Final Report Summary - REQS (Reliable quantum simulators) Many-body quantum technologies promise a variety of exciting applications in, for instance, ultra-precise quantum metrology, quantum-communication networks, quantum computing, and quantum simulators. In the quest for their large-scale realisation, impressive progress on a variety of physical platforms has recently been made. This fast pace of advance, however, makes the problem of reliable certification an increasingly pressing issue. The challenge consists of ensuring that quantum devices that are classically intractable in the ideal case perform as expected under realistic lab conditions. The problem of certification, thus, comes hand in hand with those of robustness under noise and computational hardness. From a practical viewpoint, further experimental progress on many-body quantum technologies is nowadays hindered by the lack of practical certification tools. At a fundamental level, in turn, certifying many-body quantum devices is ultimately about testing quantum mechanics in regimes where it has never been tested before. The long-term guiding vision of the project REQS has been to identify under which realistic experimental conditions quantum simulators (QSs) are reliable and if these include regimes of non-trivial classical tractability. More concretely, the specific objectives for the duration of the project have been:1) Robustness of QSs under imperfections: Our aim here has been to study the scaling behaviour, with the number of system components or lattice distance, of quantum simulations of many-body systems under the effects of realistic lab imperfections. We have also aimed at characterising the robustness of both collective properties and expectation values of local observables against imperfections, with emphasis on the classically nontrivial regimes. 2) Validation of QSs: Our main goal here has been to develop protocols for the quantum certification of realistic quantum simulations of many-body systems. We have also aimed at studying the limitations of certification using only classical resources.3)Computational power of QSs: Our objective here has been to characterise the classical hardness of quantum simulations of realistic many-body systems. We have focussed on classical simulatability results for quantum simulations that are classically hard in the ideal case when they are subject to lab imperfections.In all our studies, physical feasibility and experimental relevance have been essential. Our aim has been to develop theoretical-physics notions novel from a formal perspective but also useful to current experiments. In turn, physical concretions in collaboration with experimental groups have been achieved, as planned. In what follows, we describe in more detail the achievements of the full length of the project. During the two-year duration of the project, we have obtained concrete results on each and all of the three specific objectives of REQS.1) Robustness of QSs under imperfections: We have obtained theoretical results on the (lack of) robustness against imperfections of experimentally realistic quantum simulations of Boson-Sampling. Boson-Sampling is a classically computationally hard problem that can — in principle — be efficiently solved with quantum linear-optical networks. Apart from this, the fellow has first-authored an exhaustive review paper on open-system dynamics of entanglement. There, several aspects of the robustness/fragility of many-body entangled systems under local noise or decoherence are discussed in detail. These include formal scaling laws with the system size and characterisation of the differences in stability between collective properties, such as state fidelities or multi-partite entanglement, and expectation values of local observables. Besides, we have obtained results on the characterisation of quantum non-local correlations of many-body quantum states under local noise measuring only two-body correlations. In addition, we have developed the theory for an experimental study of the performance of a small-sized optical quantum simulation of the cooling process of a cluster-state Hamiltonian system, relevant for measurement-based quantum computing and quantum error correction, under realistic lab imperfections. Furthermore, we have also provided theory support for an experiment on the resilience of hybrid polarisation-orbital-angular-momentum photonic qubits against turbulence noise.2) Validation of QSs: We have both obtained a no-go result for a particular classical certification task and developed a complete toolbox of quantum certification for photonic quantum technologies. Namely, on the one hand, during the first half of the project, we were able to show that, with probability exponentially close to one in the number of bosons, no symmetric algorithm can distinguish the Boson-Sampling distribution from the uniform (flat) one from fewer than exponentially many samples. Symmetric algorithms define a restricted — yet relevant — class of classical distinguishers. This means that the two distributions are operationally indistinguishable under this class of distinguishers. We considered the prospects of using knowledge about the implemented unitary for devising non-symmetric algorithms that could potentially improve upon this. We conclude that, due to the very fact that Boson-Sampling is believed to be hard, efficient classical certification of Boson-Sampling devices seems to be out of reach even for non-symmetric algorithms. On the other hand, during the second half of the project, we have presented a major advance in the field of quantum certification. We have developed an experimentally friendly, yet mathematically rigorous, technique for the re liable quantum certification of a broad family of multi-mode photonic quantum simulators, including experimental preparations in certain non-Gaussian regimes. To this end, we provided formal definitions of quantum-state certification, derived a novel fidelity lower bound for generic multi-boson states and introduced the notion of non-Gaussian state nullifiers. This results constitute the central deliverable of the whole project. It will be soon published in Nature Communications and we believe it will become a key paper in many-body quantum certification, which is ultimately about testing quantum mechanics at large-scale regimes. 3) Computational power of QSs: We have obtained classical simulability results for the measurement-outcome probability distribution of quantum simulations of Boson-Sampling under realistic lab conditions. More precisely, we found that efficient classical 1-norm approximate sampling is possible if, for instance, the input Fock-basis state is replaced by a Gaussian state and the Fock-basis measurement by a bucket detector represented by a positive Wigner function. It is not claimed that the latter scenario matches exactly realistic experiments, but it does share many features. This confirms that a quantum simulation that is classically hard in the ideal scenario can, under realistic lab conditions, become classically tractable. In other words, the real experimental quantum simulation of a classically hard task can actually only be doing something that is efficiently simulatable with cheaper classical computing resources. REQS has successfully delivered results on each and all of its specific objectives. It has delivered a very important paper focussed exclusively on its most important goal, the validation of many-body quantum simulators, which will soon be published in Nature Communications, with the fellow as first author, and can become a key paper in the area. It has produced another paper that answers questions raised in all three goals, and several other papers that are directly or indirectly connected to at least one of the goals. Altogether, a total of nine papers were either published in peer reviewed scientific journals with international dissemination (including three in Nature Communications, one in Reports of Progress in Physics, one in Physical Review Letters, and one in Scientific Reports), or finished and submitted, during the entire two-year duration of the project. Furthermore, both new research projects about quantum validation as well as follow-ups of recently finished projects are now ongoing together with the host group and are expected to yield interesting new papers in a near future. In addition, apart from the strictly scientific outcomes, REQS has also had a big impact on science dissemination and a huge boost to the scientific career of the fellow. As for the former, during the project the fellow has presented his work in several international scientific events or research visits to renown research groups. As for the latter, in turn, during the project the fellow has secured a permanent professorship at the Universidade Federal do Rio de Janeiro, Brazil, where he is now based since February 2015 and from where he continues to collaborate intensely with European research groups.