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<jats:p>This review deals with restricted Boltzmann machine (RBM) under the light of statistical physics. The RBM is a classical family of machine learning (ML) models which played a central role in the development of deep learning. Viewing it as a spin glass model and exhibiting various links with other models of statistical physics, we gather recent results dealing with mean-field theory in this context. First the functioning of the RBM can be analyzed via the phase diagrams obtained for various statistical ensembles of RBM, leading in particular to identify a compositional phase where a small number of features or modes are combined to form complex patterns. Then we discuss recent works either able to devise mean-field based learning algorithms; either able to reproduce generic aspects of the learning process from some ensemble dynamics equations or/and from linear stability arguments.</jats:p>

<jats:p>We find by the wavelet transform that the classical plane light wave of linear polarization can be decomposed into a series of discrete Morlet wavelets. In the theoretical frame, the energy of the classical light wave becomes discrete; interestingly, the discretization is consistent with the energy division of <jats:italic>P</jats:italic> portions in Planck radiation theory, where <jats:italic>P</jats:italic> is an integer. It is shown that the changeable energy of a basic plane light wave packet or wave train is <jats:italic>H</jats:italic> <jats:sub>0<jats:italic>k</jats:italic> </jats:sub> = <jats:italic>np</jats:italic> <jats:sub>0<jats:italic>k</jats:italic> </jats:sub> <jats:italic>ω</jats:italic> (<jats:italic>n</jats:italic> = 1, 2, 3, … <jats:italic>k</jats:italic> = | <jats:italic> <jats:bold>k</jats:bold> </jats:italic>|), with discrete wavelet structure parameter <jats:italic>n</jats:italic>, wave vector <jats:italic> <jats:bold>k</jats:bold> </jats:italic> and idler frequency <jats:italic>ω</jats:italic>, and a constant <jats:italic>p</jats:italic> <jats:sub>0<jats:italic>k</jats:italic> </jats:sub>. The wave-particle duality from the Mach--Zehnder interference of single photons is simulated by using random basic plane light wave packets.</jats:p>

<jats:p>Quantum computers are in hot-spot with the potential to handle more complex problems than classical computers can. Realizing the quantum computation requires the universal quantum gate set {T, H, CNOT} so as to perform any unitary transformation with arbitrary accuracy. Here we first briefly review the Majorana fermions and then propose the realization of arbitrary two-qubit quantum gates based on chiral Majorana fermions. Elementary cells consist of a quantum anomalous Hall insulator surrounded by a topological superconductor with electric gates and quantum-dot structures, which enable the braiding operation and the partial exchange operation. After defining a qubit by four chiral Majorana fermions, the single-qubit T and H quantum gates are realized via one partial exchange operation and three braiding operations, respectively. The entangled CNOT quantum gate is performed by braiding six chiral Majorana fermions. Besides, we design a powerful device with which arbitrary two-qubit quantum gates can be realized and take the quantum Fourier transform as an example to show that several quantum operations can be performed with this space-limited device. Thus, our proposal could inspire further utilization of mobile chiral Majorana edge states for faster quantum computation.</jats:p>

<jats:p>We experimentally demonstrate that tomographic measurements can be performed for states of qubits before they are prepared. A variant of the quantum teleportation protocol is used as a channel between two instants in time, allowing measurements for polarization states of photons to be implemented 88 ns before they are created. Measurement data taken at the early time and later unscrambled according to the results of the protocol’s Bell measurements, produces density matrices with an average fidelity of 0.90±0.01 against the ideal states of photons created at the later time. Process tomography of the time reverse quantum channel finds an average process fidelity of 0.84±0.02. While our proof-of-principle implementation necessitates some post-selection, the general protocol is deterministic and requires no post-selection to sift desired states and reject a larger ensemble.</jats:p>

<jats:p>Verification in quantum computations is crucial since quantum systems are extremely vulnerable to the environment. However, verifying directly the output of a quantum computation is difficult since we know that efficiently simulating a large-scale quantum computation on a classical computer is usually thought to be impossible. To overcome this difficulty, we propose a self-testing system for quantum computations, which can be used to verify if a quantum computation is performed correctly by itself. Our basic idea is using some extra ancilla qubits to test the output of the computation. We design two kinds of permutation circuits into the original quantum circuit: one is applied on the ancilla qubits whose output indicates the testing information, the other is applied on all qubits (including ancilla qubits) which is aiming to uniformly permute the positions of all qubits. We show that both permutation circuits are easy to achieve. By this way, we prove that any quantum computation has an efficient self-testing system. In the end, we also discuss the relation between our self-testing system and interactive proof systems, and show that the two systems are equivalent if the verifier is allowed to have some quantum capacity.</jats:p>

<jats:p>Recent advances in quantum technology have led to the development and the manufacturing of programmable quantum annealers that promise to solve certain combinatorial optimization problems faster than their classical counterparts. Semi-supervised learning is a machine learning technique that makes use of both labeled and unlabeled data for training, which enables a good classifier with only a small amount of labeled data. In this paper, we propose and theoretically analyze a graph-based semi-supervised learning method with the aid of the quantum annealing technique, which efficiently utilizes the quantum resources while maintaining good accuracy. We illustrate two classification examples, suggesting the feasibility of this method even with a small portion (30%) of labeled data involved.</jats:p>

<jats:p>Using the innovative method of the additional Bloch vector, the electron transfer properties of a double quantum dot (DQD) system measured by a quantum point contact (QPC) in a fluctuating environment are investigated. The results show that the environmental noises in transverse and longitudinal directions play different roles in the dynamical evolution of the open quantum systems. Considering the DQD with symmetric energy level, the Fano factor exhibits a slight peak with the increase of transverse noise amplitude <jats:italic>σ</jats:italic> <jats:sub>T</jats:sub>, which provides a basis for distinguishing dynamical phenomena caused by different directional fluctuation noises in symmetric DQD structures by studying the detector output. In the case of asymmetric DQD, the dependence of a detector current involving the level displacement is distinct when increasing the transverse noise damping coefficient <jats:italic>τ</jats:italic> <jats:sub>T</jats:sub> and the longitudinal noise damping coefficient <jats:italic>τ<jats:sub>ε</jats:sub> </jats:italic> respectively. Meanwhile, the transverse noise damping coefficient <jats:italic>τ</jats:italic> <jats:sub>T</jats:sub> could significantly reduce the Fano factor and enhance the stability of the quantum system compared with the longitudinal one. The Fano factors with stable values as the enhancement of noise amplitudes show different external influences from the detector measurement, and provide a numerical reference for adjusting the noise amplitudes in both transverse and longitudinal directions appropriately in a microscopic experimental process to offset the decoherence effect caused by the measurements. Finally, the research of average waiting time provides unique insights to the development of single electron transfer theory in the short-time limit.</jats:p>

<jats:p>The effects of various notch structures on direct current (DC) and radio frequency (RF) performances of AlGaN/GaN high electron mobility transistors (HEMTs) are analyzed. The AlGaN/GaN HEMTs, each with a 0.8-μm gate length, 50-μm gate width, and 3-μm source–drain distance in various notch structures at the AlGaN/GaN barrier layer, are manufactured to achieve the desired DC and RF characteristics. The maximum drain current (<jats:italic>I</jats:italic> <jats:sub>ds,max</jats:sub>), pinch-off voltage (<jats:italic>V</jats:italic> <jats:sub>th</jats:sub>), maximum transconductance (<jats:italic>g</jats:italic> <jats:sub>m</jats:sub>), gate voltage swing (GVS), subthreshold current, gate leakage current, pulsed <jats:italic>I</jats:italic>–<jats:italic>V</jats:italic> characteristics, breakdown voltage, cut-off frequency (<jats:italic>f</jats:italic> <jats:sub>T</jats:sub>), and maximum oscillation frequency (<jats:italic>f</jats:italic> <jats:sub>max</jats:sub>) are investigated. The results show that the double-notch structure HEMT has a 30% improvement of gate voltage swing, a 42.2% improvement of breakdown voltage, and a 9% improvement of cut-off frequency compared with the conventional HEMT. The notch structure also has a good suppression of the current collapse.</jats:p>

<jats:p>A special transformation is introduced and thereby leads to the <jats:italic>N</jats:italic>-soliton solution of the (2+1)-dimensional generalized Konopelchenko–Dubrovsky–Kaup–Kupershmidt (KDKK) equation. Then, by employing the long wave limit and imposing complex conjugate constraints to the related solitons, various localized interaction solutions are constructed, including the general <jats:italic>M</jats:italic>-lumps, <jats:italic>T</jats:italic>-breathers, and hybrid wave solutions. Dynamical behaviors of these solutions are investigated analytically and graphically. The solutions obtained are very helpful in studying the interaction phenomena of nonlinear localized waves. Therefore, we hope these results can provide some theoretical guidance to the experts in oceanography, atmospheric science, and weather forecasting.</jats:p>

<jats:p>The effects of Sm doping into CuInTe<jats:sub>2</jats:sub> chalcopyrite on the cohesive energy before and after light absorption are systematically investigated by the empirical electron theory (EET) of solids and molecules. The results show that the static energy of CuIn<jats:sub>1 – <jats:italic>x</jats:italic> </jats:sub>Sm<jats:sub> <jats:italic>x</jats:italic> </jats:sub>Te<jats:sub>2</jats:sub> decreases with Sm content increasing due to the valence electronic structure modulated by doping Sm into CuIn<jats:sub>1 – <jats:italic>x</jats:italic> </jats:sub>Sm<jats:sub> <jats:italic>x</jats:italic> </jats:sub>Te<jats:sub>2</jats:sub>. The calculated optical absorption transition energy from the static state to the excited energy level in CuIn<jats:sub>1 – <jats:italic>x</jats:italic> </jats:sub>Sm<jats:sub> <jats:italic>x</jats:italic> </jats:sub>Te<jats:sub>2</jats:sub> accords well with the experimental absorption bandgap of CuIn<jats:sub>1 – <jats:italic>x</jats:italic> </jats:sub>Sm<jats:sub> <jats:italic>x</jats:italic> </jats:sub>Te<jats:sub>2</jats:sub>. Moreover, it is found that the energy bandgap of CuIn<jats:sub>1 – <jats:italic>x</jats:italic> </jats:sub>Sm<jats:sub> <jats:italic>x</jats:italic> </jats:sub>Te<jats:sub>2</jats:sub> is significantly widened with Sm content increasing due to its special valent electron structure, which is favorable for enhancing the light absorption in a wider range and also for the potential applications in solar cells.</jats:p>