Extended Data Fig. 4: Two-qubit Randomised Benchmarking.
From: Quantum error detection in a silicon quantum processor
To characterise the fidelity of the CZ gate between nuclear spins N1 ~ N4, we employ interleaved randomised benchmarking (IRB). The nuclear spins are initialised to the \(|{\Downarrow \Downarrow \Downarrow \Uparrow \Uparrow \Uparrow\rangle}\) state before executing the IRB sequence. First, we perform a standard two-qubit RB experiment as a reference. Each RB sequence of length n consists of n Clifford gates randomly selected from the two-qubit Clifford group, with each Clifford gate composed by an average of 4.99 π/2 pulses and 1.55 CZ gates. A recovery gate at the end of each RB sequence ensures that the two target qubits return to \(| {\Downarrow \Downarrow \rangle}\) state. We randomly generate RB sequences from the Clifford group for each experiment, measure each sequence 200 times and average all results. The six IRB experiments use 9, 15, 28, 15, 22, and 21 sequences, respectively. By fitting the probability of \(| {\Downarrow \Downarrow \rangle}\) by \({P}_{\Downarrow \Downarrow }(n)=A{p}_{\rm{ref}}^{n}+B\), we extract the reference depolarizing parameter pref. Subsequently, we interleave the CZ gate after each Clifford gate in the reference RB sequence, update the recovery gate, and then perform the new RB sequence to extract the depolarizing parameter pCZ. By comparing the decay curves of the reference and interleaved sequences, the fidelity of CZ gate is extracted as FCZ = (1 + 3pCZpref)/4. We characterise the two-qubit CZ gate for all pairs among the four nuclear spins, and obtain gate fidelities of 99.65 ± 1.06%, 99.20 ± 0.69%, 95.11 ± 1.41%, 99.62 ± 1.05%, 96.25 ± 0.87% and 96.73 ± 0.87%, respectively. The fidelity uncertainties are derived by the bootstrap resampling and are denoted at the 1σ confidence level. a, The experimental circuit for two-qubit randomised benchmarking. \({{{X}}}_{\mathrm\pi /2}^{1}{{{X}}}_{\mathrm\pi /2}^{2}\), virtual \({Z}_{\mathrm{\pi} /2}^{1}\), virtual \({Z}_{\mathrm{\pi} /2}^{2}\) and CZ gate are chosen as the primitives to generate two-qubit Clifford gates. b, Two-qubit Randomised Benchmarking measurement results for all pairs among the four nuclear spins. The error bars represent one standard deviation calculated using the bootstrapping method. c, Summary of quantum gate infidelities in the four-nuclear-spin system. The values inside the circles denote the averaged infidelity of single-qubit Clifford gates for the corresponding nuclear spins, and the values between the circles represent the infidelity of the CZ gates between them.
