Universal control of four singlet-triplet qubits

TL;DR

The paper demonstrates universal control of four singlet-triplet qubits in a 2×4 germanium quantum dot ladder, achieving high-fidelity single-qubit gates (~99.5–99.8%) and SWAP-style two-qubit interactions between neighboring qubits. By tuning detuning and barrier gates, the authors map the S–T_- energy spectrum, implement two-qubit gates across all adjacent pairs, and verify entanglement through Bell-state tomography and concurrence measurements, including a remote Bell state between Q1 and Q4 with fidelity 75% and concurrence 22%. The work combines randomized benchmarking and gate-set tomography to quantify gate performance and demonstrates a quantum circuit that distributes entanglement across the entire array, highlighting the viability of baseband-controlled singlet-triplet qubits in germanium for scalable quantum computing and analog quantum simulation. Future improvements in two-qubit gate fidelity and tunable barriers, possibly via feedback control and alternative gate schemes, could push this platform toward fault-tolerant operation and larger-scale quantum spin networks.

Abstract

The coherent control of interacting spins in semiconductor quantum dots is of strong interest for quantum information processing as well as for studying quantum magnetism from the bottom up. Here, we present a $2\times4$ germanium quantum dot array with full and controllable interactions between nearest-neighbor spins. As a demonstration of the level of control, we define four singlet-triplet qubits in this system and show two-axis single-qubit control of each qubit and SWAP-style two-qubit gates between all neighbouring qubit pairs, yielding average single-qubit gate fidelities of 99.49(8)-99.84(1)% and Bell state fidelities of 73(1)-90(1)%. Combining these operations, we experimentally implement a circuit designed to generate and distribute entanglement across the array. A remote Bell state with a fidelity of 75(2)% and concurrence of 22(4)% is achieved. These results highlight the potential of singlet-triplet qubits as a competing platform for quantum computing and indicate that scaling up the control of quantum dot spins in extended bilinear arrays can be feasible.

Figures (14)

• Figure 1: Device and energy spectroscopy.a, Schematic drawing showing the Ge/SiGe heterostructure and three layers of gate electrodes on top to define the quantum dot ladder and sensing dots: screening gates (purple), plunger gates (red), barrier gates (green). Ohmic contacts (gray) extend towards the Ge quantum well in which the holes are confined. The aluminum oxide dielectric between different gate layers is omitted for clarity. b, False-colored scanning electron microscope image of a device nominally identical to that used in the measurements, where the in-plane position of the 8 quantum dots is indicated with numbers 1-8 in circles. Charge sensors close to the ladder corners are labeled within larger circles. The plunger (red) and barrier (green) gates of the quantum dots are labeled outside the image. A schematic of the ladder structure of the quantum dots is shown on top, with Q1-Q4 formed by vertical double quantum dots (DQD). c-e, The energy levels of two-spin states in a DQD as a function of energy detuning $\varepsilon_{ij}$ between dot $i$ and $j$ with from left to right the case $J(\varepsilon_{ij} = 0) < \overline{E}_z$, $J(\varepsilon_{ij} = 0) = \overline{E}_z$, $J(\varepsilon_{ij} = 0) > \overline{E}_z$. The dashed black circles denote the positions of $S-T_-$ anticrossings. f-i, The measured energy spectra that probe the positions of the $S-T_-$ anticrossings as a function of the detuning and the barrier gate voltage for each vertical DQD at $B=5$ mT. The color scale shows the measured spin triplet probability $P_T$ after initializing a vertical DQD in a singlet state (in (0,2) or (2,0)) and applying a gate voltage pulse (20 ns ramp in, 40 ns wait time, 0 ns ramp out) to the detuning shown on the horizontal axis, for different $\text{v}b_{ij}$. The cartoons on top of panels f-i represent the eight dots, and the dark grey line indicates which exchange coupling is active in the panel below.
• Figure 1: Charge stability diagrams and Pauli spin blockade.a-d, Charge stability diagrams for DQD 1-5 (a), 2-6 (b), 3-7 (c), and 4-8 (d), respectively. a and b are recorded using the sensor $\text{S}_{\text{BL}}$ while c and d are recorded using the sensor $\text{S}_{\text{BR}}$. Hole numbers inside the relevant charge stability regions are indicated, showing all the DQDs can be emptied to (0,0). e-h, Charge stability diagrams measured by scanning the detuning $\varepsilon_{ij}$ and the overall chemical potential $\mu_{ij}$ of the DQD. The PSB regions inside the (2, 0) or (0, 2) area are indicated by solid white triangles and trapezoids. For outer DQD 1-5 and 4-8, we find PSB by pulsing $\varepsilon_{15}$ and $\varepsilon_{48}$ from (1,1) to (2,0) and (0,2), where within a triangular region an electron tunnels between the dots starting from the $S(1, 1)$ but no tunneling occurs (the system is in Pauli spin blockade) from $T_0(1, 1)$, $T_-(1, 1)$ and $T_+(1, 1)$. For inner DQD 2-6 and 3-7, we swap their spin states to those of DQD 5-6 and 7-8 where the sensor signals are stronger. i,j, Illustration of PSB using the energy levels in the quadruple quantum dot plaquette for DQD 2-6 (i) and 3-7 (j), respectively. The hole numbers are indicated as $\left( n_1,n_2n_5,n_6 \right)$ for i and $\left( n_3,n_4n_7,n_8 \right)$ for j, and the subscripts $S$ and $T$ show the two-spin states of holes in the quantum dots indicated by bold numbers, respectively. The solid arrows show fast spin-conserving tunneling while the dashed arrows show suppressed tunneling due to PSB. Here we take pair 2-6 as an example to explain the readout process of the inner spin pairs. First, we align DQD 1-5 at the charge stability boundary between (2,0) and (2,1), as shown by the white dot in e, and then pulse $\varepsilon_{26}$ from negative to positive. We subsequently find a shaded region between (0,1) and (0,2) in the diagram for DQD 2-6, which is caused by PSB in DQD 5-6. The mechanism is shown in i: when we pulse DQD 2-6 to the point where $S(0,2)$ is lower in energy than $(0,1)$, the holes in DQD 2-6 moves across to DQD 5-6, irrespective of the spin states. Subsequently, the conventional PSB mechanism in DQD 5-6 allows $S(1,1)$ to transition to $S(0,2)$, while the triplets $T(1,1)$ have to remain in the (1,1) charge state. In this way, we indirectly realize spin-to-charge conversion for the two spins initially in DQD 2-6. Actually, $S(1,1)$ in DQD 2-6 can also directly tunnel to $S(0,2)$ inside the same DQD, as seen by the curved arrow in i. The mechanism to measure DQD 3-7 is analogous.
• Figure 2: Universal single-qubit control of four singlet-triplet qubits.a,b, The pulse schemes used for $x$-axis control (a) and $y$-axis control (b). In the experiments, the detuning pulse in a and b has a 20 ns ramp (not shown) from (2,0) to (1,1), similar to the pulse used for the energy spectroscopy. c-f, Experimental results for $x$-axis rotations of each qubit, showing measured triplet probabilities $P_T$ as a function of $t_\text{wait}$ and the corresponding barrier voltage $\delta \text{v}b_{ij}$. g, Measured $P_T$ for the sequence shown in panel b as a function of $t_\text{wait}$ and the barrier voltage change $\delta \text{v}b_{26}$. The inset shows the numerically computed $P_T$ as a function of $t_\text{wait}$ and the ratio of $z$-axis component to the $x$-axis component, $(J-\overline{E}_z) / \Delta_{ST_-}$. The position where$\sqrt{Y}$ is properly calibrated is indicated by a white dot. h, Single-qubit randomized benchmarking data for Q1-Q4. The numbers in the legend are the extracted average gate fidelities, which are obtained from the Clifford gate fidelities using a ratio of 3.625. i, Table showing the single-qubit gate fidelities of Q1-Q4 measured by gate set tomography (GST). All the data above are measured at $B=5$ mT.
• Figure 2: Data of two-axis qubit control around the $x$- and $z$-axis, measured at $B$ = 10 mT. a,b, Pulse scheme and Bloch sphere illustration of $x$-axis and $z$-axis evolution of $S-T_-$ qubits. The straight blue and orange arrows show the corresponding rotation axis. The $x$-axis rotations are set by the $S-T_-$ coupling, $\Delta_{ST_-}$. For large $J$ such that $J-\overline{E}_z\gg \Delta_{ST_-}$, the rotation axis tilts towards the $z$-axis. The rotation is never exactly around the $z$-axis due to the presence of a finite $\Delta_{ST_-}$, yet, sufficiently orthogonal control is possible when $(J-\overline{E}_z) \gg \Delta_{ST_-}$. In b, we illustrate a Ramsey-like pulse sequence used to demonstrate $z$-axis control. We first initialize the qubit into a singlet, perform a $\pi/2$ rotation around the $x$-axis of duration $t_{\pi/2}$, and then change $J$ diabatically by pulsing the corresponding barrier gate by an amount $\delta\text{v}b_{ij}$ to implement a $z$-axis rotation. Finally, we perform another $\pi/2$ operation around the $x$-axis and project the qubit into the $S-T_-$ basis for spin readout. c-f, Experimental results for $x$-axis rotations of each qubit, showing measured triplet probabilities $P_T$ as a function of $t_\text{wait}$ and the detuning voltage $\varepsilon_{ij}$. g-j, Experimental results for $z$-axis rotations of each qubit, showing $P_T$ as a function of $t_\text{wait}$ and the barrier voltage change $\delta \text{v}b_{ij}$. The oscillation frequency is given by $f_{ST_-}=\sqrt{(J-\overline{E}_z)^2+\Delta^2_{ST_-}}/h$, where $h$ is Planck's constant. We note that the outer two barrier gates $\text{v}b_{15}$ and $\text{v}b_{48}$ have a stronger effect on the corresponding $J_{ij}$ than the inner barrier gates $\text{v}b_{26}$ and $\text{v}b_{37}$. This may be explained by additional residual resist below the inner barrier gates, which are fabricated in the last step hsiao2023exciton, and by the different fan-out routing for the outer barrier gates (see Fig. \ref{['Device']}a,b in the main text). Within the tuning range of the barrier gate, the highest ratio $(J-\overline{E}_z)/\Delta_{ST_-}$ amounts to around 20 for the outer qubits Q1 and Q4 and about 10 for the inner qubits Q2 and Q3 (see Supplementary Information section IV for more details).
• Figure 3: Two-qubit interactions across the quantum dot ladder.a, A plaquette of two connected DQDs. The $S-T_-$ qubits have a splitting of $J_{ij}-\overline{E}_{z, ij}$ and $J_{kl}-\overline{E}_{z, kl}$ (neglecting $\Delta_{ST_-}$), which are controlled by the detunings $\varepsilon_{ij}$ and $\varepsilon_{kl}$, respectivily. The qubit-qubit coupling $J_\text{coup}$ is an average of $J_{ik}$ and $J_{jl}$ between the corresponding dots, which are controlled by $\varepsilon_{ik}$ and $\varepsilon_{jl}$. b, The energy levels of two-qubit states, where we fix $\varepsilon_{kl}$ to be positively large and scan $\varepsilon_{ij}$. At the positions where $J_{ij}-\overline{E}_{z, ij}$ equals $J_{kl}-\overline{E}_{z, kl}$, an anticrossing with a gap $J_\text{coup}$ forms (black dashed circles), which can be used to induce SWAP oscillations between $\ket{ST_-}$ and $\ket{T_-S}$. The parameters used in this calculation are based on the experimental results for Q3-Q4 shown in Supplementary Fig. 4c. c, The pulse scheme for SWAP operations. We start in (0,2) or (2,0), at large positive or negative detuning, and diabatically pulse one qubit to (1,1) at modest detuning such that it remains in $\ket{S}$, and pulse the other qubit to zero detuning where a $\pi$ rotation for a time $t_X$ takes it to $\ket{T_-}$. At this point, the qubits are set to $\ket{ST_-}$ or $\ket{T_-S}$. Next, we pulse the detunings of both qubits to make their energies resonant, while at the same time activating $J_{ik}$ and $J_{jl}$. This will kickstart SWAP oscillations between the two qubits. The dashed lines in the pulse of $\varepsilon_{ij}$ show that we scan the detuning of one qubit to find the condition for SWAP operations. After an evolution time $t_\text{wait}$, we pulse the detunings to the PSB readout configuration for one of the qubits. d-f, The experimental results of SWAP oscillations, showing measured triplet probabilities $P_T$ or singlet probabilities $P_S$ as a function of operation time $t_\text{wait}$ and the detuning voltage for Q1-Q2 (d), Q2-Q3 (e) and Q3-Q4 (f). The initial states of two qubits (before the SWAP oscillations) are denoted on the top, and the qubit pair that is read out is indicated by the dashed arrow showing the readout pulse direction. g, The quantum circuit used to create a generalized Bell state between Q1 and Q2 and to characterize it via quantum state tomography (QST). h, Measured two-qubit density matrix of Q1-Q2, after removal of SPAM errors and using maximum-likelihood estimation (MLE). i, State fidelities and concurrence estimated from the density matrices of the Bell states of Q1-Q2, Q2-Q3 and Q3-Q4. The data of panels d-f and h-i is measured at $B=5$ mT.