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.
Highlights
After the prominent proposal of Shor’s algorithm [3] in 1994 which shows the apparent advantage of quantum computers in cracking the existing cryptosystems (i.e., Rivest–Shamir-Adleman (RSA) and elliptic curve-based cryptography (ECC)), substantial research efforts have been made to design an efficient quantum circuit for modular exponentiation operation, which is the essential component of the algorithm
Using the constant multiplication circuit by [24], we find that Toom 3-way multiplication still give lower asymptotic Toffoli depth and space compared to [11] but with a higher cost in the number of Toffoli gates
In terms of Toffoli count, Toom-3 circuit still scales quadratically as in the naïve method. This is heavily contributed by the division operation that exist in Toom-3, which requires quadratic number of gates for its operation
Summary
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. A few proposals [8,11,13] have explored other multiplication algorithms which traditionally offers better asymptotic time complexity, i.e., classical Karatsuba [14] and Toom–Cook multiplication [15,16] for use in quantum computation. Their work shows that the reversible version of Karatsuba is still able to maintain asymptotically lower runtime than the naïve multiplication To be precise, their analysis for sequential approach shows a similar complexity for depth, space (qubit count), and circuit size, i.e., O(nlog2 3 ) ≈ O(n1.585 ). The result of analysis shows that even though the division circuit required in Toom-3 does contributes to a higher Toffoli count, it still acquires better asymptotic complexity in terms of Toffoli depth and qubit count, still giving a competitive advantage compared to existing multiplication algorithms.
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