Quantum Circuit Design of Toom 3-Way Multiplication

Abstract

In classical computation, Toom–Cook is one of the multiplication methods for large numbers which offers faster execution time compared to other algorithms such as schoolbook and Karatsuba multiplication. For the use in quantum computation, prior work considered the Toom-2.5 variant rather than the classically faster and more prominent Toom-3, primarily to avoid the nontrivial division operations inherent in the latter circuit. In this paper, we investigate the quantum circuit for Toom-3 multiplication, which is expected to give an asymptotically lower depth than the Toom-2.5 circuit. In particular, we designed the corresponding quantum circuit and adopted the sequence proposed by Bodrato to yield a lower number of operations, especially in terms of nontrivial division, which is reduced to only one exact division by 3 circuit per iteration. Moreover, to further minimize the cost of the remaining division, we utilize the unique property of the particular division circuit, replacing it with a constant multiplication by reciprocal circuit and the corresponding swap operations. Our numerical analysis shows that the resulting circuit indeed gives a lower asymptotic complexity in terms of Toffoli depth and qubit count compared to Toom-2.5 but with a large number of Toffoli gates that mainly come from realizing the division operation.

Space and Time-Efficient Quantum Multiplier in Post Quantum Cryptography Era

“…In particular, the authors design the circuit for the Toom-Cook 2.5-way multiplication and approximate the number of Toffoli gates and qubits by looking at the recursive tree struc-ture of their method. Other research, conducted by Larasati et al, [8] referring to the classical implementation in [14], investigated the Toom-Cook 3-way multiplication design of the quantum circuit, giving an asymptotically lower depth than the Toom-Cook 2.5-way circuit. The difficulty in developing higher-degree Toom-Cook multiplication involving odd numbers is that there is a resource bottleneck because it still involves nontrivial-division operations [8].…”

“…Larasati et al [8] elaborate on the Toom-Cook 3-way by citing Bodrato et al in [14]. Both researchers continue to use the division gate, and the objective is to reduce the number of operations, particularly nontrivial ones.…”

“…Both researchers continue to use the division gate, and the objective is to reduce the number of operations, particularly nontrivial ones. Some of Larasati et al's strategies emphasize the use of division gates, which have resulted in a maximum of one exact division by three circuits per iteration [8]. In addition, the cost of the remaining division was reduced by utilizing the circuit's unique property and replacing it with a constant multiplication by reciprocal circuit and the corresponding swap operations [8].…”

“…N UMEROUS studies have attempted to explain arithmetic multiplication in classical or quantum computing environments from the Naïve Schoolbook onward, continuing through multiple multiplication development versions such as Karatsuba [1] [2] [3], Montgomery [4], and Toom-Cook [5] [6] [7] [8]. Notably, many other studies have been conducted in a variety of fields or from different perspectives on multiplication.…”

Reducing the Depth of Quantum FLT-Based Inversion Circuit

“…Additionally, for constructing a modular multiplication, we utilize the Schoolbook multiplication followed by reduction for both scenarios due to its simplicity. Employing other multiplication methods (e.g., quantum Karatsuba such as [14,30] or Toom Cook multiplication such as [31,32]) can be done accordingly by simply changing the underlying blocks since it is supported by Qiskit. Following previous works on quantum cryptanalysis [13,14,28], swap operations can be considered as free since it can be done via qubit relabeling [33,34].…”