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

Authors :
Kento Oonishi
Tomoki Tanaka
Shumpei Uno
Takahiko Satoh
Rodney Van Meter
Noboru Kunihiro
Source :
IEEE Transactions on Quantum Engineering, Vol 3, Pp 1-18 (2022)
Publication Year :
2022
Publisher :
IEEE, 2022.

Abstract

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 minimizingKQ (where K is the number of logical qubits required by the algorithm, and Q is the elementary gate step), defined by the product of the number of qubits and the depth of the circuit. In this article, 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 with 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 especially 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 $\Theta (n / \sqrt{\log n})$ $T$ 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.

Details

Language :
English
ISSN :
26891808 and 38741539
Volume :
3
Database :
Directory of Open Access Journals
Journal :
IEEE Transactions on Quantum Engineering
Publication Type :
Academic Journal
Accession number :
edsdoj.7d4e4062e387415394bcbdccfe402337
Document Type :
article
Full Text :
https://doi.org/10.1109/TQE.2021.3136195