Photonics is a promising platform for demonstrating a quantum computational advantage (QCA) by outperforming the most powerful classical supercomputers on a well-defined computational task. Despite this promise, existing proposals and demonstrations face challenges. Experimentally, current implementations of Gaussian boson sampling (GBS) lack programmability or have prohibitive loss rates. Theoretically, there is a comparative lack of rigorous evidence for the classical hardness of GBS. In this work, we make progress in improving both the theoretical evidence and experimental prospects. We provide evidence for the hardness of GBS, comparable to the strongest theoretical proposals for QCA. We also propose a QCA architecture we call high-dimensional GBS, which is programmable and can be implemented with low loss using few optical components. We show that particular algorithms for simulating GBS are outperformed by high-dimensional GBS experiments at modest system sizes. This work thus opens the path to demonstrating QCA with programmable photonic processors.

in the SQ model is often taken as strong evidence for hardness in the sample model.In summary, we study in this work the following problems, which are stated more formally in Section 3:Problems: PAC probabilistic modelling of quantum circuit Born machines (informal). Let C be the set of output distributions corresponding to a class of local quantum circuits. Given either sample-oracle or SQ-oracle access to some unknown distribution P ∈ C, output, with high probability, either generative modelling: an efficient generator, or density modelling: an efficient evaluator for a distribution P which is sufficiently close to P .If there exists either a sample or computationally efficient algorithm which, with respect to either the sample oracle or the SQ oracle, solves the generative (density) modelling problem associated with a given set of distributions C, then we say that C is sample or computationally efficiently generator (evaluator) learnable within the relevant oracle model. We are particularly interested in this work in establishing the existence or non-existence, of efficient quantum or classical learning algorithms, for the output distributions of various classes of local quantum circuits, within both the sample and statistical query model. Main resultsGiven this motivation and context, we provide two main results, which stated informally, are as follows:Result 1 (Informal version of Corollary 1). The concept class consisting of the output distributions of super-logarithmic depth nearest neighbour Clifford circuits is not sample efficiently PAC generator-learnable or evaluator-learnable, in the statistical query model.Result 2 (Informal version of Theorem 2). The concept class consisting of the output distributions of nearest neighbour Clifford circuits is both sample and computationally efficiently classically PAC generator-learnable and evaluator-learnable, in the sample model.

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