## Periodic Reporting for period 1 - CV SUBUNIQC (Sub-Universal Quantum Circuits in Continuous Variables)

Reporting period: 2016-07-01 to 2018-06-30

"Three decades ago, it was proposed that quantum computers (i.e. quantum systems where information can be encoded, processed and read out) could outperform classical devices for information processing. For instance, they may allow the factoring of integer numbers in a time which scales polynomially with the size of the input, while known classical algorithms require an exponential time. Problems of this kind are said to be hard to classically solve, while an efficient quantum solution theoretically exists for some of them, e.g. for factoring. However, in practice, it has not yet been possible to build a quantum computer large enough to beat classical machines. This has raised the question as to whether this difficulty is only technical, and will be overcome one day, or whether there are fundamental reasons why this would not be possible. In trying to answer this question, physicists and computer scientists have developed ""sub-universal"" quantum computing models, which aim at solving very specific problems, simpler than factoring, but still displaying a quantum advantage. Among those is the so-called boson sampling protocol, which enables to compute the permanent of a unitary matrix (a matrix property analogous to the determinant). In other words, scientists now seek for the observation of a minimal supremacy of quantum computers over classical ones.

Inspired by recent experimental achievements (Paris, Japan, Virginia), the goal of my project has been studying at the theoretical level new models of sub-universal quantum computers, based on original photonic architectures. Indeed, these models were only poorly studied, so far, in the promising context of the ""Continuous Variable"" (CV) encoding, which has recently allowed to reach the record-size for quantum computing resource states. This project has articulated through two main objectives: 1) The design of new sub-universal quantum circuits in CV, providing proof of their classical computational hardness 2) The study of viable experimental quantum optics platforms where these protocols may be efficiently implemented.

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Inspired by recent experimental achievements (Paris, Japan, Virginia), the goal of my project has been studying at the theoretical level new models of sub-universal quantum computers, based on original photonic architectures. Indeed, these models were only poorly studied, so far, in the promising context of the ""Continuous Variable"" (CV) encoding, which has recently allowed to reach the record-size for quantum computing resource states. This project has articulated through two main objectives: 1) The design of new sub-universal quantum circuits in CV, providing proof of their classical computational hardness 2) The study of viable experimental quantum optics platforms where these protocols may be efficiently implemented.

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In the 18 months in which I have worked on my project, I was able to provide new protocols related to sub-universal model of quantum computation in continuous variables, that were not studied before. For instance, I have proposed the Continuous-Variable Instantaneous Quantum Computing model, which as I have demonstrated yields an output probability distribution that cannot be efficiently sampled from a classical computer. Furthermore, I have proposed a further photonics architecture, that I have called Continuous Variable Sampler, which outputs another probability distribution that cannot be efficiently sampled from a classical computer, and which furthermore is particularly friendly from the experimental perspective. Therefore, the latter architecture could be soon experimentally implemented, enabling testing quantum supremacy for computation.

"The project that I have carried is important for society as it takes a step and towards solving the long-date ""Shor’s trilemma"": because of the fast quantum factorisation algorithm mentioned above, proposed by P. Shor in 1994, at least one of the following three options must hold true: a) the Extended Church-Turing thesis is wrong (a foundational hypothesis in computer science, stating that any physical model of computation can be efficiently simulated on a classical computer, modelled by a Turing machine) b) textbook quantum mechanics, on which Shor quantum factorisation algorithm is based, is wrong c) there is a fast classical factoring algorithm. Which option is true is a question of both fundamental and technological importance at the global scale. Observing an efficient solution on a quantum hardware of problems that are not efficiently solvable on a classical one (such as boson sampling or other sub-universal models) would invalidate option a). My project had the primary goal of designing new sub-universal models and addressing their experimental implementation. The protocols that I have proposed could hence yield to the first-ever experimental evidence of a quantum advantage, validating option a) and thereby solving Shor’s trilemma. Implementation of a sub-universal quantum computer also represents a major technological advance, as specific problems (such as boson sampling and problems that map onto it, e.g. some Isingtype models) would become efficiently solvable."