Quantum Multiplier Based on Exponent Adder

Quantum multiplication is a fundamental operation in quantum computing. Most existing quantum multipliers require $O(n)$ qubits to multiply two $n$-bit integer numbers, limiting their applicability to multiply large integer numbers using near-term quantum computers. In this paper, we propose the Quantum Multiplier Based on Exponent Adder (QMbead), a new approach that addresses this limitation by requiring just $\log_2(n)$ qubits to multiply two $n$-bit integer numbers. QMbead uses a so-called exponent encoding to represent two multiplicands as two superposition states, respectively, and then employs a quantum adder to obtain the sum of these two superposition states, and subsequently measures the outputs of the quantum adder to calculate the product of the multiplicands. This paper presents two types of quantum adders based on the quantum Fourier transform (QFT) for use in QMbead. The circuit depth of QMbead is determined by the chosen quantum adder, being $O(\log^2 n)$ when using the two QFT-based adders. If leveraging a logarithmic-depth quantum adder, the time complexity of QMbead is $O(n \log n)$, identical to that of the fastest classical multiplication algorithm, Harvey-Hoeven algorithm. Interestingly, QMbead maintains an advantage over the Harvey-Hoeven algorithm, given that the latter is only suitable for excessively large numbers, whereas QMbead is valid for both small and large numbers. The multiplicand can be either an integer or a decimal number. QMbead has been successfully implemented on quantum simulators to compute products with a bit length of up to 273 bits using only 17 qubits. This establishes QMbead as an efficient solution for multiplying large integer or decimal numbers with many bits.