A Hybrid Quantum-Classical Generative Adversarial Network for Near-Term Quantum Processors

In this article, we present a hybrid quantum-classical generative adversarial network (GAN) for near-term quantum processors. The hybrid GAN comprises a generator and a discriminator quantum neural network (QNN). The generator network is realized using an angle encoding quantum circuit and a variational quantum ansatz. The discriminator network is realized using multi-stage trainable encoding quantum circuits. A modular design approach is proposed for the QNNs which enables control on their depth to compromise between accuracy and circuit complexity. Gradient of the loss functions for the generator and discriminator networks are derived using the same quantum circuits used for their implementation. This prevents the need for extra quantum circuits or auxiliary qubits. The quantum simulations are performed using the IBM Qiskit open-source software development kit (SDK), while the training of the hybrid quantum-classical GAN is conducted using the mini-batch stochastic gradient descent (SGD) optimization on a classic computer. The hybrid quantum-classical GAN is implemented using a two-qubit system with different discriminator network structures. The hybrid GAN realized using a five-stage discriminator network, comprises 63 quantum gates and 31 trainable parameters, and achieves the Kullback-Leibler (KL) and the Jensen-Shannon (JS) divergence scores of 0.39 and 4.16, respectively, for similarity between the real and generated data distributions

On classical n-absorbing submodules

In this paper, we introduce the notion of classical n-absorbing submodules of a module M over a commutative ring R with identity, which is a generalization of classical prime submodules. A proper submodule N of M is said to be classical n-absorbing if whenever a1a2... an+1 m in M, for a1a2... an+1 in R and m in M, then there are n of the ai's whose product with m is in N. We give some basic results concerning classical n-absorbing submodules. Then the classical n-absorbing avoidance theorem for submodules is proved. Finally, classical n-absorbing submodules in several classes of modules are studied

CMOS Quantum Computing: Toward A Quantum Computer System-on-Chip

Quantum computing is experiencing the transition from a scientific to an engineering field with the promise to revolutionize an extensive range of applications demanding high-performance computing. Many implementation approaches have been pursued for quantum computing systems, where currently the main streams can be identified based on superconducting, photonic, trapped-ion, and semiconductor qubits. Semiconductor-based quantum computing, specifically using CMOS technologies, is promising as it provides potential for the integration of qubits with their control and readout circuits on a single chip. This paves the way for the realization of a large-scale quantum computing system for solving practical problems. In this paper, we present an overview and future perspective of CMOS quantum computing, exploring developed semiconductor qubit structures, quantum gates, as well as control and readout circuits, with a focus on the promises and challenges of CMOS implementation

When the annihilator graph of a commutative ring is planar or toroidal?

Let R be a commutative ring with identity, and let Z(R) be the set of zero-divisors of R. The annihilator graph of R is defined as the undirected graph AG(R) with the vertex set Z(R)* = Z(R) \ {0}, and two distinct vertices x and y are adjacent if and only if  ann_R(xy) \neq ann_R(x) \cup ann_R(y). In this paper, all rings whose annihilator graphs can be embedded on the plane or torus are classified

Some classes of perfect strongly annihilating-ideal graphs associated with commutative rings

summary:Let $R$ be a commutative ring with identity and $A(R)$ be the set of ideals with nonzero annihilator. The strongly annihilating-ideal graph of $R$ is defined as the graph ${\rm SAG}(R)$ with the vertex set $A(R)^*=A(R)\setminus\{0\}$ and two distinct vertices $I$ and $J$ are adjacent if and only if $I\cap {\rm Ann}(J)\neq (0)$ and $J\cap {\rm Ann}(I)\neq (0)$. In this paper, the perfectness of ${\rm SAG}(R)$ for some classes of rings $R$ is investigated

The annihilating-ideal graph of z(n) is weakly perfect

A graph is called weakly perfect if its vertex chromatic number equals its clique number. Let RR be a commutative ring with identity and A(R)A(R) be the set of ideals with non-zero annihilator. The annihilating-ideal graph of RR is defined as the graph AG(R)AG(R) with the vertex set A(R)∗=A(R)∖{0}A(R)∗=A(R)∖{0} and two distinct vertices II and JJ are adjacent if and only if IJ=0IJ=0. In this paper, we show that the graph AG(Zn)AG(Zn), for every positive integer nn, is weakly perfect. Moreover, the exact value of the clique number of AG(Zn)AG(Zn) is given and it is proved that AG(Zn)AG(Zn) is class 1 for every positive integer nn

Signature of Non-Minimal Scalar-Gravity Coupling with an Early Matter Domination on the Power Spectrum of Gravitational Waves

The signal strength of primordial gravitational waves experiencing an epoch of early scalar domination is reduced with respect to radiation domination. In this paper, we demonstrate that the specific pattern of this reduction is sensitive to the coupling between the dominant field and gravity. When this coupling is zero, the impact of early matter domination on gravitational waves is solely attributed to the alteration of the Hubble parameter and the scale factor. In the presence of non-zero couplings, on the other hand, the evolution of primordial gravitational waves is directly affected as well, resulting in a distinct step-like feature in the power spectrum of the gravitational wave as a function of frequency. This feature serves as a smoking gun signature of this model. In this paper, we provide an analytical expression of the power spectrum that illustrates the dependence of power spectrum on model parameters and initial conditions. Furthermore, we provide analytical relations that specify the frequency interval in which the step occurs. We compare the analytical estimates with numerical analysis and show they match well.Comment: 19 pages, 7 figures, 1 table, and 2 appendice

GaN Integrated Circuit Power Amplifiers: Developments and Prospects

GaN integrated circuit technologies have dramatically progressed over the recent years. The prominent feature of GaN high-electron mobility transistors (HEMTs), unparalleled output power densities, has created a paradigm shift in the established and emerging high-power applications. In this article, we present a review on the developments and prospects of GaN integrated circuit power amplifiers (PAs). The progress of GaN transistors including improvements in their important features, i.e., supply voltage, substrate material, transistor scaling approach, and device modeling are elaborated and the current state-of-the-art processes with 20-nm gate length, 450 GHz cut-off frequency, and over 600 V supply voltage are discussed. We also investigate developments in the GaN integrated circuit PA architectures and their implementation challenges including the reactive matching PAs capable of delivering over 100 W output power and operating up to 200 GHz, PA linearity, back-off efficiency enhancement, reconfigurable PAs, and distributed PA architectures. Finally, we discuss the prospects of GaN technology and possible future improvements, in transistor and circuit levels, which can advance performance and functionality of GaN integrated circuits

A note on infinite aSaS-groups

summary:Let $G$ be a group. If every nontrivial subgroup of $G$ has a proper supplement, then $G$ is called an $aS$-group. We study some properties of $aS$-groups. For instance, it is shown that a nilpotent group $G$ is an $aS$-group if and only if $G$ is a subdirect product of cyclic groups of prime orders. We prove that if $G$ is an $aS$-group which satisfies the descending chain condition on subgroups, then $G$ is finite. Among other results, we characterize all abelian groups for which every nontrivial quotient group is an $aS$-group. Finally, it is shown that if $G$ is an $aS$-group and $|G|\neq pq,p$, where $p$ and $q$ are primes, then $G$ has a triple factorization