Quantum Information Processing, Vol. 4, No. 2, June 2005 (© 2005)
Existence of the Exact CNOT on a Quantum
Computer with the Exchange Interaction
and M. Ozawa
Received October 24, 2004; accepted February 8, 2005
We prove the existence of the exact CNOT gate on a quantum computer with the
nearest-neighbor exchange interaction in the serial operation mode. Its existence
has been an open problem, though a concrete sequence of exchange operations,
which is approximately locally equivalent to the exact CNOT, has already been
found. We found the exact values of time parameters (exchange rates between
qubits) by using computer algebraic techniques such as Gr
obner bases and resul-
tants. These techniques have been widely used for ﬁnding rigorous solutions of
simultaneous algebraic equations, and here are applied to ﬁnding quantum gates
on the decoherence-free subsystem for the ﬁrst time.
KEY WORDS: Quantum computation; decoherence-free subsystem; computer
obner basis; resultant.
PACS: 02.70.Wz, 03.65.Yz, 03.67.Lx.
The study of quantum computation explosively advanced after Shor dis-
covered a quantum algorithm that solves factorization problems much
NTT Communication Science Laboratories, NTT Corporation, 3-1, Morinosato Wakamiya,
Atsugi-shi, Kanagawa, 243-0198, Japan.
Graduate School of Informatics, Kyoto University, Yoshida-Honmachi, Sakyo-ku, Kyoto,
Department of Mathematics, Kobe University, Nada-ku, Kobe, 657-8501, Japan.
Graduate School of Engineering Science, Osaka University, Toyonaka-shi, Osaka, 560-8531,
Japan; Also at NTT Communication Science Laboratories, NTT Corporation, 3-1, Mori-
nosato Wakamiya, Atsugi-shi, Kanagawa, 243-0198, Japan; Japan Science and Technology
Graduate School of Information Sciences, Tohoku University, Aoba-ku, Sendai, 980-8579,
Japan; Also at Japan Science and Technology Agency, Japan.
1570-0755/05/0600-0065/0 © 2005 Springer Science+Business Media, Inc.