Quantum Information Processing, Vol. 6, No. 1, February 2007 (© 2006)
Mathematical Theory of Duality Quantum Computers
Received April 3, 2006; accepted June 11, 2006; Published online November 28, 2006
We present a mathematical theory for a new type of quantum computer called a
duality quantum computer that has recently been proposed. We discuss the nonu-
nitarity of certain circuits of a duality quantum computer and point out a para-
doxical situation that occurs when mixed states are considered. It is shown that a
duality quantum computer can measure itself without needing a separate measure-
ment apparatus to determine its ﬁnal state.
KEY WORDS: Quantum duality; divider and combiner operators; quantum
PACS: 03.67.Lx; 03.67.-a; 02.30.Tb.
In a recent paper, Long proposed a new type of quantum computer called
a duality quantum computer.
According to Long, a duality quantum
computer is much more powerful than an ordinary quantum computer.
In fact, a duality quantum computer can solve an unstructured data-
base search problem in logarithmic time and can solve NP-complete prob-
lems in polynomial time. Moreover, Long has presented proof-in-principle
designs for two possible duality quantum computers. This indicates that if
a general purpose quantum computer can be constructed, then a duality
quantum computer can probably also be constructed.
Simply stated, a quantum computer is a series of quantum gates,
represented by unitary operators U
on a Hilbert space, that can
be used to perform a computation.
An initial state ψ
into the quantum computer and then evolves into the output state ψ =
. To gain information about the computation, we make a
Department of Mathematics, University of Denver, Denver, CO 80208, USA. E-mail:
1570-0755/07/0200-0037/0 © 2006 Springer Science+Business Media, Inc.