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.

翻译：量子乘法是量子计算中的基本运算。现有大多数量子乘法器需要$O(n)$个量子比特来相乘两个$n$比特整数，这限制了它们在近量子计算机上处理大整数相乘的适用性。本文提出基于指数加法器的量子乘法器（QMbead），该新方法通过仅需$\log_2(n)$个量子比特即可相乘两个$n$比特整数，突破了这一限制。QMbead采用所谓指数编码将两个被乘数分别表示为两个叠加态，进而利用量子加法器获取这两个叠加态之和，随后测量量子加法器的输出以计算被乘数的乘积。本文针对QMbead应用提出了两种基于量子傅里叶变换（QFT）的量子加法器。QMbead的电路深度由所选量子加法器决定，使用两种基于QFT的加法器时复杂度为$O(\log^2 n)$。若采用对数深度量子加法器，QMbead的时间复杂度为$O(n \log n)$，与最快经典乘法算法——Harvey-Hoeven算法相同。值得注意的是，QMbead相较Harvey-Hoeven算法仍具优势，因后者仅适用于超大数，而QMbead对大小数值均有效。被乘数可为整数或小数。已在量子模拟器上成功实现QMbead，仅用17个量子比特即可计算最高273比特的乘积。这确立了QMbead作为高效处理多比特大整数或小数乘法的解决方案。