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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. [1]) 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 [2]. 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 [2]. 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 [3], linear optics [4], 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 [11] 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 [12] and four [13] 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 [14].

Weconsider purestates

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 [17].

per we will make use of the one-qubit LU operations

R(k)

jj

matrices acting on qubit k.

ation R(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 [11] 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

belonging tothe2n-

As the

Throughout the pa-

(ξ) = exp(−iξσ(k)

) where the σ(k)

In particular, the oper-

j

are the Pauli