The integer multiplier with two unchanged operands reducing T and CNOT gates

Quantum circuits for multiplication are necessary for scientific computing on quantum computers. Clifford + T circuits are widely used in fault-tolerant implementations. The costs of implementing T and double-qubit gates are higher than those of other one-qubit gates in the Clifford + T group. In addition, the small number of qubits available in existing quantum devices is another constraint on quantum circuits. Therefore, we reduce T and CNOT gates and circuit width as the primary optimization goal in this paper. We propose an algorithm for multiplication with two unchanged operands. Preserving both operands of the multiplication is important for realizing some quantum algorithms, such as quantum bilinear interpolation. Using this algorithm, we design the circuit of the integer multiplier with two unchanged operands reducing CNOT gates. Next, we develop a Clifford + T circuit for the multiplier and introduce new optimization rules to reduce T gates. Comparative analysis shows that the proposed multiplier achieves the best width among existing multipliers. Compared to multipliers with two unchanged operands that use at most one ancillary qubit, our proposed multiplier has the best T-count, T-depth, and CNOT-count.