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<jats:title>Abstract</jats:title><jats:p>The dispersive sweep of fast radio bursts (FRBs) has been used to probe the ionized baryon content of the intergalactic medium<jats:sup>1</jats:sup>, which is assumed to dominate the total extragalactic dispersion. Although the host-galaxy contributions to the dispersion measure appear to be small for most FRBs<jats:sup>2</jats:sup>, in at least one case there is evidence for an extreme magneto-ionic local environment<jats:sup>3,4</jats:sup> and a compact persistent radio source<jats:sup>5</jats:sup>. Here we report the detection and localization of the repeating FRB 20190520B, which is co-located with a compact, persistent radio source and associated with a dwarf host galaxy of high specific-star-formation rate at a redshift of 0.241 ± 0.001. The estimated host-galaxy dispersion measure of approximately <jats:inline-formula><jats:alternatives><jats:tex-math>$${903}_{-111}^{+72}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mrow> <mml:mn>903</mml:mn> </mml:mrow> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>111</mml:mn> </mml:mrow> <mml:mrow> <mml:mo>+</mml:mo> <mml:mn>72</mml:mn> </mml:mrow> </mml:msubsup> </mml:math></jats:alternatives></jats:inline-formula> parsecs per cubic centimetre, which is nearly an order of magnitude higher than the average of FRB host galaxies<jats:sup>2,6</jats:sup>, far exceeds the dispersion-measure contribution of the intergalactic medium. Caution is thus warranted in inferring redshifts for FRBs without accurate host-galaxy identifications.</jats:p>

<jats:title>Abstract</jats:title><jats:p>Helium-3 has nowadays become one of the most important candidates for studies in fundamental physics<jats:sup>1–3</jats:sup>, nuclear and atomic structure<jats:sup>4,5</jats:sup>, magnetometry and metrology<jats:sup>6</jats:sup>, as well as chemistry and medicine<jats:sup>7,8</jats:sup>. In particular, <jats:sup>3</jats:sup>He nuclear magnetic resonance (NMR) probes have been proposed as a new standard for absolute magnetometry<jats:sup>6,9</jats:sup>. This requires a high-accuracy value for the <jats:sup>3</jats:sup>He nuclear magnetic moment, which, however, has so far been determined only indirectly and with a relative precision of 12 parts per billon<jats:sup>10,11</jats:sup>. Here we investigate the <jats:sup>3</jats:sup>He<jats:sup>+</jats:sup> ground-state hyperfine structure in a Penning trap to directly measure the nuclear <jats:italic>g</jats:italic>-factor of <jats:sup>3</jats:sup>He<jats:sup>+</jats:sup><jats:inline-formula><jats:alternatives><jats:tex-math>$${g}_{I}^{{\prime} }=-\,4.2550996069(30{)}_{{\rm{stat}}}(17{)}_{{\rm{sys}}}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msubsup> <mml:mrow> <mml:mi>g</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>I</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>′</mml:mo> </mml:mrow> </mml:msubsup> <mml:mo>=</mml:mo> <mml:mo>−</mml:mo> <mml:mspace /> <mml:mn>4.2550996069</mml:mn> <mml:mo>(</mml:mo> <mml:mn>30</mml:mn> <mml:msub> <mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow> <mml:mi>stat</mml:mi> </mml:mrow> </mml:msub> <mml:mo>(</mml:mo> <mml:mn>17</mml:mn> <mml:msub> <mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow> <mml:mi>sys</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula>, the zero-field hyperfine splitting <jats:inline-formula><jats:alternatives><jats:tex-math>$${E}_{{\rm{HFS}}}^{\exp }=-\,8,\,665,\,649,\,865.77{(26)}_{{\rm{stat}}}{(1)}_{{\rm{sys}}}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msubsup> <mml:mrow> <mml:mi>E</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>HFS</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>exp</mml:mi> </mml:mrow> </mml:msubsup> <mml:mo>=</mml:mo> <mml:mo>−</mml:mo> <mml:mspace /> <mml:mn>8</mml:mn> <mml:mo>,</mml:mo> <mml:mspace /> <mml:mn>665</mml:mn> <mml:mo>,</mml:mo> <mml:mspace /> <mml:mn>649</mml:mn> <mml:mo>,</mml:mo> <mml:mspace /> <mml:mn>865.77</mml:mn> <mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mrow> <mml:mn>26</mml:mn> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow> <mml:mi>stat</mml:mi> </mml:mrow> </mml:msub> <mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow> <mml:mi>sys</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula> Hz and the bound electron <jats:italic>g</jats:italic>-factor <jats:inline-formula><jats:alternatives><jats:tex-math>$${g}_{e}^{{\rm{\exp }}}=-\,2.00217741579(34{)}_{{\rm{stat}}}(30{)}_{{\rm{sys}}}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msubsup> <mml:mrow> <mml:mi>g</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>e</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>exp</mml:mi> </mml:mrow> </mml:msubsup> <mml:mo>=</mml:mo> <mml:mo>−</mml:mo> <mml:mspace /> <mml:mn>2.00217741579</mml:mn> <mml:mo>(</mml:mo> <mml:mn>34</mml:mn> <mml:msub> <mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow> <mml:mi>stat</mml:mi> </mml:mrow> </mml:msub> <mml:mo>(</mml:mo> <mml:mn>30</mml:mn> <mml:msub> <mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow> <mml:mi>sys</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula>. The latter is consistent with our theoretical value <jats:inline-formula><jats:alternatives><jats:tex-math>$${g}_{e}^{{\rm{theo}}}=-\,2.00217741625223(39)$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msubsup> <mml:mrow> <mml:mi>g</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>e</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>theo</mml:mi> </mml:mrow> </mml:msubsup> <mml:mo>=</mml:mo> <mml:mo>−</mml:mo> <mml:mspace /> <mml:mn>2.00217741625223</mml:mn> <mml:mo>(</mml:mo> <mml:mn>39</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula> based on parameters and fundamental constants from ref. <jats:sup>12</jats:sup>. Our measured value for the <jats:sup>3</jats:sup>He<jats:sup>+</jats:sup> nuclear <jats:italic>g</jats:italic>-factor enables determination of the <jats:italic>g</jats:italic>-factor of the bare nucleus <jats:inline-formula><jats:alternatives><jats:tex-math>$${g}_{I}=-\,4.2552506997(30{)}_{{\rm{stat}}}(17{)}_{{\rm{sys}}}(1{)}_{{\rm{theo}}}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi>g</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>I</mml:mi> </mml:mrow> </mml:msub> <mml:mo>=</mml:mo> <mml:mo>−</mml:mo> <mml:mspace /> <mml:mn>4.2552506997</mml:mn> <mml:mo>(</mml:mo> <mml:mn>30</mml:mn> <mml:msub> <mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow> <mml:mi>stat</mml:mi> </mml:mrow> </mml:msub> <mml:mo>(</mml:mo> <mml:mn>17</mml:mn> <mml:msub> <mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow> <mml:mi>sys</mml:mi> </mml:mrow> </mml:msub> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:msub> <mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow> <mml:mi>theo</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula> via our accurate calculation of the diamagnetic shielding constant<jats:sup>13</jats:sup><jats:inline-formula><jats:alternatives><jats:tex-math>$${\sigma }_{{}^{3}{\mathrm{He}}^{+}}=0.00003550738(3)$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi>σ</mml:mi> </mml:mrow> <mml:mrow> <mml:msup> <mml:mrow /> <mml:mn>3</mml:mn> </mml:msup> <mml:msup> <mml:mrow> <mml:mi>He</mml:mi> </mml:mrow> <mml:mo>+</mml:mo> </mml:msup> </mml:mrow> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>0.00003550738</mml:mn> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>3</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula>. This constitutes a direct calibration for <jats:sup>3</jats:sup>He NMR probes and an improvement of the precision by one order of magnitude compared to previous indirect results. The measured zero-field hyperfine splitting improves the precision by two orders of magnitude compared to the previous most precise value<jats:sup>14</jats:sup> and enables us to determine the Zemach radius<jats:sup>15</jats:sup> to <jats:inline-formula><jats:alternatives><jats:tex-math>$${r}_{Z}=2.608(24)$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi>r</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>Z</mml:mi> </mml:mrow> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>2.608</mml:mn> <mml:mo>(</mml:mo> <mml:mn>24</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula> fm.</jats:p>

<jats:title>Abstract</jats:title><jats:p>Solid-state spin qubits is a promising platform for quantum computation and quantum networks<jats:sup>1,2</jats:sup>. Recent experiments have demonstrated high-quality control over multi-qubit systems<jats:sup>3–8</jats:sup>, elementary quantum algorithms<jats:sup>8–11</jats:sup> and non-fault-tolerant error correction<jats:sup>12–14</jats:sup>. Large-scale systems will require using error-corrected logical qubits that are operated fault tolerantly, so that reliable computation becomes possible despite noisy operations<jats:sup>15–18</jats:sup>. Overcoming imperfections in this way remains an important outstanding challenge for quantum science<jats:sup>15,19–27</jats:sup>. Here, we demonstrate fault-tolerant operations on a logical qubit using spin qubits in diamond. Our approach is based on the five-qubit code with a recently discovered flag protocol that enables fault tolerance using a total of seven qubits<jats:sup>28–30</jats:sup>. We encode the logical qubit using a new protocol based on repeated multi-qubit measurements and show that it outperforms non-fault-tolerant encoding schemes. We then fault-tolerantly manipulate the logical qubit through a complete set of single-qubit Clifford gates. Finally, we demonstrate flagged stabilizer measurements with real-time processing of the outcomes. Such measurements are a primitive for fault-tolerant quantum error correction. Although future improvements in fidelity and the number of qubits will be required to suppress logical error rates below the physical error rates, our realization of fault-tolerant protocols on the logical-qubit level is a key step towards quantum information processing based on solid-state spins.</jats:p>

<jats:title>Abstract</jats:title><jats:p>Thermal insulation under extreme conditions requires materials that can withstand complex thermomechanical stress and retain excellent thermal insulation properties at temperatures exceeding 1,000 degrees Celsius<jats:sup>1–3</jats:sup>. Ceramic aerogels are attractive thermal insulating materials; however, at very high temperatures, they often show considerably increased thermal conductivity and limited thermomechanical stability that can lead to catastrophic failure<jats:sup>4–6</jats:sup>. Here we report a multiscale design of hypocrystalline zircon nanofibrous aerogels with a zig-zag architecture that leads to exceptional thermomechanical stability and ultralow thermal conductivity at high temperatures. The aerogels show a near-zero Poisson’s ratio (3.3 × 10<jats:sup>−4</jats:sup>) and a near-zero thermal expansion coefficient (1.2 × 10<jats:sup>−7</jats:sup> per degree Celsius), which ensures excellent structural flexibility and thermomechanical properties. They show high thermal stability with ultralow strength degradation (less than 1 per cent) after sharp thermal shocks, and a high working temperature (up to 1,300 degrees Celsius). By deliberately entrapping residue carbon species in the constituent hypocrystalline zircon fibres, we substantially reduce the thermal radiation heat transfer and achieve one of the lowest high-temperature thermal conductivities among ceramic aerogels so far—104 milliwatts per metre per kelvin at 1,000 degrees Celsius. The combined thermomechanical and thermal insulating properties offer an attractive material system for robust thermal insulation under extreme conditions.</jats:p>