Сomparative analysis of quantum subtraction circuits in the basis of modern quantum processors

Abstract

The paper investigates six quantum subtraction circuits implemented in the Qiskit framework: the out-of-place ripple-borrow subtractor, in-place ripple-borrow subtractor, Cuccaro ripple-carry subtractor, parallel carry-lookahead subtractor, parallel-prefix Kogge–Stone subtractor, and the QFT-based subtractor following Draper’s approach. The aim of the study is to compare these circuits in terms of qubit count, gate count, circuit depth, CNOT count, CNOT depth, T-count, and T-depth after transpilation to the {CX, T, T†, H, X, S, S†} basis. It is shown that for three-bit operands, the in-place ripple-borrow and QFT-based circuits require the fewest qubits, while the Parallel Carry-lookahead subtractor demonstrates the smallest transpiled depth. The QFT-based circuit has a low number of CNOT gates; however, after decomposing controlled phase rotations (CP) into the Clifford+T basis, its T-cost becomes several orders of magnitude higher than that of the other circuits. The obtained results confirm that the choice of subtractor architecture depends on the target computational model: for fault-tolerant Clifford+T implementations, circuits with controlled T-complexity are preferable, whereas the QFT approach is more appropriate to evaluate in native phase bases.

Overall, the results show that for small bit-widths and Clifford+T-oriented implementations, the most balanced choice is the Parallel Carry-lookahead subtractor, while ripple-borrow circuits remain attractive under qubit-limited conditions. The QFT-based approach requires separate analysis in hardware models where phase shifts are cheaper or natively supported.

From a teaching and methodological perspective, such an analysis is important for developing students’ practical understanding of the architectural diversity of quantum arithmetic circuits. Comparing different types of subtractors demonstrates that the same operation can be implemented using fundamentally different approaches, which manifest differently at both logical and physical levels. This enables students to realize that the choice of an algorithm or circuit implementation depends not only on the mathematical formulation of the problem, but also on the target quantum architecture, the native gate set, the number of available qubits, and the requirements for fault-tolerant execution.