Teleportation in a nuclear spin quantum computer
Gennady P. Berman,
1
Gustavo V. Lo
´
pez,
2
and Vladimir I. Tsifrinovich
3
1
Theoretical Division and CNLS, Los Alamos National Laboratory, Los Alamos, New Mexico 87545
2
Departamento de Fı ´sica de la Universidad de Guadalajara S.R. 500, 44420 Guadalajara, Jalisco, Mexico
3
IDS Department, Polytechnic University, Six Metrotech Center, Brooklyn, New York11201
Received 12 November 2001; published 17 October 2002
We present a procedure for quantum teleportation in a nuclear spin quantum computer in which quantum
logic gates are implemented by using selective electromagnetic pulses. A sequence of pulses is combined with
single-spin measurements in the
z
basis for fast transfer of information in a spin quantum computer. We
simulated this procedure for quantum teleportation in a nuclear spin chain with a large number 201 of spins.
The systematic errors generated in the process of teleportation due to the non-resonant effects are analyzed in
detail. We demonstrate that a ‘‘2 k ’’ method provides a significant reduction of errors.
DOI: 10.1103/PhysRevA.66.042312 PACS numbers: 03.67.-a
I. INTRODUCTION
Quantum teleportation QT is a significant part of quan-
tum computations and communications. The basic idea of
QT is the following 1. Alice and Bob share two particles
whose spins are in an entangled state,
=
1
2
| 0
A
0
B
+| 1
A
1
B
), 1
where | 0 is the ground state and | 1 is the excited state.
The axes of quantization are the same for Alice and Bob.
The left state, with subscript ‘‘A,’’ refers to Alice’s particle;
the right state, with subscript ‘‘B,’’ refers to Bob’s particle.
We will neglect the effects of decoherence which can destroy
the entangled state 1, for example, into the trivial product
of two states,
=
A
| 0
A
+
A
| 1
A
)
B
| 0
B
+
B
| 1
B
). 2
The entangled state 1, unlike state 2, describes the non-
local relation between the spins of two particles. Namely, if
Alice measures the spin of her particle and obtains, say, the
ground state, | 0 , one can be sure that Bob will subsequently
measure the same state. But before the measurement is done
no one can correctly predict the result of the measurement.
Below we shall omit the symbols A and B.
Assume that Alice has another particle which is initially
in an arbitrary state,
x
=C
0
| 0 +C
1
| 1 . 3
If Alice allows this particle to interact with the particle which
is entangled with Bob’s particle, then after a few operations
on all three particles, Alice and Bob can transfer the state
x
to Bob’s particle. These operations destroy the initial en-
tanglement.
We shall now describe a sequence of unitary transforma-
tions which provide the QT 2. Assume that the initial spin
state of three particles is
0
=
1
2
C
0
| 0 +C
1
| 1 ) | 00 +| 11 )
=
1
2
C
0
| 000 +C
0
| 011 +C
1
| 100 +C
1
| 111 ),
4
where the left two states refer to Alice’s two particles, and
the right state refers to Bob’s particle. First, Alice applies a
control-NOT CNOT transformation, which is defined as
T
CNOT
ik
| ••• n
i
••• n
k
••• =| ••• n
i
••• n
i
n
k
••• ,
where n
i
n
k
means mod(2) of the n
i
+n
k
sum, and n
i
, n
k
=0,1. We order our qubits spins starting from zero and
label from right to left, | n
2
n
1
n
0
. The first gate that Alice
applies is ( T
CNOT
)
21
,
1
= T
CNOT
21
0
=
1
2
C
0
| 000 +C
0
| 011 +C
1
| 110 +C
1
| 101 ). 5
Now, a so-called A
j
transformation is defined as
A
j
| ••• 0
j
••• =
1
2
| ••• | 0 +| 1 )
j
••• ,
6
A
j
| ••• 1
j
••• =
1
2
| ••• | 0 -| 1 )
j
••• ,
and has the property that for a single qubit, A
j
is equivalent
to a Hadamard transformation, H, which is defined by
H| n
L -1
••• n
0
=
1
2
L /2
n ' =0
2
L
-1
-1
nn '
| n
L -1
' ••• n
0
' ,
where nn ' =(
k =0
L -1
n
k
n
k
' )mod(2). So, Alice applies an A
2
gate to
1
in Eq. 5 to obtain
PHYSICAL REVIEW A 66, 042312 2002
1050-2947/2002/664/0423127/$20.00 ©2002 The American Physical Society 66 042312-1