Quantum addition circuits and unbounded fan-out

  • Authors:

  • Yasuhiro Takahashi;Seiichiro Tani;Noboru Kunihiro

  • Affiliations:

  • NTT Communication Science Laboratories, NTT Corporation, Atsugi, Kanagawa, Japan;NTT Communication Science Laboratories, NTT Corporation, Atsugi, Kanagawa, Japan ans Quantum Computation and Information Project, ERATO-SORST, JST, Bunkyo-ku, Tokyo, Japan;Graduate School of Frontier Sciences, The University of Tokyo, Kashiwa, Chiba, Japan

  • Venue:
  • Quantum Information & Computation
  • Year:
  • 2010

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Abstract

We first show how to construct an O(n)-depth O(n)-size quantum circuit for additionof two n-bit binary numbers with no ancillary qubits. The exact size is 7n-6, whichis smaller than that of any other quantum circuit ever constructed for addition withno ancillary qubits. Using the circuit, we then propose a method for constructing anO(d(n))-depth O(n)-size quantum circuit for addition with O(n/d(n)) ancillary qubitsfor any d(n) =Ω(log n). If we are allowed to use unbounded fan-out gates with lengthO(nε) for an arbitrary small positive constant", we can modify the method and constructan 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 depthO(log* n), respectively. We apply our circuits to constructing efficient quantum circuitsfor Shor's discrete logarithm algorithm.