We describe the experimental implementation of a recently proposed quantum
algorithm involving quantum entanglement at the level of two qubits using NMR.
The algorithm solves a generalisation of the Deutsch problem and distinguishes
between even and odd functions using fewer function calls than is possible
classically. The manipulation of entangled states of the two qubits is
essential here, unlike the Deutsch-Jozsa algorithm and the Grover's search
algorithm for two bits.Comment: 4 pages, two eps figure

A. Steane

Anil Kumar

Arvind, 

B. Huttner

C.H. Tseng

D. Collins

D. Deutsch

D.G. Cory

D.P. DiVincenzo

E. Knill

I.L. Chuang

J. A. Jones

J. Preskill

J.A. Jones

Kavita Dorai

L.K. Grover

Lieven. M.K. Vandersypen

M.A. Nielsen

N. Gershenfeld

N. Linden

P.W. Shor

R. Cleve

R. Nelson

S. Lloyd

S.L. Braunstein

Seth Lloyd

T. S. Mahesh

Z.L. Madi

English

arXiv

We describe the NMR implementation of a recently proposed quantum algorithm involving quantum entanglement at the level of two qubits. The algorithm solves a generalization of the Deutsch problem, and distinguishes between even and odd functions using fewer function calls than is possible classically. The manipulation of entangled states of the two qubits is essential here, unlike the Deutsch-Jozsa algorithm and Grover's search algorithm for two bits

Dorai, Kavita

Arvind, Arvind

Anil Kumar,

Publications of the IAS Fellows

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Implementation of a Deutsch-like quantum algorithm utilizing entanglement at the
two-qubit level, on an NMR quantum information processor
Kavita Dorai1†, Arvind3⋆, and Anil Kumar1,2‡
1Department of Physics, and 2Sophisticated Instruments Facility, Indian Institute of Science, Bangalore 560012 India
3Department of Physics, Guru Nanak Dev University, Amritsar 143005 India
We describe the experimental implementation of a re-
cently proposed quantum algorithm involving quantum en-
tanglement at the level of two qubits using NMR. The algo-
rithm solves a generalisation of the Deutsch problem and dis-
tinguishes between even and odd functions using fewer func-
tion calls than is possible classically. The manipulation of
entangled states of the two qubits is essential here, unlike the
Deutsch-Jozsa algorithm and the Grover’s search algorithm
for two bits.
I. INTRODUCTION
It has been demonstrated recently that a quantum
computer, exploiting quantum state superposition and
entanglement, is definitely more powerful than any ex-
isting classical computer [1]. Many “quantum” al-
gorithms that achieve the same computational tasks
much faster than their classical counterparts have been
designed [2–5]. Thus far, NMR has been the most
successful method employed to physically implement
small quantum information processors and to test the
power of intrinsically quantum algorithms [6,7]. The
visualisation of a spin-1/2 particle as a qubit, com-
bined with existing multi-dimensional NMR methods
has led to breakthroughs in pseudopure state prepara-
tion [7–10], the demonstration of universal quantum logic
gates [11–15], and the implementation of various quan-
tum algorithms [16–24].
Early on, Deutsch and Jozsa presented a simple prob-
lem to determine whether a Boolean function f is con-
stant or balanced [2]. Classically, the algorithm requires
many function calls to solve the problem without error,
but a quantum computer can solve the problem using
only a single function call. The Cleve version of the algo-
rithm used an extra qubit to encode the required unitary
transformations and the solution could be read out as the
relative phase of the qubits [3]. Recently, the Deutsch-
Jozsa (DJ) problem was modified and implemented us-
ing a lesser number of input qubits [25–27], and inter-
est continues in further generalisations of the DJ prob-
lem [28]. It was realised that, for upto two bits the DJ
problem need not invoke entangling transformations for
its solution. The problem thus allows a classical descrip-
tion for the two-qubit case, and it is only for three or
more qubits that the quantum nature of the algorithm
is displayed [25,26]. As pointed out recently by Seth
Lloyd [29], Grover’s search algorithm for two qubits also
does not require entangling transformations. Hence both
these algorithms become truly quantum only for three or
more qubits.
A quantum algorithm to distinguish between even and
odd functions using fewer function calls than a classi-
cal algorithm has been recently designed [30]. The algo-
rithm uses entangling transformations at the two-qubit
level itself and is an interesting example of the power of a
quantum computer over corresponding classical systems.
In this paper, we present the experimental implemen-
tation of the algorithm on a two-qubit NMR quantum
computer. The requisite unitary transformations have
been implemented using spin-selective and composite-
z pulses. A judicious combination of composite pulses
and evolution under the scalar coupling Hamiltonian has
been used to construct the desired entangling transfor-
mations. The algorithm requires distinguishing between
non-orthogonal states of the two qubits in order to clas-
sify the functions [32]. In our NMR implementation, such
a distinction is achieved in a single measurement. The
algorithm to evaluate the even or odd nature of a func-
tion uses entangling transformations for its implemen-
tation on two input qubits. Entangling transformations
can produce entangled states that have no classical ana-
logue [31].
Consider a Boolean function defined from a two-
bit domain space to a one-bit range space: f(x) :
{0, 1}2 → {0, 1}. There are four possible input values
(00), (01), (10) and (11), the output for each of these be-
ing either 0 or 1. The 16 possible functions can be di-
vided into sub-classes based on the number of ones and
zeros in their outputs. The functions can be categorised
in the sub-classes [0, 4], [1, 3], [2, 2], [3, 1], or [4, 0], where
the first entry indicates the number of ones and the sec-
ond indicates the number of zeros in the output. The
functions with an even number of ones in the output
(the functions belonging to the categories [0, 4], [2, 2] and
[4, 0]) are denoted “even” functions while the functions
with an odd number of ones in the output (the [1, 3] and
[3, 1] functions) are said to be “odd” functions. For the
two-qubit case, we thus have 8 even and 8 odd functions.
Classically, the classification of a function as even or
odd would require computing it at all input points. The
quantum algorithm constructed uses just two function
calls to evaluate the even or odd character of the given
function [30]. The implementation requires a quantum
gate to call the function, and a judicious use of pseudo-
Hadamard transformations (on both qubits, and selec-
1
tively on one qubit alone). The function call mechanism
is similar to the one used to solve the modified Deutsch
problem [25–27]. Each function f can be encoded by a
unitary transformation Uf , with its action on the eigen-
states of the two qubits being defined as
|x〉2-bit
Uf
−→ (−1)f(x)|x〉2-bit
Uf =


(−1)f(00) 0 0 0
0 (−1)f(01) 0 0
0 0 (−1)f(10) 0
0 0 0 (−1)f(11)

 (1)
Pseudo-Hadamard gates [13] are practically equivalent to
the Hadamard operator and have been utilised through-
out our analysis. The one-qubit pseudo-Hadamard gate
is given by
|0〉
h
→ 1√
2
(|1〉+ |0〉)
|1〉
h
→ 1√
2
(|1〉 − |0〉)
; h = 1√
2
(
1 1
−1 1
)
(2)
In our analysis, the pseudo-Hadamard gate is applied
on both qubits non-selectively as well as selectively on
one qubit alone. The corresponding gates are denoted as
h(1) = h⊗ I; h(2) = I ⊗ h; h(1,2) = h⊗ h (3)
where 1 and 2 label the qubit involved.
There are sixteen Uf matrices in all, with eight of them
being entangling and the rest non-entangling in char-
acter. For example, the Uf matrix with diagonal en-
tries [1, 1, 1,−1] cannot be written as a tensor product
of two matrices, one belonging to each qubit. This uni-
tary transformation is hence entangling in nature. It is
interesting to note that for the two-bit case, the sub-class
of functions that are either constant or balanced in the
sense of the Deutsch problem i.e. the functions belong-
ing to the (0, 4) and (2, 2) sub-classes, are all separable
in character. Therefore, the two-bit Deutsch problem
affords a classical explanation and can be implemented
using non-entangling transformations alone.
h
h
Uf
h
Uf
h-1
h-1
FIG. 1. Quantum circuit for two qubits, that implements
the algorithm to distinguish between even and odd functions,
using pseudo-Hadamard gates (denoted by h) and the Uf
transformations. Pseudo-Hadamard gates h convert an eigen-
state to a uniform superposition of all possible eigenstates
while h−1 implement the inverse operation.
A quantum circuit to evaluate the even or odd na-
ture of a function is shown in Figure 1. The algorithm
starts with both qubits in a thermal (mixed) initial state.
A pseudo-Hadamard transformation is applied on both
qubits, resulting in both of them being in a state which
is a superposition of all possible input states.
The desired unitary transformation Uf is then ap-
plied on this superposition state, followed by a pseudo-
Hadamard gate operating on the second qubit alone.
The function is called again through Uf and an inverse
pseudo-Hadamard is applied on both qubits before the
read-out operation. The result of the computation is en-
coded in the final state of the two qubits. If the func-
tion is even, both the qubits are in an unentangled state
and a signal is obtained for the qubit on which the se-
lective pseudo-Hadamard gate is applied, with no signal
being obtained for the other qubit. If the function is odd,
both qubits are in an entangled state, corresponding to
multiple-quantum coherence of the two spins, and no ob-
servable signal is obtained. The even or odd nature of
the function can thus be distinguished “pictorially”, by
looking at the final NMR spectrum.
Previously, the algorithm has been described for a pure
initial state [30]. In NMR one normally encounters mixed
states. Hence we first generalise the procedure to include
mixed initial states. Consider the two qubits in an initial
state described by the deviation density matrix
∆ρinitial =


k1 0 0 0
0 0 0 0
0 0 0 0
0 0 0 k2

 (4)
with k1 and k2 being independent variables. For k1 = 1
and k2 = 0 this density matrix represents a pseudopure
state. For k1 = 1 and k2 = −1 it represents a thermal
state. Multiples of identity have been ignored in the den-
sity matrix, as they do not contribute to the measured
signal in an NMR experiment.
We now evolve this density matrix through the se-
quence of operations h(1,2) Uf h
(2) Uf
[
h(1,2)
]−1
. After
some algebra, this leads to the deviation density matrix
∆ρeven =
1
2


k2 ζk2 0 0
ζk2 k2 0 0
0 0 k1 ζk1
0 0 ζk1 k1

 , and
∆ρodd =
1
2


k1 + k2 0 0 ζ(k1 − k2)
0 0 0 0
0 0 0 0
ζ(k1 − k2) 0 0 k1 + k2

 (5)
where ζ = +1 or −1, for different Uf transformations.
For k1 = 1 and k2 = 0 we arrive at the the pure state
result of [30] i.e., the final state is |00〉 + ζ|01〉 for an
even function and |00〉+ ζ|11〉 for an odd function. For
2
a thermal initial state i.e. k1 = −k2 = 1, the devia-
tion density matrices ∆ρeven and ∆ρodd, can be distin-
guished unambiguously by a single NMR measurement.
The spectrum for even functions will give two lines cor-
responding to the observable single quantum coherences
present in the density matrix. For odd functions there is
no signal as the only non-diagonal elements of the density
matrix are the ones corresponding to double-quantum co-
herences. We note here that a single NMR measurement
is able to distinguish between two non-orthogonal quan-
tum states which is normally not possible using other
measurement techniques [32]. We further note that the
presence of double-quantum coherences in the final state
for the odd functions shows the entangling nature of the
unitary transformations used. The use of a Hadamard in-
stead of a pseudo-Hadamard transformation would lead
to results which are qualitatively similar. The entangle-
ment in the final state would show up as a zero-quantum
coherence of the two qubits, instead of a double-quantum
coherence [30]. We have implemented the two-qubit
even/odd quantum algorithm using the molecule of 5-
Fluorouracil (dissolved in DMSO) as an NMR quantum
computer, with the fluorine and the ortho-proton being
identified as the two input qubits. This fluorine-proton
spin system is a good candidate for quantum computing
since it has good sensitivity, a resolved J-coupling of 6.1
Hz and the duration of spin-selective pulses can be rel-
atively short. All experiments have been performed on
a Bruker AMX-400 spectrometer at room temperature.
The pseudo-hadamard gate has been achieved by apply-
ing a (900)y pulse selectively on a spin, or non-selectively
on both spins, as the case maybe.
Even Function U4:
pi
2
x
pi
2
-x
pi
y
pi
2
x
pi
2
-x
pi
y
Odd Function U9:
pi
2
x
pi
2
y
pi
2
-x
τ/2 pi
y
τ/2
pi
2
x
pi
2
y
pi
2
-x
τ/2 pi
y
τ/2
qubit 2
qubit 2
qubit 1
qubit 1
FIG. 2. NMR pulse sequences to implement the even func-
tion U4 and the odd function U9. The pulses are represented
by boxes, with the phase of each pulse written above it.
Composite-z pulses are implemented by three pulses, applied
back-to-back. The time period τ is set to 1/2J , J being the
value of the spin-spin coupling. The pulse schemes for the
other functions are similar and can be constructed by varying
the phase of the composite-z pulses.
We have used pulse schemes consisting of sandwiches of
composite and spin-selective pulses (described schemat-
ically in Figure 2), to implement the even and odd Uf
transformations.
As an illustration, the unitary transformation U4
(given by the diagonal matrix [1,−1,−1, 1]) corresponds
to a pi-rotation about the z-axis of both the qubits, upto
a global phase factor. Global phase changes are not de-
tectable in NMR and are hence ignored for the purposes
of the experiment. The z-rotation can be implemented
using a composite-pulse sandwich, as a set of rotations
about the x and y axes [θ]z ≡ [pi/2]x[θ]y[pi/2]−x The Uf ’s
encoding the other even functions U1, U2 and U3 corre-
spond to the do-nothing operation (the transformation is
the unity matrix), a [pi]z rotation in the single-spin sub-
space of the first qubit, and a [pi]z rotation on the second
qubit respectively, and have been constructed using sim-
ilar pulse schemes. All these even transformations can
be decomposed into transformations in the subspaces of
each individual qubit and have hence been implemented
experimentally without invoking quantum entanglement.
The result of applying all the even transformations on
the two qubits is shown in Fig 3. Spin-selective pulses of
12.7µ secs on the proton and 22.1µ secs on the fluorine
have been used to achieve good selectivity. The spectra
obtained reveal a retention of the fluorine spin single-
quantum coherence, while no lines are seen for the proton
(whose coherence has been converted back to unobserv-
able z-magnetization by the selective pseudo-Hadamard
gate).
2U
3U
4U
1U
F19 H1
FIG. 3. The implementation of the even transformations
U1 − U4 on the two qubits (
19F and 1H) of 5-Fluorouracil.
The corresponding unitary matrices have entries along the
diagonal of [1,1,1,1], [1,1,-1,-1], [1,-1,1,-1] and [1,-1,-1,1] re-
spectively. The transformations U5 − U8 (Ui+4 = −Ui,
i = 1, 2, 3, 4), lead to exactly the same results (spectra not
shown). All spectra have been plotted to the same scale.
We have used entangling transformations to implement
the odd functions. Consider the non-separable U9 matrix
with the entries [1,−1,−1,−1] along its diagonal. The
transformation is achieved experimentally by a [pi/2]z ro-
3
tation on the first spin, followed by a [pi/2]z rotation on
the second spin, and then a free evolution for a time inter-
val τ tailored to τ = 1/2J , J being the value of the scalar
coupling. A non-selective pi pulse has been applied in the
middle of the τ interval, to refocus the chemical shift evo-
lution. The [pi/2]z rotations in the single-spin subspaces
have been achieved by the composite pulse sandwiches
[pi/2]x[pi/2]y[pi/2]−x, selective on the spin concerned. The
other odd functions have been similarly implemented.
The spectra corresponding to the experimental imple-
mentation of the odd functions is shown in Fig. 4. The
states of the two qubits are completely entangled, leading
to no observable spectral lines for all the these functions.
All the spectra in both Figs. 3 and 4 have been plotted
to the same scale.
10U
11U
12U
9U
F19 H1
FIG. 4. The implementation of odd functions U9 − U12 on
5-Fluorouracil, using entangling transformations. The corre-
sponding unitary matrices have entries along the diagonal of
[1,-1,-1,-1],[-1,1,-1,-1],[-1,-1,1,-1] and [-1,-1,-1,1] respectively.
The odd functions U13 − U16 (Ui+4 = Ui, i = 9, 10, 11, 12)
lead to the same results (spectra not shown). A τ period of
82.24 ms has been used to implement the evolution under the
scalar coupling Hamiltonian. All the spectra shown have been
plotted to the same scale as in Fig. 3.
In conclusion, we have shown the NMR implemen-
tation of a quantum algorithm that uses the entangled
states of two input qubits to distinguish between even
and odd functions. We have used spin-selective, compos-
ite pulse sandwiches and evolution under scalar coupling
to implement the required non-trivial entangling trans-
formations. It is interesting that NMR experiments are
able to distinguish between two non-orthogonal quantum
mechanical states in a single measurement.
SIF at IISc Bangalore is acknowledged for the use of
the AMX-400 spectrometer and IISc for financial sup-
port.
⋆ arvind@physics.iisc.ernet.in
† kavita@physics.iisc.ernet.in
‡ anilnmr@physics.iisc.ernet.in
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5


Implementation of a Deutsch-like quantum algorithm utilizing entanglement at the two-qubit level on an NMR quantum-information processor

null Arvind

Crossref

Physical Review A

Indian Academy of Sciences

ar
X
iv
:q
ua
nt
-p
h/
00
06
10
3v
1 
 2
2 
Ju
n 
20
00
Implementation of a Deutsch-like quantum algorithm utilizing entanglement at the
two-qubit level, on an NMR quantum information processor
Kavita Dorai1†, Arvind3⋆, and Anil Kumar1,2‡
1Department of Physics, and 2Sophisticated Instruments Facility, Indian Institute of Science, Bangalore 560012 India
3Department of Physics, Guru Nanak Dev University, Amritsar 143005 India
We describe the experimental implementation of a re-
cently proposed quantum algorithm involving quantum en-
tanglement at the level of two qubits using NMR. The algo-
rithm solves a generalisation of the Deutsch problem and dis-
tinguishes between even and odd functions using fewer func-
tion calls than is possible classically. The manipulation of
entangled states of the two qubits is essential here, unlike the
Deutsch-Jozsa algorithm and the Grover’s search algorithm
for two bits.
I. INTRODUCTION
It has been demonstrated recently that a quantum
computer, exploiting quantum state superposition and
entanglement, is definitely more powerful than any ex-
isting classical computer [1]. Many “quantum” al-
gorithms that achieve the same computational tasks
much faster than their classical counterparts have been
designed [2–5]. Thus far, NMR has been the most
successful method employed to physically implement
small quantum information processors and to test the
power of intrinsically quantum algorithms [6,7]. The
visualisation of a spin-1/2 particle as a qubit, com-
bined with existing multi-dimensional NMR methods
has led to breakthroughs in pseudopure state prepara-
tion [7–10], the demonstration of universal quantum logic
gates [11–15], and the implementation of various quan-
tum algorithms [16–24].
Early on, Deutsch and Jozsa presented a simple prob-
lem to determine whether a Boolean function f is con-
stant or balanced [2]. Classically, the algorithm requires
many function calls to solve the problem without error,
but a quantum computer can solve the problem using
only a single function call. The Cleve version of the algo-
rithm used an extra qubit to encode the required unitary
transformations and the solution could be read out as the
relative phase of the qubits [3]. Recently, the Deutsch-
Jozsa (DJ) problem was modified and implemented us-
ing a lesser number of input qubits [25–27], and inter-
est continues in further generalisations of the DJ prob-
lem [28]. It was realised that, for upto two bits the DJ
problem need not invoke entangling transformations for
its solution. The problem thus allows a classical descrip-
tion for the two-qubit case, and it is only for three or
more qubits that the quantum nature of the algorithm
is displayed [25,26]. As pointed out recently by Seth
Lloyd [29], Grover’s search algorithm for two qubits also
does not require entangling transformations. Hence both
these algorithms become truly quantum only for three or
more qubits.
A quantum algorithm to distinguish between even and
odd functions using fewer function calls than a classi-
cal algorithm has been recently designed [30]. The algo-
rithm uses entangling transformations at the two-qubit
level itself and is an interesting example of the power of a
quantum computer over corresponding classical systems.
In this paper, we present the experimental implemen-
tation of the algorithm on a two-qubit NMR quantum
computer. The requisite unitary transformations have
been implemented using spin-selective and composite-
z pulses. A judicious combination of composite pulses
and evolution under the scalar coupling Hamiltonian has
been used to construct the desired entangling transfor-
mations. The algorithm requires distinguishing between
non-orthogonal states of the two qubits in order to clas-
sify the functions [32]. In our NMR implementation, such
a distinction is achieved in a single measurement. The
algorithm to evaluate the even or odd nature of a func-
tion uses entangling transformations for its implemen-
tation on two input qubits. Entangling transformations
can produce entangled states that have no classical ana-
logue [31].
Consider a Boolean function defined from a two-
bit domain space to a one-bit range space: f(x) :
{0, 1}2 → {0, 1}. There are four possible input values
(00), (01), (10) and (11), the output for each of these be-
ing either 0 or 1. The 16 possible functions can be di-
vided into sub-classes based on the number of ones and
zeros in their outputs. The functions can be categorised
in the sub-classes [0, 4], [1, 3], [2, 2], [3, 1], or [4, 0], where
the first entry indicates the number of ones and the sec-
ond indicates the number of zeros in the output. The
functions with an even number of ones in the output
(the functions belonging to the categories [0, 4], [2, 2] and
[4, 0]) are denoted “even” functions while the functions
with an odd number of ones in the output (the [1, 3] and
[3, 1] functions) are said to be “odd” functions. For the
two-qubit case, we thus have 8 even and 8 odd functions.
Classically, the classification of a function as even or
odd would require computing it at all input points. The
quantum algorithm constructed uses just two function
calls to evaluate the even or odd character of the given
function [30]. The implementation requires a quantum
gate to call the function, and a judicious use of pseudo-
Hadamard transformations (on both qubits, and selec-
1
tively on one qubit alone). The function call mechanism
is similar to the one used to solve the modified Deutsch
problem [25–27]. Each function f can be encoded by a
unitary transformation Uf , with its action on the eigen-
states of the two qubits being defined as
|x〉2-bit
Uf
−→ (−1)f(x)|x〉2-bit
Uf =




(−1)f(00) 0 0 0
0 (−1)f(01) 0 0
0 0 (−1)f(10) 0
0 0 0 (−1)f(11)




(1)
Pseudo-Hadamard gates [13] are practically equivalent to
the Hadamard operator and have been utilised through-
out our analysis. The one-qubit pseudo-Hadamard gate
is given by
|0〉
h
→ 1√
2
(|1〉 + |0〉)
|1〉
h
→ 1√
2
(|1〉 − |0〉)
; h = 1√
2
(
1 1
−1 1
)
(2)
In our analysis, the pseudo-Hadamard gate is applied
on both qubits non-selectively as well as selectively on
one qubit alone. The corresponding gates are denoted as
h(1) = h ⊗ I; h(2) = I ⊗ h; h(1,2) = h ⊗ h (3)
where 1 and 2 label the qubit involved.
There are sixteen Uf matrices in all, with eight of them
being entangling and the rest non-entangling in char-
acter. For example, the Uf matrix with diagonal en-
tries [1, 1, 1,−1] cannot be written as a tensor product
of two matrices, one belonging to each qubit. This uni-
tary transformation is hence entangling in nature. It is
interesting to note that for the two-bit case, the sub-class
of functions that are either constant or balanced in the
sense of the Deutsch problem i.e. the functions belong-
ing to the (0, 4) and (2, 2) sub-classes, are all separable
in character. Therefore, the two-bit Deutsch problem
affords a classical explanation and can be implemented
using non-entangling transformations alone.
h
h
Uf
h
Uf
h-1
h-1
FIG. 1. Quantum circuit for two qubits, that implements
the algorithm to distinguish between even and odd functions,
using pseudo-Hadamard gates (denoted by h) and the Uf
transformations. Pseudo-Hadamard gates h convert an eigen-
state to a uniform superposition of all possible eigenstates
while h−1 implement the inverse operation.
A quantum circuit to evaluate the even or odd na-
ture of a function is shown in Figure 1. The algorithm
starts with both qubits in a thermal (mixed) initial state.
A pseudo-Hadamard transformation is applied on both
qubits, resulting in both of them being in a state which
is a superposition of all possible input states.
The desired unitary transformation Uf is then ap-
plied on this superposition state, followed by a pseudo-
Hadamard gate operating on the second qubit alone.
The function is called again through Uf and an inverse
pseudo-Hadamard is applied on both qubits before the
read-out operation. The result of the computation is en-
coded in the final state of the two qubits. If the func-
tion is even, both the qubits are in an unentangled state
and a signal is obtained for the qubit on which the se-
lective pseudo-Hadamard gate is applied, with no signal
being obtained for the other qubit. If the function is odd,
both qubits are in an entangled state, corresponding to
multiple-quantum coherence of the two spins, and no ob-
servable signal is obtained. The even or odd nature of
the function can thus be distinguished “pictorially”, by
looking at the final NMR spectrum.
Previously, the algorithm has been described for a pure
initial state [30]. In NMR one normally encounters mixed
states. Hence we first generalise the procedure to include
mixed initial states. Consider the two qubits in an initial
state described by the deviation density matrix
∆ρinitial =




k1 0 0 0
0 0 0 0
0 0 0 0
0 0 0 k2




(4)
with k1 and k2 being independent variables. For k1 = 1
and k2 = 0 this density matrix represents a pseudopure
state. For k1 = 1 and k2 = −1 it represents a thermal
state. Multiples of identity have been ignored in the den-
sity matrix, as they do not contribute to the measured
signal in an NMR experiment.
We now evolve this density matrix through the se-
quence of operations h(1,2) Uf h
(2) Uf
[
h(1,2)
]−1
. After
some algebra, this leads to the deviation density matrix
∆ρeven =
1
2




k2 ζk2 0 0
ζk2 k2 0 0
0 0 k1 ζk1
0 0 ζk1 k1




, and
∆ρodd =
1
2




k1 + k2 0 0 ζ(k1 − k2)
0 0 0 0
0 0 0 0
ζ(k1 − k2) 0 0 k1 + k2




(5)
where ζ = +1 or −1, for different Uf transformations.
For k1 = 1 and k2 = 0 we arrive at the the pure state
result of [30] i.e., the final state is |00〉 + ζ|01〉 for an
even function and |00〉+ ζ|11〉 for an odd function. For
2
a thermal initial state i.e. k1 = −k2 = 1, the devia-
tion density matrices ∆ρeven and ∆ρodd, can be distin-
guished unambiguously by a single NMR measurement.
The spectrum for even functions will give two lines cor-
responding to the observable single quantum coherences
present in the density matrix. For odd functions there is
no signal as the only non-diagonal elements of the density
matrix are the ones corresponding to double-quantum co-
herences. We note here that a single NMR measurement
is able to distinguish between two non-orthogonal quan-
tum states which is normally not possible using other
measurement techniques [32]. We further note that the
presence of double-quantum coherences in the final state
for the odd functions shows the entangling nature of the
unitary transformations used. The use of a Hadamard in-
stead of a pseudo-Hadamard transformation would lead
to results which are qualitatively similar. The entangle-
ment in the final state would show up as a zero-quantum
coherence of the two qubits, instead of a double-quantum
coherence [30]. We have implemented the two-qubit
even/odd quantum algorithm using the molecule of 5-
Fluorouracil (dissolved in DMSO) as an NMR quantum
computer, with the fluorine and the ortho-proton being
identified as the two input qubits. This fluorine-proton
spin system is a good candidate for quantum computing
since it has good sensitivity, a resolved J-coupling of 6.1
Hz and the duration of spin-selective pulses can be rel-
atively short. All experiments have been performed on
a Bruker AMX-400 spectrometer at room temperature.
The pseudo-hadamard gate has been achieved by apply-
ing a (900)y pulse selectively on a spin, or non-selectively
on both spins, as the case maybe.
Even Function U4:
π
2
x
π
2
-x
π
y
π
2
x
π
2
-x
π
y
Odd Function U9:
π
2
x
π
2
y
π
2
-x
τ/2 π
y
τ/2
π
2
x
π
2
y
π
2
-x
τ/2 π
y
τ/2
qubit 2
qubit 2
qubit 1
qubit 1
FIG. 2. NMR pulse sequences to implement the even func-
tion U4 and the odd function U9. The pulses are represented
by boxes, with the phase of each pulse written above it.
Composite-z pulses are implemented by three pulses, applied
back-to-back. The time period τ is set to 1/2J , J being the
value of the spin-spin coupling. The pulse schemes for the
other functions are similar and can be constructed by varying
the phase of the composite-z pulses.
We have used pulse schemes consisting of sandwiches of
composite and spin-selective pulses (described schemat-
ically in Figure 2), to implement the even and odd Uf
transformations.
As an illustration, the unitary transformation U4
(given by the diagonal matrix [1,−1,−1, 1]) corresponds
to a π-rotation about the z-axis of both the qubits, upto
a global phase factor. Global phase changes are not de-
tectable in NMR and are hence ignored for the purposes
of the experiment. The z-rotation can be implemented
using a composite-pulse sandwich, as a set of rotations
about the x and y axes [θ]z ≡ [π/2]x[θ]y[π/2]−x The Uf ’s
encoding the other even functions U1, U2 and U3 corre-
spond to the do-nothing operation (the transformation is
the unity matrix), a [π]z rotation in the single-spin sub-
space of the first qubit, and a [π]z rotation on the second
qubit respectively, and have been constructed using sim-
ilar pulse schemes. All these even transformations can
be decomposed into transformations in the subspaces of
each individual qubit and have hence been implemented
experimentally without invoking quantum entanglement.
The result of applying all the even transformations on
the two qubits is shown in Fig 3. Spin-selective pulses of
12.7 µ secs on the proton and 22.1 µ secs on the fluorine
have been used to achieve good selectivity. The spectra
obtained reveal a retention of the fluorine spin single-
quantum coherence, while no lines are seen for the proton
(whose coherence has been converted back to unobserv-
able z-magnetization by the selective pseudo-Hadamard
gate).
2U
3U
4U
1U
F19 H1
FIG. 3. The implementation of the even transformations
U1 − U4 on the two qubits (
19F and 1H) of 5-Fluorouracil.
The corresponding unitary matrices have entries along the
diagonal of [1,1,1,1], [1,1,-1,-1], [1,-1,1,-1] and [1,-1,-1,1] re-
spectively. The transformations U5 − U8 (Ui+4 = −Ui,
i = 1, 2, 3, 4), lead to exactly the same results (spectra not
shown). All spectra have been plotted to the same scale.
We have used entangling transformations to implement
the odd functions. Consider the non-separable U9 matrix
with the entries [1,−1,−1,−1] along its diagonal. The
transformation is achieved experimentally by a [π/2]z ro-
3
tation on the first spin, followed by a [π/2]z rotation on
the second spin, and then a free evolution for a time inter-
val τ tailored to τ = 1/2J , J being the value of the scalar
coupling. A non-selective π pulse has been applied in the
middle of the τ interval, to refocus the chemical shift evo-
lution. The [π/2]z rotations in the single-spin subspaces
have been achieved by the composite pulse sandwiches
[π/2]x[π/2]y[π/2]−x, selective on the spin concerned. The
other odd functions have been similarly implemented.
The spectra corresponding to the experimental imple-
mentation of the odd functions is shown in Fig. 4. The
states of the two qubits are completely entangled, leading
to no observable spectral lines for all the these functions.
All the spectra in both Figs. 3 and 4 have been plotted
to the same scale.
10U
11U
12U
9U
F19 H1
FIG. 4. The implementation of odd functions U9 − U12 on
5-Fluorouracil, using entangling transformations. The corre-
sponding unitary matrices have entries along the diagonal of
[1,-1,-1,-1],[-1,1,-1,-1],[-1,-1,1,-1] and [-1,-1,-1,1] respectively.
The odd functions U13 − U16 (Ui+4 = Ui, i = 9, 10, 11, 12)
lead to the same results (spectra not shown). A τ period of
82.24 ms has been used to implement the evolution under the
scalar coupling Hamiltonian. All the spectra shown have been
plotted to the same scale as in Fig. 3.
In conclusion, we have shown the NMR implemen-
tation of a quantum algorithm that uses the entangled
states of two input qubits to distinguish between even
and odd functions. We have used spin-selective, compos-
ite pulse sandwiches and evolution under scalar coupling
to implement the required non-trivial entangling trans-
formations. It is interesting that NMR experiments are
able to distinguish between two non-orthogonal quantum
mechanical states in a single measurement.
SIF at IISc Bangalore is acknowledged for the use of
the AMX-400 spectrometer and IISc for financial sup-
port.
⋆ arvind@physics.iisc.ernet.in
† kavita@physics.iisc.ernet.in
‡ anilnmr@physics.iisc.ernet.in
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5


Implementation of a Deutsch-like quantum algorithm utilizing
  entanglement at the two-qubit level, on an NMR quantum information processor

Implementation of a Deutsch-like quantum algorithm utilizing entanglement at the two-qubit level, on an NMR quantum information processor

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