No Scratch Quantum Computing by Reducing Qubit Overhead for Efficient Arithmetics

Quantum arithmetic computation requires a substantial number of scratch qubits to stay reversible. These operations necessitate qubit and gate resources equivalent to those needed for the larger of the input or output registers due to state encoding. Quantum Hamiltonian Computing (QHC) introduces a novel approach by encoding input for logic operations within a single rotating quantum gate. This innovation reduces the required qubit register $ N $ to the size of the output states $ O $, where $ N = \log_2 O $. Leveraging QHC principles, we present reversible half-adder and full-adder circuits that compress the standard Toffoli + CNOT layout [Vedral et al., PRA, 54, 11, (1996)] from three-qubit and four-qubit formats for the Quantum half-adder circuit and five sequential Fredkin gates using five qubits [Moutinho et al., PRX Energy 2, 033002 (2023)] for full-adder circuit; into a two-qubit, 4$\times $4 Hilbert space. This scheme, presented here, is optimized for classical logic evaluated on quantum hardware, which due to unitary evolution can bypass classical CMOS energy limitations to certain degree. Although we avoid superposition of input and output states in this manuscript, this remains feasible in principle. We see the best application for QHC in finding the minimal qubit and gate resources needed to evaluate any truth table, advancing FPGA capabilities using integrated quantum circuits or photonics.