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Topological indices and coindices are numerical invariants that relate to quantitative structure property/activity connections. The purpose of topological indices and coindices were introduced to draw the data related to chemical graphs with respect to adjacent & non adjacent pairs of vertex degrees respectively. These indices equip the researchers with a lot of information related to the properties and structure of the chemical compound. In this article, CoM-polynomials for molecular graph of linear and multiple Anthracene are computed from which eleven degree based topological coindices are derived. [ABSTRACT FROM AUTHOR]

To perform a quantum brute force attack on a cryptosystem based on Grover's algorithm, it is necessary to implement a quantum circuit of the cryptographic algorithm. Therefore, an efficient quantum circuit design of a given cryptographic algorithm is essential, especially in terms of quantum security analysis, and it is well known that T‐depth should be reduced for time complexity efficiency. In this paper, the authors propose a novel technique to reduce T‐depth (and T‐count) when some quantum circuits located in between two Toffoli‐gates are interchangeable with a controlled phase gate (CP gate), and the authors apply this technique to five types of quantum adders, reducing T‐depth by more than 33%. The authors also present new SHA‐256 quantum circuits which have a critical path with only three quantum adders while the critical paths of quantum circuits in the previous studies consist of seven or 10 quantum adders, and the authors also apply our technique to the proposed SHA‐256 quantum circuits. Four versions of SHA‐256 quantum circuit are presented. Among the previous results, T‐depth of the circuit with the smallest Width (the number of qubits) 801 was approximately 109,104. On the other hand, T‐depth of the proposed SHA‐256 quantum circuit with the Width 797 is 16,055, which is remarkably reduced by about 85%. Another proposed quantum circuit only requires 768 qubits, which is the smallest Width compared to the previous results to the best of our knowledge. Furthermore, one other version is the most time‐efficient circuit with an overall Toffoli‐depth (and T‐depth) that is less than 5000. [ABSTRACT FROM AUTHOR]

Addition is the most basic operation of computing based on a bit system. There are various addition algorithms considering multiple number systems and hardware, and studies for a more efficient addition are still ongoing. Quantum computing based on qubits as the information unit asks for the design of a new addition because it is, physically, wholly different from the existing frequency-based computing in which the minimum information unit is a bit. In this paper, we propose an efficient quantum circuit of modular addition, which reduces the number of gates and the depth. The proposed modular addition is for the Galois Field G F (2 n − 1) , which is important as a finite field basis in various domains, such as cryptography. Its design principle was from the ripple carry addition (RCA) algorithm, which is the most widely used in existing computers. However, unlike conventional RCA, the storage of the final carry is not needed due to modifying existing diminished-1 modulo 2 n − 1 adders. Our proposed adder can produce modulo sum within the range 0 , 2 n − 2 by fewer qubits and less depth. For comparison, we analyzed the proposed quantum addition circuit over G F (2 n − 1) and the previous quantum modular addition circuit for the performance of the number of qubits, the number of gates, and the depth, and simulated it with IBM's simulator ProjectQ. [ABSTRACT FROM AUTHOR]