In this paper, we report efficient quantum circuits for integer multiplication using the Toom-Cook algorithm. By analyzing the recursive tree structure of the algorithm, we obtained a bound on the count of Toffoli gates and qubits. These bounds are further improved by employing reversible pebble games through uncomputing the intermediate results. The asymptotic bounds for different performance metrics of the proposed quantum circuit are superior to the prior implementations of multiplier circuits using schoolbook and Karatsuba algorithms.
In this paper, we report efficient quantum circuits for integer multiplication using the Toom-Cook algorithm. By analyzing the recursive tree structure of the algorithm, we obtained a bound on the count of Toffoli gates and qubits. These bounds are further improved by employing reversible pebble games through uncomputing the intermediate results. The asymptotic bounds for different performance metrics of the proposed quantum circuit are superior to the prior implementations of multiplier circuits using schoolbook and Karatsuba algorithms.
Integer arithmetic is the underpinning of many quantum algorithms, with applications ranging from Shor's algorithm over HHL for matrix inversion to Hamiltonian simulation algorithms. A basic objective is to keep the required resources to implement arithmetic as low as possible. This applies in particular to the number of qubits required in the implementation as for the foreseeable future this number is expected to be small. We present a reversible circuit for integer multiplication that is inspired by Karatsuba's recursive method. The main improvement over circuits that have been previously reported in the literature is an asymptotic reduction of the amount of space required from O(n1.585) to O(n1.427). This improvement is obtained in exchange for a small constant increase in the number of operations by a factor less than 2 and a small asymptotic increase in depth for the parallel version. The asymptotic improvement are obtained from analyzing pebble games on complete ternary trees.
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.
This paper examines the asymptotic performance of multiplication and the cost of quantum implementation for the Naive schoolbook, Karatsuba, and Toom-Cook methods in the classical and quantum cases and provides insights into multiplication roles in the post-quantum cryptography (PQC) era. Further, considering that the lattice-based PQC algorithm is based on polynomial multiplication algorithms, including the Toom-Cook 4-way multiplier as its fundamental building block, we propose a higher-degree multiplier, the Toom-Cook 8-way multiplier, which has the lowest asymptotic performance and implementation cost. Additionally, the designed multiplication will include additional sub-operations to complete the multiplication of large integers in order to prevent side-channel attacks. To design our Toom-Cook 8-way in detail, we employ detailed step computations such as splitting, evaluation, point-wise multiplication, interpolation, and recomposition, as well as several strategies to reduce space and time requirements. Existing asymptotic performance and quantum implementation cost multipliers are compared with our 2-way, 4-way, and 8-way Toom-Cook multiplier designs. Our Toom-Cook 8-way quantum multiplier has the lowest asymptotic performance analysis or qubit count in terms of space efficiency, withn( 15/8 )log 15 /(2 log 15–log 8) log8nor asymptoticallyO(n1.245). The design has the lowest logical Toffoli counts bound at 112nlog815– 128nand Toffoli depth ofn( 15/8 )1– log 15 /(2 log 15–log 8) log8n, asymptotically close toO(n1.0569), which corresponds to a space- and time-efficient multiplication.
This paper studies the asymptotic bounds of quantum resources required for high-degree multiplication based on the Toom–Cook algorithm. This work extends prior investigations on high- and half-degree quantum multiplication and proposes an optimized Toom–Cook 25.5-way quantum multiplication architecture to establish optimal asymptotic bounds on quantum resource consumption for Toom–Cook-based multiplication schemes. We provide asymptotic expressions for qubit complexity, Toffoli gate count, and Toffoli depth. The proposed Toom–Cook 25.5-way architecture achieves improved asymptotic performance, requiring a qubit count of n1.176, approximately 648nlog2651 − 666n Toffoli count, and n1.03025 Toffoli depth. Compared with existing classical and quantum Toom–Cook-based methods, the proposed 25.5-way Toom–Cook algorithm achieves lower asymptotic complexity and reduced quantum resource costs, yielding tighter bounds for optimal quantum multiplication.
We present a detailed resource analysis of recursive quantum multiplication circuits based on the Toom-Cook family, emphasizing exact qubit and gate counts across multiple split parameters. By limiting recursive steps to Karatsuba and Toom-3 at each level, we show closed-form recurrences for both logical qubits and T-gate counts, and validate them up to 2048-bit instances. We further characterize the padding overhead from non-divisible limb sizes and discuss the impact of different recursion strategies on circuit depth. These results establish new benchmarks for quantum arithmetic and lay the groundwork for extensions to modular and fault-tolerant architectures.
This paper presents a method for constructing quantum circuits for schoolbook multiplication using controlled add-subtract circuits, asymptotically halving the Toffoli count compared to traditional controlled-adder-based constructions. Controlled n-qubit add-subtract circuits, which perform an addition when the control qubit is one and a subtraction when it is zero, require only n-1 Toffoli gates, instead of the 2n-1 needed for controlled adders. Despite the existence of multiplication circuits with better asymptotic scaling, schoolbook multiplication yields the lowest Toffoli counts for small register sizes, making it advantageous in practical applications. For example, the presented approach reduces the Toffoli count by up to around 30% in circuits for breaking 256-bit elliptic curve keys compared to circuits with standard schoolbook multipliers.