Boolean Function Decomposition Based on Grover Algorithm and Its Simulation Using Quantum Language Quipper

Author(s):  
Dong Wang ◽  
Yiwei Li ◽  
Edison Tsai ◽  
Xiaoyu Song ◽  
Marek Perkowski ◽  
...  
2019 ◽  
Vol 29 (12) ◽  
pp. 1950172
Author(s):  
Qinbin He ◽  
Fangyue Chen

By using the universal perceptron theory, a decomposition algorithm called improved distance and DNA-like decomposition algorithm (ID-DNA-like DA) is proposed to investigate the linear decomposition structures of [Formula: see text]-bit Boolean functions. The specific linear decomposition structure of an [Formula: see text]-bit Boolean function is generated by the proposed decomposition algorithm ([Formula: see text]). Especially, the linear decomposition structure of 4-bit Boolean function is shown to have the following properties: the total number of 4-bit Boolean functions is 65[Formula: see text]536 and the number of linearly separable Boolean functions is 1882; the number of 2-length decomposition Boolean functions is not less than 41[Formula: see text]366; the number of 3-length decomposition Boolean functions does not exceed 22[Formula: see text]286; and the number of 4-length decomposition Boolean functions is only 2. At the same time, it is shown that the decomposition length of [Formula: see text]-bit Boolean function does not exceed [Formula: see text] [Formula: see text]. In addition, a concept of the [Formula: see text]-bit Boolean function decomposition structure figure is proposed. By decomposition calculation, the related results of decomposition structure figures show that there is some symmetry with a strong self-similar fractal feature.


VLSI Design ◽  
1995 ◽  
Vol 3 (3-4) ◽  
pp. 289-300 ◽  
Author(s):  
Tadeusz Łuba ◽  
Henry Selvaraj

An effective logic synthesis procedure based on parallel and serial decomposition of a Boolean function is presented in this paper. The decomposition, carried out as the very first step of the .synthesis process, is based on an original representation of the function by a set of r-partitions over the set of minterms. Two different decomposition strategies, namely serial and parallel, are exploited by striking a balance between the two ideas. The presented procedure can be applied to completely or incompletely specified, single- or multiple-output functions and is suitable for different types of FPGAs including XILINX, ACTEL and ALGOTRONIX devices. The results of the benchmark experiments presented in the paper show that, in several cases, our method produces circuits of significantly reduced complexity compared to the solutions reported in the literature.


Author(s):  
Ana Maria Ariciu ◽  
David H. Woen ◽  
Daniel N. Huh ◽  
Lydia Nodaraki ◽  
Andreas Kostopoulos ◽  
...  

Using electron spins within molecules for quantum information processing (QIP) was first proposed by Leuenberger and Loss (1), who showed how the Grover algorithm could be mapped onto a Mn12 cage (2). Since then several groups have examined two-level (S = ½) molecular spin systems as possible qubits (3-12). There has also been a report of the implementation of the Grover algorithm in a four-level molecular qudit (13). A major challenge is to protect the spin qubit from noise that causes loss of phase information; strategies to minimize the impact of noise on qubits can be categorized as corrective, reductive, or protective. Corrective approaches allow noise and correct for its impact on the qubit using advanced microwave pulse sequences (3). Reductive approaches reduce the noise by minimising the number of nearby nuclear spins (7-11), and increasing the rigidity of molecules to minimise the effect of vibrations (which can cause a fluctuating magnetic field via spin-orbit coupling) (9,11); this is essentially engineering the ligand shell surrounding the electron spin. A protective approach would seek to make the qubit less sensitive to noise: an example of the protective approach is the use of clock transitions to render spin states immune to magnetic fields at first order (12). Here we present a further protective method that would complement reductive and corrective approaches to enhancing quantum coherence in molecular qubits. The target is a molecular spin qubit with an effective 2S ground state: we achieve this with a family of divalent rare-earth molecules that have negligible magnetic anisotropy such that the isotropic nature of the electron spin renders the qubit markedly less sensitive to magnetic noise, allowing coherent spin manipulations even at room temperature. If combined with the other strategies, we believe this could lead to molecular qubits with substantial advantages over competing qubit proposals.<br>


2021 ◽  
Vol 11 (14) ◽  
pp. 6300
Author(s):  
Igor Smolyar ◽  
Daniel Smolyar

Patterns found among both living systems, such as fish scales, bones, and tree rings, and non-living systems, such as terrestrial and extraterrestrial dunes, microstructures of alloys, and geological seismic profiles, are comprised of anisotropic layers of different thicknesses and lengths. These layered patterns form a record of internal and external factors that regulate pattern formation in their various systems, making it potentially possible to recognize events in the formation history of these systems. In our previous work, we developed an empirical model (EM) of anisotropic layered patterns using an N-partite graph, denoted as G(N), and a Boolean function to formalize the layer structure. The concept of isotropic and anisotropic layers was presented and described in terms of the G(N) and Boolean function. The central element of the present work is the justification that arbitrary binary patterns are made up of such layers. It has been shown that within the frame of the proposed model, it is the isotropic and anisotropic layers themselves that are the building blocks of binary layered and arbitrary patterns; pixels play no role. This is why the EM can be used to describe the morphological characteristics of such patterns. We present the parameters disorder of layer structure, disorder of layer size, and pattern complexity to describe the degree of deviation of the structure and size of an arbitrary anisotropic pattern being studied from the structure and size of a layered isotropic analog. Experiments with arbitrary patterns, such as regular geometric figures, convex and concave polygons, contour maps, the shape of island coastlines, river meanders, historic texts, and artistic drawings are presented to illustrate the spectrum of problems that it may be possible to solve by applying the EM. The differences and similarities between the proposed and existing morphological characteristics of patterns has been discussed, as well as the pros and cons of the suggested method.


2017 ◽  
Vol 27 (2) ◽  
Author(s):  
Stanislav V. Smyshlyaev

AbstractThe paper is concerned with relations between the correlation-immunity (stability) and the perfectly balancedness of Boolean functions. It is shown that an arbitrary perfectly balanced Boolean function fails to satisfy a certain property that is weaker than the 1-stability. This result refutes some assertions by Markus Dichtl. On the other hand, we present new results on barriers of perfectly balanced Boolean functions which show that any perfectly balanced function such that the sum of the lengths of barriers is smaller than the length of variables, is 1-stable.


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