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<jats:title>Abstract</jats:title><jats:p>Quantum many-body systems away from equilibrium host a rich variety of exotic phenomena that are forbidden by equilibrium thermodynamics. A prominent example is that of discrete time crystals<jats:sup>1–8</jats:sup>, in which time-translational symmetry is spontaneously broken in periodically driven systems. Pioneering experiments have observed signatures of time crystalline phases with trapped ions<jats:sup>9,10</jats:sup>, solid-state spin systems<jats:sup>11–15</jats:sup>, ultracold atoms<jats:sup>16,17</jats:sup> and superconducting qubits<jats:sup>18–20</jats:sup>. Here we report the observation of a distinct type of non-equilibrium state of matter, Floquet symmetry-protected topological phases, which are implemented through digital quantum simulation with an array of programmable superconducting qubits. We observe robust long-lived temporal correlations and subharmonic temporal response for the edge spins over up to 40 driving cycles using a circuit of depth exceeding 240 and acting on 26 qubits. We demonstrate that the subharmonic response is independent of the initial state, and experimentally map out a phase boundary between the Floquet symmetry-protected topological and thermal phases. Our results establish a versatile digital simulation approach to exploring exotic non-equilibrium phases of matter with current noisy intermediate-scale quantum processors<jats:sup>21</jats:sup>.</jats:p>

<jats:title>Abstract</jats:title><jats:p>Understanding the direct transformation from graphite to diamond has been a long-standing challenge with great scientific and practical importance. Previously proposed transformation mechanisms<jats:sup>1–3</jats:sup>, based on traditional experimental observations that lacked atomistic resolution, cannot account for the complex nanostructures occurring at graphite−diamond interfaces during the transformation<jats:sup>4,5</jats:sup>. Here we report the identification of coherent graphite−diamond interfaces, which consist of four basic structural motifs, in partially transformed graphite samples recovered from static compression, using high-angle annular dark-field scanning transmission electron microscopy. These observations provide insight into possible pathways of the transformation. Theoretical calculations confirm that transformation through these coherent interfaces is energetically favoured compared with those through other paths previously proposed<jats:sup>1–3</jats:sup>. The graphite-to-diamond transformation is governed by the formation of nanoscale coherent interfaces (diamond nucleation), which, under static compression, advance to consume the remaining graphite (diamond growth). These results may also shed light on transformation mechanisms of other carbon materials and boron nitride under different synthetic conditions.</jats:p>

<jats:title>Abstract</jats:title><jats:p>To impart directionality to the motions of a molecular mechanism, one must overcome the random thermal forces that are ubiquitous on such small scales and in liquid solution at ambient temperature. In equilibrium without energy supply, directional motion cannot be sustained without violating the laws of thermodynamics. Under conditions away from thermodynamic equilibrium, directional motion may be achieved within the framework of Brownian ratchets, which are diffusive mechanisms that have broken inversion symmetry<jats:sup>1–5</jats:sup>. Ratcheting is thought to underpin the function of many natural biological motors, such as the F<jats:sub>1</jats:sub>F<jats:sub>0</jats:sub>-ATPase<jats:sup>6–8</jats:sup>, and it has been demonstrated experimentally in synthetic microscale systems (for example, to our knowledge, first in ref. <jats:sup>3</jats:sup>) and also in artificial molecular motors created by organic chemical synthesis<jats:sup>9–12</jats:sup>. DNA nanotechnology<jats:sup>13</jats:sup> has yielded a variety of nanoscale mechanisms, including pivots, hinges, crank sliders and rotary systems<jats:sup>14–17</jats:sup>, which can adopt different configurations, for example, triggered by strand-displacement reactions<jats:sup>18,19</jats:sup> or by changing environmental parameters such as pH, ionic strength, temperature, external fields and by coupling their motions to those of natural motor proteins<jats:sup>20–26</jats:sup>. This previous work and considering low-Reynolds-number dynamics and inherent stochasticity<jats:sup>27,28</jats:sup> led us to develop a nanoscale rotary motor built from DNA origami that is driven by ratcheting and whose mechanical capabilities approach those of biological motors such as F<jats:sub>1</jats:sub>F<jats:sub>0</jats:sub>-ATPase.</jats:p>