Due to the reversibility constraints of quantum circuits, the development of modular operations in quantum modular multiplication has been significantly limited. In this work, we propose an optimized modular operation scheme based on Barrett reduction and implement quantum circuits for three versions of Barrett reduction, including our optimized approach. The proposed quantum modular operation circuit can be integrated with quantum-quantum and quantum-classical multiplication circuits. Additionally, we analyze the quantum resource requirements of the proposed circuit and compare the results across the three versions. Our optimized folding Barrett reduction scheme demonstrates superior performance. Furthermore, general quantum modular reduction relies on recursive subtractors and controlled adders to avoid division operation, resulting in a T-depth of approximately [Formula: see text], where n is the bit length of the modulus. In contrast, our approach achieves a T-depth of only [Formula: see text] by utilizing the Barrett reduction.
Modular arithmetic is central to quantum algorithms for cryptographic problems, including Shor's algorithm and Grover-based cryptanalysis, with modular reduction contributing substantially to circuit cost. Pseudo-Mersenne moduli
q=2n−c allow classical Crandall reduction to replace division with folding and constant arithmetic, providing a structural opportunity for more efficient quantum modular reduction than Barrett reduction. We translate this advantage into a reversible quantum setting by deriving explicit folding and normalization conditions for
2n-bit inputs. To the best of our knowledge, this constitutes the first exact reversible quantum circuit formulation of Crandall reduction. Based on this formulation, we develop two variants: Crandall reduction-1 is designed to minimize execution cost through one-step normalization, whereas Crandall reduction-2 uses two-step normalization to support a wider range of
c with limited overhead. Logical resource estimates show that both variants require fewer qubits and lower T-count and T-depth than optimized folding Barrett reduction. At
n=10, Crandall reduction-1 reduces both T-count and T-depth by approximately 46.9% relative to optimized folding Barrett reduction. Surface-code analysis further shows that, at
n=20 under the Sparse Blossom decoder, the estimated runtimes of the two variants are 30.05 ms and 35.39 ms, respectively, compared with 53.77 ms for optimized folding Barrett reduction. These results demonstrate the practical value of exploiting modulus-specific arithmetic structure in fault-tolerant quantum circuit design.
Quantum modular multipliers are a promising development in the field of quantum computing that aim to perform modular multiplication faster and more efficiently than classical algorithms. This is achieved by encoding the numbers being multiplied into quantum bits (qubits) and using quantum gates to manipulate them in a way that represents the calculation. Since quantum physics principles like superposition and entanglement dramatically speed up processing compared to conventional computing, many calculations may be run simultaneously. Quantum modular multipliers have the potential to have a significant impact in a range of fields, including cryptography and beyond, since they offer a faster and more efficient way to do modular multiplications. Several fields, including simulation, optimization, and cryptography, can benefit from the use of quantum computing.One of the crucial operations in many of these applications is modular multiplication, a sort of multiplication that requires obtaining the remainder of the product of two numbers when divided by a third number. In this paper, we provide a novel modular multiplication method based on a quantum algorithm. To examine its complexity and compare it to that of current adders, we employ a novel form of adder that is intended to not retain the final carry. This adder's decrease percentage in terms of gates and depth is roughly 70%. Our method takes advantage of the special qualities of qubits to complete the computation significantly more quickly than conventional algorithms. It is based on the fundamental ideas of quantum physics, such as superposition and entanglement.
The design of arithmetic and logical unit is a crucial block schematic of quantum computing, which has gained popularity in recent years. While division is a successive subtraction, a logical multiplier employs the technique of successive addition, which are the essential blocks of the quantum computing. The key components including comparator design, adders, and subtractors are considered with the help of Quantum-Dot Cellular Automata (QCA) in this work. Such essential blocks of the Arithmetic Logic Unit (ALU) are designed, implemented, and tested in the simulation using QCADesigner version 2.0.3. The results are found to be appealing with the standard truth tables of the above logical circuits. Hence, the designs could be utilized in the practical applications of Internet of Things, Block chains and Cryptographic solutions catering to the challenges of industry, healthcare and finance sectors. Addressing challenges such as a reduced number of qubits and complexity reduction could make way for more robust and practical ALU designs in QCA-based quantum systems. In the proposed novel work, the number of qubits required has been reduced to 24, offering a notable reduction compared to several recent approaches. Hence, the novelty of the work lies in the architectural-level reorganization and logic-level minimization strategy, which collectively enable a significant reduction in design complexity (up to 93%) without introducing additional fabrication or simulation overhead, optimizing clock zone assignments to enhance timing efficiency.
Quantum modular adders are one of the most fundamental yet versatile quantum computation operations. They help implement the functions of higher complexity, such as subtraction and multiplication, which are used in applications, such as quantum cryptanalysis, quantum image processing, and securing communication. To the best of our knowledge, there is no existing design of quantum modulo (
2n+1) adder (QMA). In this work, we propose four quantum adders targeted specifically for modulo (
2n+1) addition. These adders can provide both regular and modulo (
2n+1) sum concurrently, enhancing their application in residue number system-based arithmetic. Our first design, QMA1, is a novel quantum modulo (
2n+1) adder. The second proposed adder, QMA2, optimizes the utilization of quantum gates within the QMA1, resulting in 37.5% reduced CNOT gate count, 46.15% reduced CNOT depth, and 26.5% decrease in both Toffoli gates and depth. We propose a third adder QMA3 that uses zero resets, a dynamic circuits-based feature that reuses qubits, leading to 25% savings in qubit count. Our fourth design, QMA4, demonstrates the benefit of incorporating additional zero resets to achieve a purer
∣0$\rangle $state, reducing quantum state preparation errors. Notably, we conducted experiments using 5-qubit configurations of the proposed modulo (
2n+1) adders on the IBM Washington, a 127-qubit quantum computer based on the Eagle R1 architecture, to demonstrate a 28.8% reduction in QMA1’s error of which do the following: 1) 18.63% error reduction happens due to gate/depth reduction in QMA2; 2) 2.53% drop in error due to qubit reduction in QMA3; and 3) 7.64% error decreased due to application of additional zero resets in QMA4.
Shor's factorization algorithm, as one of the most significant achievements in quantum computing, exhibits an exponential speedup compared to the corresponding classical algorithm. In Shor's factorization algorithm, modular exponentiation is one of the most computationally intensive components, which relies on the modular addition building block. This work aims to explore novel designs for enhancing the efficiency of quantum modular addition. In particular, we introduce a novel quantum modular addition framework based on carry-save architecture, which facilitates the conversion of multiple 2-addend quantum operations within modular addition into a single 3-addend operation, thereby reducing the computational depth. Compared to the most efficient existing quantum modular addition, our design has achieved an impressive result - a reduction in Toffoli Depth by up to 33.33%, while maintaining comparable Toffoli Count and Qubit Count. This research underscores the potential of carry-save architecture as a promising technique for accelerating quantum modular arithmetic as well as advancing the development of quantum computing in general.
We present a quantum algorithm for multiplying two
n-bit integers with overall circuit depth and
T-depth both bounded by
O(logn), while using
O(n2) gates and ancillary qubits. Our construction generates partial products via indicator-controlled copying and adds them using a tree of reversible
3-to-
2 adders followed by one quantum carry-lookahead adder. Building on the carry-save adders of Kim et al., we combine full input fanout, workspace reuse, and phase-corrected gate implementations to reduce
T-depth. We provide explicit circuit constructions, proofs of correctness, and concrete closed-form resource bounds.