arXiv:0711.4021v1 [quant-ph] 26 Nov 2007
How many CNOT gates does it take to generate a three-qubit state ?
MarkoˇZnidariˇ c1, Olivier Giraud2and Bertrand Georgeot2
1Department of Physics, Faculty of Mathematics and Physics,
University of Ljubljana, SI-1000 Ljubljana, Slovenia
2Laboratoire de Physique Th´ eorique, Universit´ e de Toulouse, CNRS, 31062 Toulouse, France
(Dated: November 26, 2007)
The number of two-qubit gates required to transform deterministically a three-qubit pure quantum
state into another is discussed. We show that any state can be prepared from a product state
using at most three CNOT gates, and that, starting from the GHZ state, only two suffice. As a
consequence, any three-qubit state can be transformed into any other using at most four CNOT
gates. Generalizations to other two-qubit gates are also discussed.
PACS numbers: 03.67.-a, 03.67.Ac, 03.67.Bg
Quantum information and computation (see e.g. ) is
usually described using qubits as elementary units of in-
formation which are manipulated through quantum oper-
ators. In most practical implementations, such operators
have to be realized as sequences of local transformations
acting on a few qubits at a time. Whereas one-qubit gates
alone cannot create entanglement, it has been shown that
together with two-qubit gates they can form universal
sets, from which the set of all unitary transformations of
any number of qubits can be generated . The complex-
ity of a quantum algorithm is usually measured by assess-
ing the number of elementary gates needed to perform
the computation. The Controlled-Not (CNOT) gate, a
two-qubit gate whose action can be written |00? → |00?,
|01? → |01?, |10? → |11?, |11? → |10?, is one of the most
widely used both for theory and implementations. It can
be shown that the CNOT gate together with one-qubit
gates is a universal set . Experimental implementa-
tions of a CNOT gate (or the equivalent controlled phase-
flip) have been recently reported using e.g. atom-photon
interaction in cavities , linear optics , superconduct-
ing qubits [5, 6] or ion traps [7, 8].
quantum computers are still far away, small platforms of
a few qubits exist or can be envisioned in the framework
of these existing experimental techniques. In most such
implementations, two-qubit gates such as the CNOT are
much more demanding that one-qubit gates.
While large size
Theoretical quantum computation has been usually fo-
cused on assessing the number of elementary gates to
build a given unitary operator performing a given com-
putation. Some works have tried to focus on two-qubit
gates and to minimize their number in order to build a
given unitary transformation for several qubits [9, 10].
Still, unitary transformations in many applications are
a tool to transform an initial state to a given state. It
seems therefore natural to try and assess how costly this
process is in itself. In this paper, we thus study the min-
imal number of two-qubit gates needed to change a given
quantum state to obtain another one. Of course, this
number is necessarily upper bounded by the number re-
quired for a general unitary transformation. We focus
on the case of two and three qubits. For two qubits, we
generalize the result of  and show that one CNOT
is enough to go from any given pure state to any other
one. For three qubits, we show that three CNOTs are
enough to go from |000? to any other pure state, and
that two CNOTs suffice if one starts from the GHZ state
(|000?+|111?)/√2. A corollary of the latter is thus that
four CNOTs are enough to go from any pure state to any
other pure state. The number of CNOT gates required to
go from a state to another defines a discrete distance on
the Hilbert space. Given any fixed state |ψ?, the Hilbert
space can be partitioned according to the distance to |ψ?.
It is known that if stochastic one-qubit operations are
used, entanglement of three  and four  qubits fall
into respectively two and nine different classes. Our clas-
sification according to the number of CNOTs is different,
although there are some relations. Our results generalize
to other universal two-qubit gates, in particular to the
iSWAP gate which has been shown to be implementable
for superconducting qubits .
dimensional Hilbert space C2n. The space of normal-
ized quantum states is the sphere S2n+1−1.
cost of one-qubit gates is negligible, we are interested
only in equivalence classes of states modulo local uni-
tary transformations (LU). We thus consider the sets
En= S2n+1−1/U(2)nof states nonequivalent under LU.
In the case of two and three qubits the dimension of
E2 and E3 was determined in [15, 16] and their topol-
ogy has been described in .
per we will make use of the one-qubit LU operations
matrices acting on qubit k.
y (ξ) corresponds to a rotation of the qubit
cos(ϕ)|0? + sin(ϕ)|1? ?→ cos(ϕ + ξ)|0? + sin(ϕ + ξ)|1?.
Two-qubit states. Let us first consider the two-qubit
case. It has been shown in  that one can transform
an arbitrary two-qubit state |ψ? to |00? by using only
one CNOT. Here we prove that the same holds for two
general two-qubit states |ψ? and |ψ′?.
Proof: Since E2 is homeomorphic to [0,1], only one
Throughout the pa-
(ξ) = exp(−iξσ(k)
) where the σ(k)
In particular, the oper-
are the Pauli