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The electronic structure evolutions of few-layer black phosphorus (BP) under pressure shows a wealth of phenomena, such as the nonmonotonic change of direct gap at the {\Gamma} point, the layer-number dependence, and the distinct responses to normal and hydrostatic pressures. A full and unified understanding to these rich phenomena remains lacking. Here, we provide a unified understanding from the competition between interlayer quasi-bonding (QB) interactions and intralayer chemical bonding interactions. The former decreases while the latter increases the band gap under pressure and the origin can be correlated to different combinations of inter- and intra-layer antibonding or bonding interactions at the band edges. More interestingly, the interlayer QB interactions are a coexistence of two categories of interactions, namely, the coexistence of interactions between bands of the same occupancy (occupied-occupied and empty-empty interactions) and of different occupancies (occupied-empty interaction); and, the overall effect is a four-level interaction, which explains the anomalous interlayer-antibonding feature of the conduction band edge of bilayer BP. Our current study lay the foundation for the electronic structure tuning of two-dimensional (2D) BP, and, our analysis method for multi-energy-level interactions can be applied to other 2D semiconductor homo- and hetero-structures that have occupied-empty interlayer interactions.

The quantum circuit model is the most commonly used model for implementing quantum computers and quantum neural networks whose essential tasks are to realize certain unitary operations. The circuit model usually implements a desired unitary operation by a sequence of single-qubit and two-qubit unitary gates from a universal set. Although this certainly facilitates the experimentalists as they only need to prepare several different kinds of universal gates, the number of gates required to implement an arbitrary desired unitary operation is usually large. Hence the efficiency in terms of the circuit depth or running time is not guaranteed. Here we propose an alternative approach; we use a simple discrete-time quantum walk (DTQW) on a cycle graph to model an arbitrary unitary operation without the need to decompose it into a sequence of gates of smaller sizes. Our model is essentially a quantum neural network based on DTQW. Firstly, it is universal as we show that any unitary operation can be realized via an appropriate choice of coin operators. Secondly, our DTQW-based neural network can be updated efficiently via a learning algorithm, i.e., a modified stochastic gradient descent algorithm adapted to our network. By training this network, one can promisingly find approximations to arbitrary desired unitary operations. With an additional measurement on the output, the DTQW-based neural network can also implement general measurements described by positive-operator-valued measures (POVMs). We show its capacity in implementing arbitrary 2-outcome POVM measurements via numeric simulation. We further demonstrate that the network can be simplified and can overcome device noises during the training so that it becomes more friendly for laboratory implementations. Our work shows the capability of the DTQW-based neural network in quantum computation and its potential in laboratory implementations., Comment: This a duplicated upload by mistake. This article is already uploaded as 2210.14450

The quantum circuit model is the most commonly used model for implementing quantum computers and quantum neural networks whose essential tasks are to realize certain unitary operations. Here we propose an alternative approach; we use a simple discrete-time quantum walk (DTQW) on a cycle graph to model an arbitrary unitary operation $U(N)$ without the need to decompose it into a sequence of gates of smaller sizes. Our model is essentially a quantum neural network based on DTQW. Firstly, it is universal as we show that any unitary operation $U(N)$ can be realized via an appropriate choice of coin operators. Secondly, our DTQW-based neural network can be updated efficiently via a learning algorithm, i.e., a modified stochastic gradient descent algorithm adapted to our network. By training this network, one can promisingly find approximations to arbitrary desired unitary operations. With an additional measurement on the output, the DTQW-based neural network can also implement general measurements described by positive-operator-valued measures (POVMs). We show its capacity in implementing arbitrary 2-outcome POVM measurements via numeric simulation. We further demonstrate that the network can be simplified and can overcome device noises during the training so that it becomes more friendly for laboratory implementations. Our work shows the capability of the DTQW-based neural network in quantum computation and its potential in laboratory implementations., Comment: 13 pages, 13 figures

In the numerical modeling of the GRB afterglow, some approximations have been made for simplicity, and different groups developed their codes. A robust test of these models/approaches is challenging because of the lack of directly measured physical parameters. Fortunately, the viewing angle inferred from the afterglow modeling is widely anticipated to be the same as the inclination angle of the binary neutron star (BNS) mergers that can be evaluated with the gravitational wave (GW) data. Therefore in the future, it is possible to calibrate the afterglow modeling with the GW inclination angle measurements. We take three methods, including both analytical estimations and direct simulations, to project the uncertainties of the inclination angle measurements. For some BNS mergers accompanied with electromagnetic counterparts detected in the O4/O5 runs of LIGO/Virgo/KAGRA/LIGO-India detectors, we show that the inclination angle can be determined within an uncertainty of $\leq 0.1$ rad, supposing that the Hubble constant is known with an accuracy of $\leq 3\%$ and the uncertainty of Hubble flow velocity is within $\sim 1 \%$. The off-axis GRB outflow will give rise to afterglow emission, and the most energetic ones may be detectable at the distance of $\sim$ 100-200 Mpc even for a viewing angle of $\geq 0.3$ rad. Such events can thus serve as a robust test of the afterglow modeling approach. We have also evaluated the prospect of resolving the so-called Hubble tension with a single GW/GRB association event. We find out that a $\sim 3\%$ precision Hubble constant is obtainable if the uncertainty of the viewing angle can be constrained to be within $\sim 0.1$ rad, which is expected to be the case for some nearby ($\leq 250$ Mpc) bright/on-axis GRBs with a well-behaved afterglow light curve displaying a clear achromatic break at early times., Comment: 18 pages, 11 figures, and 1 table; PRD in press