Rapid High-Fidelity Spin-State Readout in $\mathrm{Si}$/$\mathrm{Si}$-$\mathrm{Ge}$ Quantum Dots via rf Reflectometry

Silicon spin qubits show great promise as a scalable qubit platform for fault-tolerant quantum computing. However, fast high-fidelity readout of charge and spin states, which is required for quantum error correction, has remained elusive. Radio-frequency reflectometry enables rapid high-fidelity readout of $\mathrm{Ga}\mathrm{As}$ spin qubits, but the large capacitances between accumulation gates and the underlying two-dimensional electron gas in accumulation-mode $\mathrm{Si}$ quantum-dot devices, as well as the relatively low two-dimensional electron gas mobilities, have made radio-frequency reflectometry challenging in these platforms. In this work, we implement radio-frequency reflectometry in a $\mathrm{Si}$/$\mathrm{Si}\text{−}\mathrm{Ge}$ quantum-dot device with overlapping gates by making minor device-level changes that eliminate these challenges. We demonstrate charge-state readout with a fidelity above $99.9\mathrm{\%}$ in an integration time of $300\mathrm{ns}$. We measure the singlet and triplet states of a double quantum dot via both conventional Pauli spin blockade and a charge latching mechanism, and we achieve maximum fidelities of $82.9$ and $99.0\mathrm{\%}$ in $2.08$- and $1.6$-$\mu \mathrm{s}$ integration times, respectively. We also use radio-frequency reflectometry to perform single-shot readout of single-spin states via spin-selective tunneling in microsecond-scale integration times.

Figure 1

Device and design. (a) False color scanning electron microscope image of a device with a nominally identical geometry to the ones tested. Gates are colored according to their purpose, with accumulation, screening, plunger, and tunneling gates shown in purple, black, blue, and yellow, respectively. The rf reflectometry circuit is connected to an Ohmic contact to the 2DEG indicated by a square with an “$x$” in it. (b) Schematic of the rf reflectometry circuit. The on-chip portion of the circuit is highlighted where the region in pink is the contribution from the 2DEG under the accumulation gate, and represents a distributed network of $R_S$ and $c_g$ components where $C_G=\sum c_g$ is the total gate capacitance. The region in blue is the contribution from the dot, and $R_d$ is the resistance of the dot. (c) Schematic of the device used showing the underlying 2DEG formed in typical device operation. The accumulation area (outlined in black) is the region of the 2DEG that corresponds to the reflectometry circuit.

Figure 2

Rf reflectometry performance in prototype devices. (a) Measurement of $S_{21}$ in a quantum-dot device that is not optimized for rf reflectometry (QD 0) made using a network analyzer connected to the Ohmic contact corresponding to LA2 via a bias tee. The reflected rf signal is sensitive to the accumulation gate voltage, but no change is observed when the tunneling gates are used to modulate the channel conductivity. The large shift in the resonance frequency from approximately $150\mathrm{MHz}$ to approximately $70\mathrm{MHz}$ occurs when the accumulation gate is held above threshold thereby inducing a large $C_G$. (b) Demonstration of in situ impedance matching in a two-gate FET. $V_A$ is the voltage on the accumulation gate (A) with the depletion gate (D) held below threshold. (c) Response of the reflected rf signal as the channel conductance is varied using the depletion gate. The $-16\mathrm{dB}$ offset in (a)–(c) arises from the combination of room-temperature attenuation and amplification used on the send and return signals, respectively. The oscillations in the baseline arise from stray reflections in the measurement setup. The insets in (b) and (c) show a schematic of the two-gate FET tested. All measurements in (b) and (c) are made at approximately 4 K.

Figure 3

Charge sensing via rf reflectometry. (a) Measurement of conductance peaks of the sensor dot using rf reflectometry. (b) Charge-stability diagram of a double quantum dot acquired via rf reflectometry measurement of the sensor dot. A plane has been subtracted to remove cross-talk between the double dot plunger gates and the sensor dot. (c) Plot of $1-F_C$ as a function of integration time $T_{\mathrm{int}}$. The inset shows a representative time series with $T_{\mathrm{int}}=300\mathrm{ns}$. (d) Histogram of the time series shown in (c) analyzed with an integration time $T_{\mathrm{int}}=100\mathrm{ns}$ and having a measurement fidelity of $F_C=98.8\mathrm{\%}$.

Figure 4

Singlet-triplet readout via Pauli spin blockade. (a) Average charge-sensor signal acquired at each measurement point across many repetitions of a pulse sequence that initializes a random spin state prior to pulsing to the measurement point. A trapezoid near the interdot transition in the (4,2) charge configuration indicates the Pauli spin-blockade region. The limits of the color bar are adjusted such that the signal in the (4,3) and (3,2) charge configurations is saturated so that the spin-blockade region is more easily visible. Positions L, R, E, and M, are the ground-state initialization, random-state initialization, empty, and measure positions, respectively. (b) Difference of the average signals measured at position M after initializing a random state and after initializing a singlet state as a function of measurement time exhibiting a decay with $T_1=11.0\mu \mathrm{s}$. (c) Plot of the singlet-triplet readout fidelity as a function of integration time. A dashed line indicates the integration time at which the maximum fidelity is achieved. (d) Histogram of single-shot measurements with an integration time of $T_{\mathrm{int}}=2.08\mu \mathrm{s}$ and a fidelity of $F=82.9\mathrm{\%}$. The signal acquired on the DAQ is inverted from the signal shown in (a), resulting in the singlet states having the lower voltage signal. Data shown in (a)–(d) are taken at $B_{\mathrm{ext}}=50\mathrm{mT}$, where $B_{\mathrm{ext}}$ is the externally applied in-plane magnetic field.

Figure 5

Singlet-triplet readout utilizing a latching mechanism. (a) Average charge-sensor signal acquired at each measurement point across many repetitions of a pulse sequence that initializes a random spin state before pulsing to the measurement point. Positions L, R, E, and M, are the ground-state initialization, random-state initialization, empty, and measure positions, respectively. (b) Difference of the average signals measured at position M after initializing a random state and after initializing a singlet state as a function of measurement time exhibiting a decay with $T_1=51.9\mu \mathrm{s}$. (c) Singlet-triplet readout fidelity as a function of integration time. A dashed line indicates the integration time at which the maximum fidelity is achieved. (d) Histogram of single-shot measurements with an integration time of $T_{\mathrm{int}}=800\mathrm{ns}$ and a fidelity of $F=98.2\mathrm{\%}$. Data shown in (a)–(d) are taken at $B_{\mathrm{ext}}=50\mathrm{mT}$.

Figure 6

Single-spin readout. (a) Averaged DAQ signal during the single-spin readout pulse sequence (pink). The sequence empties the dot, loads an electron with a random spin, and then pulses to the measurement point. Empty, load, and measurement sections of the pulse are separated by dashed gray lines, and are labeled with E, L, and M, respectively, above the plot. A spin bump corresponding to spin-up tunneling events is visible. A control pulse that initializes a spin-down electron prior to measurement is shown in black. The difference between the control and measurement pulses during the measurement window is shown in the inset. (b) 100 single-shot spin-selective tunneling measurements demonstrating individual electron tunneling events. The plunger gate is pulsed to the measurement point at $T=48\mu \mathrm{s}$ in the pulse sequence. (c) Two representative single-shot traces. The top trace shows a spin-up electron tunneling out and back in as a spin down. The bottom trace shows no spike and indicates a spin-down electron. The traces are offset for clarity. (d) Histogram of the maximum single-point value of the signal in the range $T=48\mu \mathrm{s}$ to $T=66\mu \mathrm{s}$. Data shown in (a)–(d) are taken at $B_{\mathrm{ext}}=1.5\mathrm{T}$ and a sampling rate of 10 MHz.