenthe two quantum dots. The exchange interaction be-tween the quantum dots is controlled using a virtual bar-rier gate (details in Materials and Methods), coupling the two quantum dots while keeping the detuning andon-site energy of the quantum dots constant and closeto the charge-symmetry point. We demonstrate CROTgates between all four pairs of quantum dots in Fig. 2,proving that spin qubits can be coupled in two dimen-sions. A sequence of qubit pulses is applied, as indi-cated in the diagram, consisting of a single qubit controlpulse (green) and a target qubit two-qubit pulse (red).We vary the length of both the control pulse θ control aswell as the length of the target qubit pulse φ Q1-Q4 , with t p ( φ = π ) = 50 − ns (details in Table S1). Theconditional rotations are performed on all four targetqubits (main four columns) for both the horizontally in-teracting qubits (rows 1 and 2), as well as the verticallyinteracting qubits (rows 3 and 4), by driving the |↓↓(cid:105) - |↑↓(cid:105) transitions with f low (rows 1 and 3), as well as theinverse |↓↑(cid:105) - |↑↑(cid:105) transitions with f high (rows 2 and 4),with (cid:12)(cid:12) Q target Q control (cid:11) . We then perform a measurementon both readout pairs by sequentially pulsing the Q1Q2(left sub-columns), and the Q3Q4 qubit pairs (right sub- P up P up P up P up X XY Y U CZ (t) U CZ (t) X XX XXXY Y idle(t) idle(t) J onJ offQ1
Q2 Q4 Q1 Q2 Q3 Q3 Q4 τ on = 130 ns τ on, echo = 220 ns τ off = 200 ns τ off, echo = 2100 nsXX X X Z(θ) Q target Q control = U CZ Z(φ ) Z(φ ) U CZ = V ba rr i e r time A CDB t CZ < 10 ns Q3Q2Q3Q2
Figure 4.
Controlled phase gate and dynamical decoupling. ( A ) Circuit diagram of the experiment performed in panel B . The controlled phase gate is probed by performing a Ramsey sequence on the target qubit for both basis states of the controlqubit. The phase of the second π/ (X) gate is swept by performing an update of the microwave phase through quadraturemodulation. Additionally, a phase update is performed on both the target and control qubit to compensate for any single qubitphases picked up as a result of the gate pulsing to achieve a controlled-Z (CZ) gate. ( B ) The spin-up probability of the targetqubit (in bold) as a function of the phase θ of the second X gate for the control qubit initialized in the |↓(cid:105) (blue) and |↑(cid:105) (red)state. Measurements for the inverted target and control qubits in Fig. S10. By applying an exchange pulse and single qubitphase updates, we achieve a CZ gate at θ = 0 rad. ( C ) Circuit diagrams of the experiment performed in panel D . The phasecoherence throughout the two-qubit experiment is probed using a Ramsey sequence, both for the case with J on (top) andoff (bottom) and both with (orange) and without (blue) applying an echo pulse. ( D ) Spin-up probability as a function of theexperiment length, for the situation with exchange on (left, triangles) and off (right, circles). From the decay data we extractcharacteristic decay times τ of τ on = 130 ns, τ on, echo = 220 ns, τ off = 200 ns, and τ off, echo = 2100 ns (details in Materials andMethods). Q3Q2 Y Y Y Y Q1Q4
X X XX XX X X XXXX(t prep ) X X AB P no r m t prep (ns) t prep (ns) t prep (ns) t prep (ns) t prep (ns) t prep (ns) t prep (ns) t prep (ns) t prep (ns)Q3Q4Q1Q2 I II III IV V VI VII VIII IX
Figure 5.
Coherent generation of a four-qubit Greenberger-Horne-Zeilinger (GHZ) state. ( A-B ) A four-qubit GHZstate is created by applying three sequential two-qubit gates, each consisting of an X-CZ-X gate circuit. Next, a Y decouplingpulse is applied, after which we disentangle the GHZ state again (circuit diagram in A ). Pulses pictured in the same columnare applied simultaneously. The initial state of Q3 is varied by applying a preparation rotation of length t . For different stagesthroughout the algorithm (dashed lines), we measure the non-blocked state probability as a function of t for both the Q1Q2 andQ3Q4 readout system, normalized to their respective readout visibility. At the end of the algorithm the qubit states correspondto the initial single qubit rotation, and the clear oscillations confirm the coherent evolution of the algorithm from isolated qubitstates to a four-qubit GHZ state. ( B ). columns) to their respective readout points. Because thetarget qubit resonance frequency depends on the controlqubit state, the conditional rotation is characterized bythe fading in and out of the target qubit rotations as afunction of the control qubit pulse length. The patternis therefore shifted by a π rotation on the control qubit,for driving the two separate transitions. When drivingthe |↓↓(cid:105) - |↑↓(cid:105) transition of the qubit pairs used for read-out (row 1), we apply an additional single-qubit π -pulseto the preparation qubit for symmetry, since the controlqubit also serves as the readout ancilla. When the con-trol qubit is in a different readout pair as the target qubit(rows 3 and 4), we can independently observe the singlequbit control, and two-qubit target qubit rotations in thetwo readout systems. By setting the pulse length equalto φ Q = π , a fast CX gate can be obtained within ap-proximately t p = 100 ns between all of the four qubitpairs.To demonstrate full control over the coupling betweenthe different qubits, we measure the qubit resonance fre-quency as a function of the eight possible permutationsof the different basis states of the other three qubits, asillustrated in Fig. 3A-B. Without any exchange present,the resonance frequency of the target qubit should be in-dependent on the preparation of the other three qubits,as schematically depicted in Fig. 3C. When the exchangeto one of the neighboring quantum dots is enabled bypulsing the virtual barrier gate, the resonance line splitsin two, allowing for the operation of the CROT gate, asis shown for both the Q1-Q2 and Q2-Q3 interactions inFig. 3D and E respectively. When both barriers to thenearest-neighbors are pulsed open at the same time, weobserve a fourfold splitting of the resonance line (Fig. 3F-I). This allows the performance of a resonant i -Toffolithree-qubit gate (Fig. 3K and Fig. S8), which has theo-retically been proposed as an efficient manner to createthe Toffoli, Deutsch, and Fredkin gates [42]. We observea difference in the efficiency at which the different condi-tional rotations can be driven, as can also be seen fromthe width of the resonance peaks in Fig. 3F-I. This isexpected to happen when the exchange energy is compa-rable to the difference in Zeeman splitting and is causedby the mixing of the basis states due to the exchangeinteraction between the holes [43] (details in Materialsand Methods). Finally, we open three of the four virtualbarriers and observe the resonance line splitting in eight,being different for all eight permutations of the control-qubit preparation states (Fig. 3J). This enables us to ex-ecute a resonant four-qubit gate and in Fig. 3L we showthe coherent operation of a three-fold conditional rota-tion (see Fig. S8 for the coherent operation of the otherresonance lines).While the demonstration of these conditional rotationscan be beneficial for the simulation of larger coupled spinsystems, the ability to dynamically control the exchangeinteraction allows for faster two-qubit operations [14, 16].We efficiently implement controlled phase (CPHASE) gates between the different qubit pairs by adiabaticallypulsing the exchange interaction using the respective vir-tual barrier gate. Increasing the exchange interaction, theantiparallel spin states will shift in energy with respect tothe parallel spin states, giving rise to a conditional phaseaccumulation. We control the length and size of the volt-age pulse (Fig S9) to acquire a CZ gate, in which theantiparallel spin states accumulate a phase of exactly π with respect to the parallel spin states. We demonstratethis in Fig. 4A-B, where we employ a Ramsey sequence tomeasure the conditional phase. After the exchange pulse U CZ we apply a software Z gate to both the target andcontrol qubits to compensate for individual single qubitphases. As a result of the large range over which the ex-change interaction can be controlled, we achieve fast CZgates that are executed well within 10 ns for all qubitpairs (details in Table S2).To prepare our system for quantum algorithms, we im-plement decoupling pulses into the multi-qubit sequencesto extend phase coherence, as demonstrated in Fig. 4C-D.To probe the effect of a decoupling pulse when exchange ison (Fig. 4D, left, triangles), we perform a CPHASE gatebetween qubits Q2 and Q3 and compare the decay of theresulting exchange oscillations as a function of the oper-ation time for the situations with (orange) and without(blue) a Y echo pulse. We observe an extended durationfor the conditional phase rotations of τ = 220 ns whenapplying a decoupling pulse, compared to τ = 130 nsfor a standard CPHASE gate. A more relevant situa-tion however, is the coherence of the two-qubit entangledstate. We probe this by entangling Q2 and Q3 by form-ing the | Ψ + (cid:105) Bell state and letting the system evolve fortime t (Fig. 4D, right, circles). Next, we disentanglethe system again and measure the spin-up probability ofQ3 as a function of the evolution time. Without the de-coupling pulse, we observe the loss of coherence after acharacteristic time τ = 200 ns. However, by applyingthe additional echoing pulse on both Q2 and Q3, we cansignificantly extend this time scale beyond µ s, enoughto perform a series of single and multi-qubit gates, owingto our short operation times.We show this by coherently generating and disentan-gling a four-qubit Greenberger-Horne-Zeilinger (GHZ)state (Fig. 5). Making use of the fast two-qubit CZ gates,as well as a decoupling pulse on all qubits, we can main-tain phase coherence throughout the experiment. Weperform a parity readout on both the Q1Q2 (red) andQ3Q4 (green) at different stages of the algorithm and nor-malize the observed blocked state fraction to the readoutvisibility. We prepare a varying initial state by applying amicrowave pulse of length t to Q3, as observed in I. Afterapplying CZ gates between all four qubits, the systemresides in an entangled GHZ type state at IV/V, for a π/ preparation pulse on Q3. The effective spin state os-cillates between the antiparallel | (cid:105) and | (cid:105) statesas a function of t prep , resulting in a high state readoutfor all t . The small oscillation that can still be observedfor the Q1Q2 system, is caused by a small difference inreadout visibility for the two distinct antiparallel spinstates. Next, we deploy a Y decoupling pulse to echoout all single qubit phase fluctuations during the exper-iment (Fig. S11). After disentangling the system again,we project the Q3 qubit state by applying a final X ( π/ )gate, and indeed recover the initial Rabi rotation.The demonstration of a two-by-two four-qubit arrayshows that quantum dot qubit systems can be scaled intwo-dimensions and multi-qubit logic can be executed.The hole states used are subject to strong spin-orbit cou-pling, enabling all-electrical driving of the spin state, ben-eficial for scaling up to even larger systems. Making useof a latched readout mechanism overcomes fast spin re-laxation due to the spin-orbit coupling. Furthermore,the ability to freely couple one, two, three and four spinsusing fast gate pulses has great prospects both for per-forming high-fidelity quantum gates as well as studyingexotic spin systems using analog quantum simulations.While the execution of relevant quantum algorithms willrequire many more qubits, the germanium platform hasthe potential to leverage the enormous advancements insemiconductor manufacturing techniques for the realiza-tion of fault-tolerant quantum processors. ACKNOWLEDGEMENTS
We thank L.M.K. Vandersypen for useful discussionsand thank S.G.J. Philips for his contributions to softwaredevelopment. M.V. acknowledges support through a Vidigrant, two projectruimte grants, and an NWA grant, allassociated with the Netherlands Organization of Scien-tific Research (NWO).
COMPETING INTERESTS
The authors declare no competing interests.Correspondence should be addressed to M.V.([email protected]).
DATA AVAILABILITY
All data underlying this study will be available fromthe 4TU ResearchData repository.
SUPPLEMENTARY MATERIALS
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