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 qubitefficient
solutions, laying a foundation for further development
of 2’s complement-based quantum algorithms.