Efficient Construction of a Control Modular Adder on a Carry-Lookahead Adder Using Relative-Phase Toffoli Gates

DOI DOI PDF Research Data 58 References Open Access
  • Kento Oonishi
    Graduate School of Information Science and Technology, The University of Tokyo, Tokyo, Japan
  • Tomoki Tanaka
    Mitsubishi UFJ Financial Group (MUFG), Inc. and MUFG Bank, Ltd., Tokyo, Japan
  • Shumpei Uno
    Quantum Computing Center, Keio University, Yokohama, Japan
  • Takahiko Satoh
    Quantum Computing Center, Keio University, Yokohama, Japan
  • Rodney Van Meter
    Quantum Computing Center, Keio University, Yokohama, Japan
  • Noboru Kunihiro
    University of Tsukuba, Tsukuba, Japan

Description

Control modular addition is a core arithmetic function, and we must consider the computational cost for actual quantum computers to realize efficient implementation. To achieve a low computational cost in a control modular adder, we focus on minimizing KQ, defined by the product of the number of qubits and the depth of the circuit. In this paper, we construct an efficient control modular adder with small KQ by using relative-phase Toffoli gates in two major types of quantum computers: Fault-Tolerant Quantum Computers (FTQ) on the Logical layer and Noisy Intermediate-Scale Quantum Computers (NISQ). We give a more efficient construction compared to Van Meter and Itoh's, based on a carry-lookahead adder. In FTQ, $T$ gates incur heavy cost due to distillation, which fabricates ancilla for running $T$ gates with high accuracy but consumes a lot of specially prepared ancilla qubits and a lot of time. Thus, we must reduce the number of $T$ gates. We propose a new control modular adder that uses only 20% of the number of $T$ gates of the original. Moreover, when we take distillation into consideration, we find that we minimize $\text{KQ}_{T}$ (the product of the number of qubits and $T$-depth) by running $��\left(n / \sqrt{\log n} \right)$ $T$ gates simultaneously. In NISQ, CNOT gates are the major error source. We propose a new control modular adder that uses only 35% of the number of CNOT gates of the original. Moreover, we show that the $\text{KQ}_{\text{CX}}$ (the product of the number of qubits and CNOT-depth) of our circuit is 38% of the original. Thus, we realize an efficient control modular adder, improving prospects for the efficient execution of arithmetic in quantum computers.

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