Computational complexity in high-dimensional quantum computing

We study an efficiency for operating a full adder/half adder by quantum-gated computing. Fortunately, we have two typical arithmetic calculations discussed in Nakamura and Nagata (Int J Theor Phys 60:70, 2021). The two typical arithmetic calculations are (1 + 1) and (2 + 3). We demonstrate some evaluations of three two-variable functions which are elements of a boolean algebra composed of the four-atom set utilizing the d-dimensional Bernstein–Vazirani algorithm. This is faster than that of its classical apparatus which would require d12 steps. Using the three two-variable functions evaluated here, we demonstrate a typical arithmetic calculation (2 + 3) in the binary system using the full adder operation. The typical arithmetic calculation is faster than that of its classical apparatus which would require d12 steps when we introduce the full adder operation. Another typical arithmetic calculation (1 + 1) is faster than that of its classical apparatus which would require d8 steps when we introduce only the half adder operation. The concrete and specific calculation (2n + 1),(n > 1) is faster than that of a classical apparatus which would require n × d12 steps. The quantum advantage increases when two numbers we treat become very large.