Quantum addition circuits and unbounded fan-out (pp0872-0890)
          Yasuhiro Takahashi, Seiichiro Tani, and Noboru Kunihiro
          doi: https://doi.org/10.26421/QIC10.9-10-12
Abstracts: We first show how to construct an O(n)-depth O(n)-size quantum circuit for addition of two n-bit binary numbers with no ancillary qubits. The exact size is 7n − 6, which is smaller than that of any other quantum circuit ever constructed for addition with no ancillary qubits. Using the circuit, we then propose a method for constructing an O(d(n))-depth O(n)-size quantum circuit for addition with O(n/d(n)) ancillary qubits for any d(n) = Ω(log n). If we are allowed to use unbounded fan-out gates with length O(n ε ) for an arbitrary small positive constant ε, we can modify the method and construct an O(e(n))-depth O(n)-size circuit with o(n) ancillary qubits for any e(n) = Ω(log* n). In particular, these methods yield efficient circuits with depth O(log n) and with depth O(log* n), respectively. We apply our circuits to constructing efficient quantum circuits for Shor’s discrete logarithm algorithm.
Key words: quantum circuits, addition, unbounded fan-out, Shor’s discrete logarithm algorithm