# Quantum Modular Adder over \(GF(2^{n}-1)\) without Saving the Final Carry
## 1 Introduction  
Recently, quantum computers have been actively researched and developed by Google, IBM, Microsoft, and Rigetti, and each has reported that they have reached quantum supremacy (quantum superiority) that exceeds the performance of existing supercomputers [1, 2, 3, 4, 5]. IBM announced its plan to commercialize it as a cloud service, and Amazon and Intel have opened a quantum cloud service for research and development [6, 7]. Such a quantum computer is a device operated by the principle of quantum mechanics and quantum phenomena using quantum photons [1, 8, 9]. It processes information in it, which is 0, 1, or a superposition of the two states in a quantum register as a qubit. This superposition is associated with the \"uncertainty\" of the quantum state. The unit for quantum information processing is a qubit, where 0 and 1 can exist simultaneously. However, existing computers are devices that operate by electronic phenomena using semiconductor devices such as transistors, and process information in a deterministic system that produces one output for one input based on 0 or 1 bit as the minimum unit for information processing. Since quantum computing processes information on a device that is entirely different from today's computer, logic gates, basic operations, data structures, and algorithms forquantum information processing must be newly designed and implemented according to the characteristics of quantum computers. Because the most basic logic gates are open as needed, researchers have designed quantum circuits for various operations, data structures, and algorithms using logic gates. In a quantum computer, the algorithm is represented by a quantum circuit using a qubit and a gate, and the number of qubits, the number of gates, and the depth are significant elements for evaluating the performance of the quantum circuit, and the smaller the number, the better. As the newly designed addition circuit is the most fundamental operation, developing an efficient addition circuit leads to the practical design of other essential operations such as multiplication, division, and modular addition, which are primitives for solving various problems [10; 11; 12]. Thus far, various quantum modular adders for specific fields have been proposed based on existing classical addition algorithms [13], utilizing quantum adders, such as Quantum Ripple Carry Adder (QRCA), Quantum Carry Save Adder (QCSA), and Quantum Carry Lookahead Adder (QCLA), classified by how to handle the carry propagation [14]. A more special addition circuit, such as [PERSON]'s quantum adder for superposition states, as one of the quantum principles has also been proposed [12]. However, a primary quantum algorithm, such as [PERSON]'s algorithm, still uses general quantum adders based on the current adders [15; 16; 17; 18]. The representative quantum adder is the adder modulo N proposed by [PERSON] et al., among the QRCAs based on the Ripple Carry Adder (RCA), which is the most widely applied in classical computing [10]. The RCA is the simplest adder with the lowest power, area, and design time suitable for various ultra-low-power IoT (Internet of Things) applications such as implantable biomedical devices, RADAR system, linear convolution, Harr transformation, and fast Fourier transformation [19; 20; 21; 22]. Moreover, it is usually used to design a hybrid adder with other faster adders such as the Carry Lookahead Adder (CLA) and Kogge Stone Adder (KSA). A single adder cannot optimally operate to improve the speed, area, leakage current, overall power dissipation, and the design time because there is a trade-off among various adders [23; 24]. Although the CLA or the KSA are used for higher speeds, the power consumption, the energy dissipation, and the area consumption for the RCA are far lower than those at the same speed [20]. Nevertheless, the RCA is slower than the CLA or the KSA because of solving the carry propagation problem.