Quantum Arithmetic for Real-Number 2’s Complement Multiplication

In classical computing, 2’s complement representation is essential for handling signed binary arithmetic, enabling efficient operations for both positive and negative numbers. However, many quantum multiplication circuits typically lack native support for 2’s complement, limiting their application to signed arithmetic tasks. Prior quantum multiplication algorithms capable of handling 2’s complement often demand an impractically large number of qubits. This paper introduces 3 novel quantum circuit designs for real-number 2’s complement multiplication, inspired by classical signed multiplication algorithms. These designs achieve a quadratic time complexity, comparable to RCA-based quantum unsigned multiplication algorithms, while significantly reducing qubit requirements compared to prior quantum signed multiplication schemes. The first model requires the largest number of qubits but applies to a broad range of cases; the second uses fewer qubits but requires specific multiplier conditions; and the third uses the fewest qubits but operates under stricter multiplier constraints and circuit positioning conditions. Benchmarking results demonstrate their ability to perform 2’s complement multiplication correctly. This work advances qubit-efficient solutions, laying a foundation for further development of 2’s complement-based quantum algorithms.