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<jats:title>Abstract</jats:title><jats:p>The central technological appeal of quantum science resides in exploiting quantum effects, such as entanglement, for a variety of applications, including computing, communication and sensing<jats:sup>1</jats:sup>. The overarching challenge in these fields is to address, control and protect systems of many qubits against decoherence<jats:sup>2</jats:sup>. Against this backdrop, optical photons, naturally robust and easy to manipulate, represent ideal qubit carriers. However, the most successful technique so far for creating photonic entanglement<jats:sup>3</jats:sup> is inherently probabilistic and, therefore, subject to severe scalability limitations. Here we report the implementation of a deterministic protocol<jats:sup>4–6</jats:sup> for the creation of photonic entanglement with a single memory atom in a cavity<jats:sup>7</jats:sup>. We interleave controlled single-photon emissions with tailored atomic qubit rotations to efficiently grow Greenberger–Horne–Zeilinger (GHZ) states<jats:sup>8</jats:sup> of up to 14 photons and linear cluster states<jats:sup>9</jats:sup> of up to 12 photons with a fidelity lower bounded by 76(6)% and 56(4)%, respectively. Thanks to a source-to-detection efficiency of 43.18(7)% per photon, we measure these large states about once every minute, which is orders of magnitude faster than in any previous experiment<jats:sup>3,10–13</jats:sup>. In the future, this rate could be increased even further, the scheme could be extended to two atoms in a cavity<jats:sup>14,15</jats:sup> or several sources could be quantum mechanically coupled<jats:sup>16</jats:sup>, to generate higher-dimensional cluster states<jats:sup>17</jats:sup>. Overcoming the limitations encountered by probabilistic schemes for photonic entanglement generation, our results may offer a way towards scalable measurement-based quantum computation<jats:sup>18,19</jats:sup> and communication<jats:sup>20,21</jats:sup>.</jats:p>

<jats:title>Abstract</jats:title><jats:p>Future large-scale quantum computers will rely on quantum error correction (QEC) to protect the fragile quantum information during computation<jats:sup>1,2</jats:sup>. Among the possible candidate platforms for realizing quantum computing devices, the compatibility with mature nanofabrication technologies of silicon-based spin qubits offers promise to overcome the challenges in scaling up device sizes from the prototypes of today to large-scale computers<jats:sup>3–5</jats:sup>. Recent advances in silicon-based qubits have enabled the implementations of high-quality one-qubit and two-qubit systems<jats:sup>6–8</jats:sup>. However, the demonstration of QEC, which requires three or more coupled qubits<jats:sup>1</jats:sup>, and involves a three-qubit gate<jats:sup>9–11</jats:sup> or measurement-based feedback, remains an open challenge. Here we demonstrate a three-qubit phase-correcting code in silicon, in which an encoded three-qubit state is protected against any phase-flip error on one of the three qubits. The correction to this encoded state is performed by a three-qubit conditional rotation, which we implement by an efficient single-step resonantly driven iToffoli gate. As expected, the error correction mitigates the errors owing to one-qubit phase-flip, as well as the intrinsic dephasing mainly owing to quasi-static phase noise. These results show successful implementation of QEC and the potential of a silicon-based platform for large-scale quantum computing.</jats:p>