Abstract
Quantum error detection is essential for large-scale universal quantum computation, particularly for quantum error correction. However, the key elements of fault-tolerant quantum computing with silicon qubits, including error detection with stabilizers, remain challenging. Here we report quantum error detection in a donor-based silicon quantum processor comprising four nuclear spin qubits and one electron spin auxiliary qubit. The entanglement capability of this system is validated through the establishment of two-qubit Bell-state entanglement between the nuclear spins and the generation of a four-qubit Greenberger–Horne–Zeilinger state with a state fidelity of 88.5 ± 2.3%. We use a four-qubit error detection circuit with stabilizers to detect arbitrary single-qubit errors. We recover the encoded Bell-state entanglement information by performing the Pauli frame update via postprocessing; on the basis of the detected errors, we identify strongly biased noise in our system.
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Data availability
The data relevant to this study are available via Zenodo at https://zenodo.org/records/15348336 (ref. 55). Source data are provided with this paper.
Code availability
All the code relevant to this work is available from the corresponding authors upon reasonable request.
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Acknowledgements
We acknowledge C. Pan and L. Han for the dilution refrigerator technique support, X. Deng and Y. Lu for their insightful discussions, Y. Deng and M. Gao for their help polishing the figures and L. W. Zhang for assistance with the references. This work was supported by the National Natural Science Foundation of China (grant numbers 92165210 to Y.H., 62174076 to Y.H., 11904157 to P.H. and 12275117 to T.X.), Quantum Science and Technology—National Science and Technology Major Project (numbers 2021ZD0302300 to Y.H. and 2024ZD0300400 to T.X.), Shenzhen Science and Technology Program (grant number KQTD20200820113010023 to Y.H.), Guangdong Basic and Applied Basic Research Foundation (grant numbers 2022A1515011348 to T.P. and 2022B1515020074 to T.X.).
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Z.T., H.W., Y.-N.Z., B.Z., H.S., J.L., K.W., K.S., M.D. and T.P. fabricated the device, under the supervision of G.W., Y.H., S.L. and D.Y. C.Z., C.L., Y.J., F.X., X.B., Y.-F.Z. and C.H. performed the experiments and analysed the data, under the supervision of G.H. and Y.H. S.Z. developed the tools to calculate the quantum dynamics for error detection and error correction, under the supervision of P.H. F.X. developed and applied computational tools to calculate gate benchmarking and tomography, under the supervision of T.X. C.L., C.Z., G.H. and Y.H. wrote the manuscript, with input from all co-authors. The manuscript was revised by all authors.
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Extended data
Extended Data Fig. 1 Rabi oscillation and Ramsey interferometry measurement for the electron.
The Rabi oscillation and Ramsey interferometry for the electron are characterised with the nuclear spins initialised in \(| {\Downarrow \Downarrow \Downarrow \Uparrow \rangle}\) state. a, For Rabi measurement, a resonance ESR pulse with varying width is applied, and the extracted Rabi frequency is indicated. b, For Ramsey measurement, the detuning frequency is set at 100 kHz, and the integration time for each point is ∼128 s.
Extended Data Fig. 2 Rabi oscillation and Ramsey interferometry measurement for four nuclear spin qubits.
We characterise the dephasing time \({T}_{2}^{* }\) for each nuclear spin qubit using the Ramsey interferometry experiment. Here we sweep the evolution time between two π/2 pulses, and the oscillations are obtained by tuning the phase of the final π/2 pulse with τ at detunings of 7 kHz, 8 kHz, 5 kHz, and 50 Hz, respectively. The measurement sequence is repeated 40 times for each τ, and 10 repetitions are performed for each nuclear spin. The equivalent integration time for τ is around 3 min for the qubits N1N2N3, and 12 min for N4. The average data is fitted by the function \(P(t)=A{\mathrm{exp}}{(-t/{T}_{2}^{*})}^{2}\cos (2{\uppi} \Delta ft+\phi )+B\), where Δf is the frequency detuning and ϕ is the initial phase, yielding dephasing times of \({T}_{2,{{\rm{N}}}_{1}}^{* }=441\pm 11\,\upmu {\rm{s}},\) \({T}_{2,{{\rm{N}}}_{2}}^{* }=349\pm 8\,\upmu {\rm{s}},\) \({T}_{2,{{\rm{N}}}_{3}}^{* }=788\pm 23\,\upmu {\rm{s}},\) \({T}_{2,{{\rm{N}}}_{4}}^{* }=24.8\pm 0.8\,{\mathrm{ms}}\). The uncertainties for \({T}_{2}^{* }\) are derived by the bootstrap resampling and are denoted at the 1σ confidence level. For N4, the long dephasing time is likely due to the small hyperfine interaction of A4 = 226.0 kHz. a, Rabi oscillation measurement results for each nucleus, performed using RF powers matching those applied in the quantum circuits of the main experiments. b, Ramsey interferometry measurement results for each nucleus.
Extended Data Fig. 3 Single-qubit gate Randomised Benchmarking.
We characterise the single-qubit gate fidelity for four nuclear spins N1 ~ N4 by performing a Randomised Benchmarking (RB) experiment. First, nuclear spins are initialised to the \(| {\Downarrow \Downarrow \Downarrow \Uparrow \Uparrow \Uparrow \rangle}\) state. For each qubit we apply a sequence of n Clifford gates which are randomly chosen from the single-qubit Clifford group, followed by a final recovery gate to return the target qubit to its initial state. Each sequence of length n is repeated 100 times, and the results are averaged over 9 randomised sequences. Data are presented as mean ± s.d. We measure the recovery probability as a function of the number of Clifford gates, n, following the decay model \({P}_{\Downarrow }(n)=A{p}_\mathrm{C}^{n}+B\), where A and B are constants that account for the state preparation and measurement (SPAM) errors. The depolarizing parameter pC is related to the average Clifford gate fidelity FC = (1 + pC)/2. RB measurements performed on the nuclear spin qubits N1 ~ N4 within their respective single-qubit subspaces yield Clifford gate fidelities of 99.37 ± 0.03%, 99.73 ± 0.02%, 99.84 ± 0.01%, and 99.33 ± 0.04%, respectively. The fidelity uncertainties are derived by the bootstrap resampling and denoted at the 1σ confidence level. a, The experimental circuit for single-qubit randomised benchmarking. All 24 single-qubit Clifford gates are generated using Xπ/2 and virtual Zπ/2 gates. b, Single-qubit randomised benchmarking measurement results for four nuclear spins N1 ~ N4.
Extended Data Fig. 4 Two-qubit Randomised Benchmarking.
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.
Extended Data Fig. 5 Bell states between each pair of the nuclear spins.
a-f, the tomography of Bell state \(| {\Phi }^{+}\rangle\) for nuclear spin pairs N1N2, N1N3, N1N4, N2N3, N2N4, and N3N4, with measured Bell state fidelities of \({F}_{| {\Phi }^{+}\rangle ,12}=95.2\pm 1.2 \% ,\)\(\,{F}_{| {\Phi }^{+}\rangle ,13}=96.3\pm 1.1 \% ,\)\(\,{F}_{| {\Phi }^{+}\rangle ,14}=91.6\pm 1.6 \% ,\)\(\,{F}_{| {\Phi }^{+}\rangle ,23}=95.2\pm 1.0 \% ,\)\(\,{F}_{| {\Phi }^{+}\rangle ,24}=91.7\pm 1.5 \%\), and \({F}_{| {\Phi }^{+}\rangle ,34}=90.3\pm 1.2 \%\).
Extended Data Fig. 6 Bell state error correction from all the error subspaces.
a, Experimental circuit used to recover \(| {\Phi }^{+}\rangle\) from the four error subspaces. This is realised by preparing \(\frac{1}{\sqrt{2}}(| {\Downarrow \Downarrow \rangle} +| {\Downarrow \Uparrow \rangle} )=\frac{1}{2}(| {\Phi }^{+}\rangle +| {\Phi }^{-}\rangle +| {\Psi }^{+}\rangle +| {\Psi }^{-}\rangle )\), which is an equal superposition of the four Bell states. Quantum state tomography is then performed on the data qubits corresponding to different error syndromes, where nearly equal probabilities are observed for the four possible measurement outcomes of the syndromes: {0,0}, {0,1}, {1,0}, and {1,1}. The data qubits are mapped to corresponding Bell state with fidelities of \({F}_{| {\Phi }^{+}\rangle }=81.6\pm 2.6 \% ,{F}_{| {\Psi }^{+}\rangle }=80.6\pm 2.7 \% ,{F}_{| {\Phi }^{-}\rangle }=75.7\pm 2.8 \%\), and \({F}_{| {\Psi }^{-}\rangle }=80.4\pm 2.9 \%\), respectively, as illustrated in b-e. Based on the error syndromes, the error state ρerror can be corrected by applying a corresponding single-qubit operation U through ρcorrect = UρerrorU† by postprocessing. As shown in f, \(| {\Phi }^{+}\rangle\) is finally recovered with a fidelity of \({F}_{| {\Phi }^{+}\rangle }=79.0\pm 1.4 \%\).
Extended Data Fig. 7 Nuclear spin lifetime.
a, Nuclear spin N1 lifetime measurement, which is realised by toggling the two ESR frequencies corresponding to \(| {\Uparrow \rangle}\) and \(| {\Downarrow} \rangle\) of N1, followed by electron spin single-shot measurement. This procedure essentially constitutes a nuclear spin quantum non-demolition measurement. Each data point represents the electron spin-up fraction difference Δf↑, comparing \(| {\Uparrow \rangle}\) and \(| {\Downarrow \rangle}\). Each spin-up fraction is averaged over 40 single-shot measurements and conducted over ~ 80 ms. The red dotted line displays the discrimination threshold for determining whether the nuclear is in the \(| {\Uparrow \rangle}\) or \(| {\Downarrow \rangle}\). b, The quantum jumps58 are observed throughout the measurement, with the time resolution limited by the electron spin measurement speed. In the histogram of integrated signal versus Δf↑, the two major peaks fitted by two Gaussian curves correspond to the two nuclear spin states. The resulting error rate for nuclear spin discrimination is drawn in c, giving an error rate less than 0.1% for both \(| {\Uparrow \rangle}\) and \(| {\Downarrow \rangle}\) states assertions, with a discrimination threshold around -0.2. d, The lifetimes for \(| {\Uparrow \rangle}\) and \(| {\Downarrow \rangle}\) of N1 are measured over 6 hours, which is fitted using P(t) = Aexp(− t/T1) + B, yielding \({T}_{| \Uparrow \rangle }^{\rm{N}_{1}} \sim 174\,\rm{s}\) and \({T}_{| \Downarrow \rangle }^{\rm{N}_{1}} \sim 300\,\rm{s}\). The lifetimes of nuclear spins N2 and N3 are characterised similarly, giving \({T}_{| \Uparrow \rangle }^{\rm{N}_{2}} \sim 100\,\rm{s}\) and \({T}_{| \Downarrow \rangle }^{\rm{N}_{2}} \sim 196\,\rm{s}\), \({T}_{| \Uparrow \rangle }^{\rm{N}_{3}} \sim 153\,\rm{s}\) and \({T}_{| \Downarrow \rangle }^{\rm{N}_{3}} \sim 144\,\rm{s}\), and the nuclear spin flips are mostly caused by the ionization process during electron spin readout8.
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Zhang, C., Li, C., Tian, Z. et al. Quantum error detection in a silicon quantum processor. Nat Electron 9, 295–303 (2026). https://doi.org/10.1038/s41928-025-01557-1
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DOI: https://doi.org/10.1038/s41928-025-01557-1
