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. This paper proposes a new approach, the Quantum Multiplier Based on Exponent Adder (QMbead), which addresses this issue by requiring only $log_2(n)$ qubits to multiply two $n$-bit integer numbers. QMbead uses a so-called exponent encoding to represent the 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. The 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 $O(log_2^2 n)$ when using the two QFT-based adders. The multiplicand can be either an integer or a decimal number. QMbead has been 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。现有的大多数量子乘法器需要$O(n)$个量子比特来相乘两个$n$位整数，这限制了其在近期量子计算机上相乘大整数的适用性。本文提出一种新方法——基于指数加法器的量子乘法器（QMbead），通过仅需$log_2(n)$个量子比特即可相乘两个$n$位整数，解决了这一问题。QMbead采用所谓的指数编码分别将两个被乘数表示为两个叠加态，然后利用量子加法器获得这两个叠加态之和，随后测量量子加法器的输出以计算被乘数的乘积。本文提出了两种基于量子傅里叶变换（QFT）的量子加法器用于QMbead。QMbead的电路深度由所选量子加法器决定，使用两种基于QFT的加法器时为$O(log_2^2 n)$。被乘数可为整数或小数。QMbead已在量子模拟器上实现，使用仅17个量子比特即可计算位长高达273位的乘积。这确立了QMbead作为相乘多位大整数或小数的有效方案。