We present a novel set of reversible modular multipliers applicable to quantum computing, derived from three classical techniques: 1) traditional integer division, 2) Montgomery residue arithmetic, and 3) Barrett reduction. Each multiplier computes an exact result for all binary input values, while maintaining the asymptotic resource complexity of a single (non-modular) integer multiplier. We additionally conduct an empirical resource analysis of our designs in order to determine the total gate count and circuit depth of each fully constructed circuit, with inputs as large as 2048 bits. Our comparative analysis considers both circuit implementations which allow for arbitrary (controlled) rotation gates, as well as those restricted to a typical fault-tolerant gate set.
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.
Arithmetic operations are an important component of many quantum algorithms. Optimizing quantum circuits for these operations therefore leads to more efficient implementations of the corresponding algorithms. In this paper, we develop new fault-tolerant quantum circuits for various integer division algorithms (both reversible and non-reversible). These circuits, when implemented in the Clifford+T gate set, achieve improvements in the asymptotic T-count and CNOT-count from
O(n2) to
O(nm), and in the asymptotic T-depth from
O(n2) to
O(nlogm), where
n and
m are the bit lengths of the dividend and divisor, respectively. The qubit counts are significantly lower than those in previous works. We achieve this through careful modifications of the division algorithms and by expressing them in terms of a primitive we call COMP-N-SUB that compares two integers and conditionally subtracts them. We show that this primitive can be implemented at a cost, in terms of both Clifford and non-Clifford gates, comparable to that of one addition. By contrast, performing comparison and conditional subtraction separately would cost roughly as much as a controlled addition plus a regular addition.
Quantum modular multiplication circuit is one of the basic quantum computation circuits which are basic functions in quantum algorithms. However, since quantum-quantum modular multipliers require a high cost reversible modular inversion routine for modular multiplication, researchers have been unable to propose a feasible quantum-quantum modular multiplier. In this paper, we proposed efficient quantum-classical modular multipliers and the first quantum-quantum modular multipliers that do not require a reduction stage by transforming the partial product used in multiplication utilizing bit-shift operation. Then, we calculated quantum resource complexity and analyzed it compared to other quantum modular multipliers and utilized ETRI (Electronics and Telecommunications Research Institute) Qcrypton to analyze quantum resource complexity in the practical quantum computing situation. The proposed quantum modular multipliers show an improvement of 50% in terms of gates and circuit depth compared to the most recently proposed high-performance quantum modular multipliers.
Reversible computing is an emerging technique to achieve ultra-low-power circuits. Reversible arithmetic circuits allow for achieving energy-efficient high-performance computational systems. Residue number systems (RNS) provide parallel and fault-tolerant additions and multiplications without carry propagation between residue digits. The parallelism and fault-tolerance features of RNS can be leveraged to achieve high-performance reversible computing. This paper proposed RNS full reversible circuits, including forward converters, modular adders and multipliers, and reverse converters used for a class of RNS moduli sets with the composite form {2k, 2p-1}. Modulo 2n-1, 2n, and 2n+1 adders and multipliers were designed using reversible gates. Besides, reversible forward and reverse converters for the 3-moduli set {2n-1, 2n+k, 2n+1} have been designed. The proposed RNS-based reversible computing approach has been applied for consecutive multiplications with an improvement of above 15% in quantum cost after the twelfth iteration, and above 27% in quantum depth after the ninth iteration. The findings show that the use of the proposed RNS-based reversible computing in convolution results in a significant improvement in quantum depth in comparison to conventional methods based on weighted binary adders and multipliers.
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.