Efficient quantum circuits for arithmetic operations and their applications

Heng Fan

Hai-Sheng Li, Ping Fan, Haiying Xia +2 more

Yasuhiro Takahashi

Quantum circuits for elementary arithmetic operations are important not only for implementing Shor's factoring algorithm on a quantum computer but also for understanding the computational power of small quantum circuits, such as linear-size or logarithmic-depth quantum circuits. This paper surveys some recent approaches to constructing efficient quantum circuits for elementary arithmetic operations and their applications to Shor's factoring algorithm. It covers addition, comparison, and the quantum Fourier transform used for addition.

Hai-Sheng Li, Yusi Xu, Yunbai Qin +2 more

The efficient quantum circuits of arithmetic operations are important to perform quantum algorithms. To implement efficient matrix operations, we first modify the generalized model of the novel enhanced quantum representation of digital images (GNEQR) to store unsigned and signed integer matrices. Next, we design the circuits of the circuits of quantum addition, quantum modulo addition, quantum subtraction, and quantum modulo subtraction, these operations all keeping two operands unchanged. Then, we propose the circuits of quantum matrix addition, quantum matrix modulo addition, quantum matrix subtraction, and quantum matrix modulo subtraction for the first time. Furthermore, we present a simulation method to verify the correctness of the proposed arithmetic operations of matrix. The results of simulation experiment show that the propose arithmetic operations of matrix are efficient and correct.

Hafiz Md. Hasan Babu

Amit Saha, Turbasu Chatterjee, Anupam Chattopadhyay +1 more

Efficient quantum realizations of arithmetic operations, such as multiplication/division, addition/subtraction, square root, and arcsine — are of utmost importance for several quantum algorithms of practical importance. In the literature, such implementations are reported with the objective of minimizing qubit count, Toffoli/T/CNOT gate count, logical depth, quantum volume, and other commonly used metrics. By extending the realm of quantum states to higher dimensions, it has been demonstrated recently that intermediate qutrits can be leveraged to reduce ancilla qubits, thereby reducing qubit count. In this paper, we have incorporated this approach to prepare efficient implementation for several quantum arithmetic operations, obtaining significantly lowered qubit count. As an exemplary application, we study derivative pricing, in which, quantum arithmetic circuits are necessary for path loading using the re-parameterization method, as well as the payoff calculation. The intermediate qutrit approach requires to access higher energy levels, making the design prone to errors. Nevertheless, we show that the overall decrease in the error probability of error is significant owing to the fact that we achieve circuit robustness compared to the qubit-only approach in this NISQ era. We also study how we may achieve possible effectiveness with fewer gate counts of the proposed approach in the fault-tolerant setting.