Quantum random-number generators (QRNGs) can offer a means to generate information-theoretically provable random numbers, in principle. In practice, unfortunately, the quantum randomness is inevitably mixed with classical randomness due to classical noises. To distill this quantum randomness, one needs to quantify the randomness of the source and apply a randomness extractor. Here, we propose a generic framework for evaluating quantum randomness of real-life QRNGs by min-entropy, and apply it to two different existing quantum random-number systems in the literature. Moreover, we provide a guideline of QRNG data postprocessing for which we implement two information-theoretically provable randomness extractors: Toeplitz-hashing extractor and Trevisan's extractor.

Variational quantum algorithm (VQA) accesses the centralized data to train the model, and using distributed computing can significantly improve the training overhead; however, the data is privacy sensitive. In this paper, we propose communication-efficient learning of VQA from decentralized data, which is so-called quantum federated learning (QFL). Motivated by the classical federated learning algorithm, we improve data privacy by aggregating updates from local computation to share model parameters. Here, aiming to find approximate optima in the parameter landscape, we develop an extension of the conventional VQA. Finally, we deploy on the TensorFlow Quantum processor within variational quantum tensor networks classifiers, approximate quantum optimization for the Ising model, and variational quantum eigensolver for molecular hydrogen. Our algorithm demonstrates model accuracy from decentralized data, which have higher performance on near-term processors. Importantly, QFL may inspire new investigations in the field of secure quantum machine learning.

We construct two families of orthogonal product quantum states that cannot be exactly distinguished by local operation and classical communication (LOCC) in the quantum system of "Equation missing"<!-- image only, no MathML or LaTex -->^2k+i ⊗ "Equation missing"<!-- image only, no MathML or LaTex -->^2l+j (i, j ∈ {0, 1} and i ≥ j ) and "Equation missing"<!-- image only, no MathML or LaTex -->^3k+i ⊗ "Equation missing"<!-- image only, no MathML or LaTex -->^3l+j (i, j ∈ {0, 1, 2}). And we also give the tiling structure of these two families of quantum product states where the quantum states are unextendible in the first family but are extendible in the second family. Our construction in the quantum system of "Equation missing"<!-- image only, no MathML or LaTex -->^3k+i ⊗ "Equation missing"<!-- image only, no MathML or LaTex -->^3l+j is more generalized than the other construction such as Wang et al.’s construction and Zhang et al.’s construction, because it contains the quantum system of not only "Equation missing"<!-- image only, no MathML or LaTex -->^2k ⊗ "Equation missing"<!-- image only, no MathML or LaTex -->^2l and "Equation missing"<!-- image only, no MathML or LaTex -->^2k+1 ⊗ "Equation missing"<!-- image only, no MathML or LaTex -->^2l but also "Equation missing"<!-- image only, no MathML or LaTex -->^2k ⊗ "Equation missing"<!-- image only, no MathML or LaTex -->^2l+1 and "Equation missing"<!-- image only, no MathML or LaTex -->^2k+1 ⊗ "Equation missing"<!-- image only, no MathML or LaTex -->^2l+1. We calculate the non-commutativity to quantify the quantumness of a quantum ensemble for judging the local indistinguishability. We give a general method to judge the indistinguishability of orthogonal product states for our two constructions in this paper. We also extend the dimension of the quantum system of "Equation missing"<!-- image only, no MathML or LaTex -->^2k ⊗ "Equation missing"<!-- image only, no MathML or LaTex -->^2l in Wang et al.’s paper. Our work is a necessary complement to understand the phenomenon of quantum nonlocality without entanglement.

Variational quantum algorithms (VQAs) use classical computers as the quantum outer loop optimizer and update the circuit parameters to obtain an approximate ground state. In this article, we present a meta-learning variational quantum algorithm (meta-VQA) by recurrent unit, which uses a technique called “meta-learner.” Motivated by the hybrid quantum-classical algorithms, we train classical recurrent units to assist quantum computing, learning to find approximate optima in the parameter landscape. Here, aiming to reduce the sampling number more efficiently, we use the quantum stochastic gradient descent method and introduce the adaptive learning rate. Finally, we deploy on the TensorFlow Quantum processor within approximate quantum optimization for the Ising model and variational quantum eigensolver for molecular hydrogen (H2), lithium hydride (LiH), and helium hydride cation (HeH+). Our algorithm can be expanded to larger system sizes and problem instances, which have higher performance on near-term processors.