Motivated by the recent experimental demonstrations of quantum supremacy, proving the hardness of the output of random quantum circuits is an imperative near term goal. We prove under the complexity theoretical assumption of the non-collapse of the polynomial hierarchy that approximating the output probabilities of random quantum circuits to within$\exp(-\Omega(m\log m))$additive error is hard for any classical computer, where$m$is the number of gates in the quantum computation. More precisely, we show that the above problem is #P-hard under BPPNPreduction. In the recent experiments, the quantum circuit has n-qubits and the architecture is a two-dimensional grid of size$\sqrt{n}\times\sqrt{n}$[1]. Indeed for constant depth circuits approximating the output probabilities to within$2^{-\Omega(n\log n)}$is hard. For circuits of depth$\log n$or$\sqrt{n}$for which the anti-concentration property holds, approximating the output probabilities to within$2^{-\Omega(n\log^{2}n)}$and$2^{-\Omega(n^{3/2}\log n)}$is hard respectively. We then show that the hardness results extend to any open neighborhood of an arbitrary (fixed) circuit including the trivial circuit with identity gates. We made an effort to find the best proofs and proved these results from first principles, which do not use the standard techniques such as the Berlekamp–Welch algorithm, the usual Paturi's lemma, and Rakhmanov's result.

The project, design and development of the diachronic corpus of Japanese began in 2009 at the Department of Corpus Study, the National Institute of Japanese Language and Linguistics, Japan (NINJAL), as a collaborative research project by linguists and literature scholars of NINJAL and the University of Oxford. Its focus is on collecting representative Japanese literary works and classical documents from the tenth century to the nineteenth century. We are currently working on the development of the prototype version of the diachronic Japanese corpus: i.e. selection of materials, digitization of texts, addition of alternative texts (containing different orthography) to original texts, compilation of a basic thesaurus that differentiates between different spellings, and word segmentation. This paper addresses the discussion of the basic concepts encountered during our work on the project: synchronic and diachronic analysis, which led us to the design of a serial comparison model which allows us to examine language change between documents or literary works with respect to time.