Low-qubit Quantum Circuits for Efficient Integer Squaring

Quantum squaring circuits play a critical role in many quantum algorithms; however, most existing designs incur a significant qubit overhead due to the loss of input states and excessive use of ancillary qubits. In this work, we introduce a qubit-efficient quantum circuit for integer squaring that achieves a linear qubit cost of only 3N qubits for an N-bit input, significantly outperforming state-of-the-art designs that scale quadratically in terms of qubits. Our approach reintegrates the input operand after computation, enabling the uncomputation of intermediate results and efficient recycling of ancilla qubits. This reversible strategy prevents the retention of redundant information, which is a common limitation of prior works. The comparative analysis confirms the scalability and practicality of our design for qubit-constrained quantum hardware, offering a promising solution for arithmetic operations in resource-limited quantum environments.