Periodic Reporting for period 2 - QUNET (A quantum network for distributed quantum computation) Reporting period: 2021-05-01 to 2022-10-31 Summary of the context and overall objectives of the project A key question for quantum science is: Can quantum systems be protected from decoherence? This is not just a fundamental question; if we can reliably control large quantum states, it becomes possible to perform certain important computational and information tasks that go beyond classical physics. A promising approach is to protect quantum states by distributing quantum error correction over quantum networks. These networks consist of nodes that contain quantum bits to store and process quantum states, and that are connected by entanglement links based on photons. This approach is naturally scalable to larger sizes by connecting independent modules.The goal of this project is to realize a proof-of-principle demonstration for such an error-corrected distributed quantum system. Our qubits are based on an atomic defect in diamond, the nitrogen-vacancy center. We use the electron spin of this defect to control multiple nuclear spin qubits in its vicinity. These spin qubits in diamond combine high operation temperatures, good quantum coherence and the ability to optically link them together into a quantum network. The final goal of this project is to realize a system consisting of multiple nodes, and demonstrate a quantum error detection distributed over that small-scale network. This network functions as a unit cell for large-scale quantum computation. Reaching this goal will be a potentially decisive step towards large quantum networks and distributed quantum computations: we will approach a new territory in which quantum states can be made more stable by making networks larger and larger, ultimately completely overcoming decoherence. Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far In the project so far, we have realized important building blocks for distributed quantum computing networks. First, we have realized quantum memory that can store quantum states robustly [Phys. Rev. X 12, 011048, 2022, NPJ Quantum Information 8, 122, 2022]. This is a key advance as faithfully storing quantum states while establishing new network links is the way to connect nodes into larger networks and to overcome imperfections. Second, we improved our understanding and control over the quantum bits. To demonstrate this we have realized the fault-tolerant operation of a logical qubit [Nature 606, 884, 2022] and the first quantum simulation of a time crystal, a new phase of matter [Science 374, 1474, 2021]. Together, these results set the stage for high quality control over multi-qubit systems that can be connected optically into a quantum network, and thus lay the foundation for the second part of this project. Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far) The targeted demonstration of a unit cell of a distributed quantum computer would be the first of its kind. It would serve as a proof-of-principle demonstration that lays out a blueprint for overcoming imperfections and decoherence, ultimately enabling large-scale quantum computation distributed over quantum networks.