Benchmarking Gate Fidelities in a $\mathrm{Si}/\mathrm{SiGe}$ Two-Qubit Device

We report the first complete characterization of single-qubit and two-qubit gate fidelities in silicon-based spin qubits, including cross talk and error correlations between the two qubits. To do so, we use a combination of standard randomized benchmarking and a recently introduced method called character randomized benchmarking, which allows for more reliable estimates of the two-qubit fidelity in this system, here giving a 92% fidelity estimate for the controlled-$Z$ gate. Interestingly, with character randomized benchmarking, the two-qubit gate fidelity can be obtained by studying the additional decay induced by interleaving the two-qubit gate in a reference sequence of single-qubit gates only. This work sets the stage for further improvements in all the relevant gate fidelities in silicon spin qubits beyond the error threshold for fault-tolerant quantum computation.

Popular Summary

The feasibility of quantum computation crucially depends on high-quality quantum gates, the manipulations that implement logic operations. While spin-based quantum bits (qubits) in silicon show great promise, researchers have yet to determine a key metric—the fidelity of the two-qubit gate. Here, we address this question using a recently introduced randomized benchmarking protocol, called character randomized benchmarking.

We obtain the fidelity of the two-qubit controlled-phase gate by studying how it accelerates the decay of qubit coherence under a random sequence of ${90}^{\circ }$ and ${180}^{\circ }$ single-qubit rotations, simultaneously applied to both qubits. This gives an estimated two-qubit gate fidelity of 92%. Using the same protocol, we find that cross talk of the one-qubit operations is asymmetric and that error correlations in single-qubit gates are negligible.

This work has two important implications. First, it shows the usefulness of the character randomized benchmarking protocol in an experimental setting. Second, the observed 92% two-qubit fidelity provides an encouraging starting point for future work on intermediate-scale quantum technology and removing errors from decoherence faster than they occur.

Figure 1

Device schematic. A double quantum dot is formed in the $\mathrm{Si}/\mathrm{SiGe}$ quantum well, where two spin qubits $Q1$ (blue spin) and $Q2$ (red spin) are defined. The green-shaded areas show the locations of the accumulation gates on top of the double dot and the reservoir. The blue dashed lines indicate the positions of three Co micromagnets, which form a magnetic field gradient along the qubit region. MW1 and MW2 are connected to two vector microwave sources to perform EDSR for single-qubit gates. The yellow ellipse shows the position of a larger sensing quantum dot (SQD) which is used as a charge sensor for single-shot readout. Plunger gates $P1$ and $P2$ are used to pulse to different positions in the charge stability diagram as needed for initialization, manipulation, and readout, as well as for pulsing the detuning for controlling the two-qubit gate.

Figure 2

Individual and simultaneous standard randomized benchmarking. (a) Circuit diagrams for individual single-qubit RB on $Q1$ (left) and $Q2$ (right), and simultaneous single-qubit RB (middle). (b) Probability for obtaining outcome 1 upon measurement in the ${\sigma }_z\otimes I$ basis as a function of the number of single-qubit Clifford operations. For the blue solid stars, $Q2$ is idle while a Clifford operation is applied to $Q1$ ($C\otimes I$). For the blue open stars, random Clifford operations are applied to $Q1$ and $Q2$ simultaneously ($C\otimes C$). For each data point, we sample 32 different random sequences, which are each repeated 100 times. Dashed lines are fit to the data with a single exponential. A constant offset of $-0.06$ is added to the open data points in order to compensate for a change in readout fidelities between the two data sets, making comparison of the two traces easier. Without SPAM errors, the data points would decay from 1 to 0.5. (c) Analogous single-qubit RB data for $Q2$, with $Q1$ idle (red solid stars) and subject to random Clifford operations (red open stars). A constant offset of $-0.05$ is added to the open data points. Throughout, single-qubit Clifford operations are generated by the native gate set $\{I,X(\pi ),Y(\pm \pi ),X(\pm \pi /2),Y(\pm \pi /2)\}$.

Figure 3

Two-qubit Clifford randomized benchmarking. Probability for obtaining outcome 11 upon measurement in the ${\sigma }_z\otimes {\sigma }_z$ basis, starting from the initial state $|11⟩$, as a function of the number of two-qubit Clifford operations. As the native gate set, we use $\{I,X(\pi ),Y(\pm \pi ),X(\pm \pi /2),Y(\pm \pi /2),\mathrm{CZ}\}$. The elements of the two-qubit Clifford group fall in four classes of operations: the parallel single-qubit Clifford class, the cnot-like class, the iswap-like class, and the swap-like class. They are compiled by single-qubit gates plus 0, 1, 2, and 3 cz gates respectively. For each data point, we sample 30 random sequences, which are each repeated 100 times. The dashed line is a fit to the data with a single exponential. The data point for a single Clifford operation is missing because we absorb the recovery Clifford gate into the last random Clifford gate, in order to minimize dephasing effects. Here the probability has been normalized to remove initialization and readout errors.

Figure 4

Character randomized benchmarking. (a) Reference CRB experiment. The probabilities $P_1$ (blue triangles), $P_2$ (red stars), and $P_3$ (green diamonds), obtained starting from the initial state $|00⟩$ followed by a Pauli operation, as a function of the number of subsequent single-qubit Clifford operations simultaneously applied to both qubits (see the schematic of the pulse sequence, note that the Clifford gates applied on $Q1$ and $Q2$ are sampled independently and are thus generally different). As the native gate set, we use $\{I,X(\pi ),Z(\pm \pi ),X(\pm \pi /2),Z(\pm \pi /2),\mathrm{CZ}\}$. For each of the 16 Pauli operators, we apply 40 different random sequences, each with 20 repetitions. The dashed lines are fits to the data with a single exponential. Without SPAM errors, the data points would decay from 1 to 0. (b) Interleaved CRB experiment. This experiment is performed in an analogous way to the reference CRB experiment, but with a two-qubit cz gate interleaved after each Clifford pair, as seen in the schematic of the pulse sequence. In both panels, the traces are offset by an increment of 0.1 for clarity and the probabilities have been normalized to remove initialization and readout errors.

Figure 5

Interleaved randomized benchmarking projected in single-qubit space. (a) Probability for obtaining outcome 0 upon measurement in the ${\sigma }_z\otimes I$ basis as a function of the number of single-qubit Clifford operations, interleaved with the cz operation. For the red circles (blue squares), $Q2$ is in $|0⟩$ ($|1⟩$) so $Q1$ is expected to undergo the identity operation [a $Z(\pi )$ rotation]. For each data point, we sample 30 different random sequences for each Clifford number, which are each repeated 100 times. Dashed lines are fits to the data with a single exponential. A constant offset of $+0.045$ is added to the blue data points. (b) Analogous data for $Q2$.

Figure 6

Rabi oscillations of $Q2$. (a) Probability that measurement of $Q2$ returns spin-up ($|1⟩$) as a function of the duration of the resonant microwave burst driving the qubit. (b) Analogous data for $Q2$ when $Q1$ is simultaneously being driven.