Suppression of 1/f noise in one-qubit systems
Suppression of 1/fα noise in one-qubit systems
Pekko Kuopanportti,
Mikko Möttönen,
Ville Bergholm,
Olli-Pentti Saira,
Jun Zhang,
and K. Birgitta Whaley
Laboratory of Physi
s, Helsinki University of Te
hnology P. O. Box 4100, 02015 TKK, Finland
Low Temperature Laboratory, Helsinki University of Te
hnology, P.O. Box 3500, 02015 TKK, Finland
Department of Chemistry and Pitzer Center for Theoreti
al Chemistry, University of California, Berkeley, CA 94720
(Dated: O
tober 26, 2018)
We investigate the generation of quantum operations for one-qubit systems under 
lassi
al noise
with 1/fα power spe
trum, where 2 > α > 0. We present an e�
ient way to approximate the
noise with a dis
rete multi-state Markovian �u
tuator. With this method, the average temporal
evolution of the qubit density matrix under 1/fα noise 
an be feasibly determined from re
ently
derived deterministi
 master equations. We obtain qubit operations su
h as quantum memory and
the NOT gate to high �delity by a gradient based optimization algorithm. For the NOT gate, the

omputed �delities are qualitatively similar to those obtained earlier for random telegraph noise.
In the 
ase of quantum memory however, we observe a nonmonotoni
 dependen
y of the �delity on
the operation time, yielding a natural a

ess rate of the memory.
I. INTRODUCTION
In solid-state realization of qubits, material spe
i�
�u
tuations typi
ally indu
e the major 
ontribution to
the intrinsi
 noise. Mu
h e�ort has been fo
used on
the preservation of the state in a quantum memory in
the presen
e of 1/fα noise sin
e this is a ubiquitous
form of noise en
ountered in solid-state qubit appli
a-
tions [1, 2, 3℄. Both 
harge and spin qubits are sus-

eptible to noise of this form. For Josephson jun
tions,
both 
harge noise [4, 5℄ and 
riti
al 
urrent noise [6, 7℄
have been measured to have 1/fα power spe
tral densi-
ties. Similar 
harge �u
tuations are responsible for the
well-known 1/fα nature of low frequen
y noise in sin-
gle ele
tron transistors [8℄. Ba
kground 
harge �u
tua-
tions resulting in 1/fα noise spe
tra are 
onsidered to
be the most important sour
e of dephasing in Joseph-
son jun
tion qubits [4, 5, 9℄. Spin qubits su
h as those
formed from donor spins in semi
ondu
tors are sus
epti-
ble to nu
lear spin noise deriving from dipolar 
oupling
between environmental nu
lear spins. The nu
lear spin
bath 
ouples to the donor spins by hyper�ne intera
-
tions, whi
h renders the dynami
s of the nu
lear spins
to 
ause dephasing. Re
ent 
al
ulations for a phospho-
rus donor in sili
on show that the high frequen
y 
om-
ponent of the nu
lear spin noise is approximately de-
s
ribed by a 1/fα power spe
trum [10℄. Ele
tron spin
qubits implanted into sili
on [11℄ are also a�e
ted by
relaxation of dangling bonds deriving from oxygen va-

an
ies at the Si/SiO2 interfa
e. This gives rise to a
magneti
 noise with a 1/fα spe
trum that is the dom-
inant me
hanism for phase �u
tuations of donor spins
near the surfa
e [12℄. Another form of noise 
losely re-
lated to 1/fα noise is random telegraph noise (RTN),
whi
h arises from 
oupling of individual bistable �u
tu-
Ele
troni
 address: pekko.kuopanportti�tkk.�
ators to a qubit [2, 13, 14, 15, 16, 17, 18, 19℄.
Several approa
hes to suppress de
oheren
e based on
pulse design have been proposed in the literature. Among
them, dynami
al de
oupling s
hemes average out the un-
wanted e�e
ts of the environmental intera
tion through
the appli
ation of suitable 
ontrol pulses [20, 21℄. Appli-

ation of these s
hemes often involves hard pulses with
instantaneous swit
hings and unbounded 
ontrol ampli-
tudes, resulting in a range of validity restri
ted to time
s
ales for whi
h the pulse duration is mu
h less than the
noise 
orrelation time [22, 23℄. In Ref. [24℄, a dire
t pulse
optimization method restri
ted to bounded 
ontrol pulses
was developed for implementing one-qubit operations in a
noisy environment. This initial work on noise suppression
addressed the example of a single qubit system under the
in�uen
e of 
lassi
ally modeled random telegraph noise,
su
h as might arise from a single bistable �u
tuator.
In this paper, we extend the work of Ref. [24℄ to
the physi
ally relevant situation of 1/fα noise where
2 > α > 0. This kind of noise is known to result, for ex-
ample, from a set of bistable �u
tuators [25, 26, 27, 28℄,
i.e., RTN sour
es. We investigate two ways to approxi-
mate the 1/fα noise for 
omputer simulations, namely,
the sum of independent RTN �u
tuators and a single
dis
rete multi-state Markovian noise sour
e. We show
that the single �u
tuator provides a mu
h more e�
ient
way to model 1/fα noise than independent RTN �u
tu-
ators. Furthermore, the average temporal evolution of
the density matrix under this Markovian noise 
an be
exa
tly des
ribed by a set of deterministi
 master equa-
tions derived in Ref. [29℄. Using this approa
h, we avoid
the heavy 
omputational task arising from the numeri
al
evaluation of the density matrix averaged over a large
number of di�erent sample paths of the noise as 
om-
puted in Ref. [24℄. This framework will not only signif-
i
antly a

elerate the 
onvergen
e of the 
ontrol pulse
sequen
e optimization, but also allows further theoreti-

al analysis. Using these master equations, we employ
gradient based optimization pro
edures to obtain pulse
sequen
es that suppress 1/fα noise for quantum mem-
http://arxiv.org/abs/0704.0771v1
mailto:pekko.kuopanportti@tkk.fi
ory and for a NOT gate. Comparisons with 
omposite
pulses designed to eliminate systemati
 errors and with
refo
using pulses demonstrate that the numeri
ally opti-
mized pulse sequen
es yield the highest �delities.
The remainder of this paper is organized as follows.
In Se
. II, we show how to e�
iently approximate the
1/fα noise by a multi-state Markovian �u
tuator. In
Se
. III, we de�ne the �delity of qubit operations, re-
view the master equations des
ribing the average evolu-
tion of the qubit density matrix in the presen
e of the
noise and des
ribe the numerial optimization pro
edure.
Se
tions IV and V present optimized 
ontrol pulse se-
quen
es and the a
hieved �delities for quantum memory
and for the NOT gate, respe
tively. Finally, Se
. VI 
on-

ludes and indi
ates further appli
ations of the method.
II. ONE-QUBIT SYSTEM SUBJECT TO 1/fα
NOISE
We 
onsider a one-qubit system des
ribed by the e�e
-
tive Hamiltonian
a(t)σx +
η(t)σz , (1)
where a(t) ∈ [−amax, amax] is the external 
ontrol �eld
applied along the x dire
tion and η(t) is the 
lassi
al
noise signal perturbing the system along the z dire
tion.
The noise sour
e η(t) 
an be 
hara
terized by its auto-

orrelation fun
tion
C(t) ≡ 〈η(0)η(t)〉 = lim
∫ T/2
η(s)η(s+ t) ds, (2)
the Fourier transformation of whi
h de�nes the noise
power spe
tral density as
S(f) =
C(t)e−i2πftdt. (3)
For a single RTN sour
e with the amplitude ∆ and 
or-
relation time τc, the auto
orrelation fun
tion is given
by [30℄
(t) = ∆2e−2|t|/τc, (4)
and the 
orresponding power spe
tral density by
(f) =
1 + (πfτc)2
. (5)
A standard way to simulate 1/fα noise is to use
an ensemble of K independent un
orrelated RTN pro-

esses [1, 25, 27℄. Let ηk(t) be a symmetri
 RTN signal
swit
hing between values −∆k and ∆k with the 
orre-
lation time τk ≡ 1/γk, where γk is the transition rate
between the two states. The total noise pro
ess appears
in the Hamiltonian (1) as η(t) =
k=1 ηk(t). Sin
e the
RTN sour
es are independent, Eqs. (2) and (4) yield the
auto
orrelation fun
tion
C(t) =
−2|t|/τk =
−2γk|t|, (6)
and the 
orresponding power spe
tral density is given by
S(f) =
∆2kγk
γ2k + (πf)
. (7)
Introdu
ing the density of transition rates g(γ) and ex-
pressing the noise strength ∆ as a fun
tion of the tran-
sition rate, we 
an repla
e the summation in Eq. (7) by
an integration, whi
h yields
S(f) =
∫ γmax
∆2(γ)g(γ)γ
γ2 + (πf)2
dγ, (8)
where γmin and γmax are minimal and maximal transition
rates, respe
tively. Provided that
∆2(γ)g(γ) = 2A/γ, (9)
where A is a 
onstant, the power spe
tral density in
Eq. (8) be
omes [27℄
S(f) =
arctan
− arctan
, γmin ≪ πf ≪ γmax. (10)
Thus Eq. (10) yields an approximation to the 1/f power
spe
trum. To generate a general 1/fα power spe
tral
density for 2 > α > 0, we 
an 
hoose
∆2(γ)g(γ) = 2Aγ−α (11)
as shown in [27℄.
Although the above method yields a valid approxima-
tion for the 1/fα spe
trum, it is 
omputationally ine�-

ient. In parti
ular, the number of distin
t noise states
in
reases exponentially with the number of RTN �u
tua-
torsK, i.e., the number of terms in the sum of Eq. (7) ap-
proximating the 1/fα noise. Sin
e the size of the di�eren-
tial equation system des
ribing the average qubit dynam-
i
s in
reases linearly with the number of noise states [29℄,
in pra
ti
e one has to restri
t the 
omputation to a rather
small number of independent RTN �u
tuators.
To over
ome this problem, we present a 
on
eptually
di�erent way of generating the desired 1/fα noise spe
-
trum using a single multi-state Markovian �u
tuator.
Consider a 
ontinuous-time Markovian noise pro
ess with
M dis
rete noise states. Let Γkj denote the transition
rate from the jth state to the kth one. In order to pre-
serve total probability, we must have
Γjk = 0 for all k = 1, 2, . . . ,M. (12)
Let us assume that the transition rates are symmetri
,
i.e., Γ = ΓT . Under this assumption the noise pro
ess
has a steady-state solution in whi
h the di�erent noise
states are equally probable. In order for the noise to be
unbiased, i.e., 〈η〉 = 0, the amplitudes bk asso
iated with
the noise states must satisfy
bk = 0. (13)
Thus the auto
orrelation is given by
C(t) = 〈η(t)η(0)〉 =
bT eΓ|t|b. (14)
Sin
e Γ is symmetri
, we 
an diagonalize it with an or-
thogonal matrix V as Γ = V ΛV T , where the real diagonal
matrix Λ = diag{λk}Mk=1 
arries the eigenvalues of Γ in
a des
ending order. De�ning χ := 1√
V T b, we rewrite
Eq. (14) in the form of Eq. (6) as
C(t) = χT eΛ|t|χ =
λk|t|. (15)
In order to use this multi-state Markovian �u
tuator to
approximate 1/fα noise, we have to 
hoose the eigen-
values λk and the amplitudes χk su
h that Eq. (11) is
ful�lled. Moreover, we must 
onstru
t the orthogonal
matrix V su
h that Γ = V ΛV T satis�es Eq. (12), the
amplitudes bk satisfy Eq. (13), and the o�-diagonal ele-
ments of Γ must be non-negative.
One way to satisfy these requirements is to pi
k an
integer m ≥ 2 and set M = 2m and to 
hoose the eigen-
values as
{λk}Mk=1 = −2{0, γmin, γmin + δ, γmin + 2δ, . . . , γmax},
where γmax = (M − 2)δ+ γmin and 0 < δ ≤ γmin. Hen
e,
the distribution of the transition rates g(γ) is uniform
on [γmin, γmax]. Then we set V = H
, where H is the
Hadamard matrix
Expli
it 
al
ulation shows that these 
hoi
es ensure that
Eq. (12) is satis�ed. To ful�ll Eqs. (11) and (13), we set
χ1 = 0 and χk = γ
k for k = 2, . . . , M , where γk
is equal to γmin + (k − 2)δ. It 
an be shown that this

onstru
tion will also produ
e transition matri
es Γ with
non-negative o�-diagonal elements. Hen
e we have pro-
vided an e�
ient way to implement 1/fα noise. Note
that the M -state Markovian �u
tuator, Eq. (15), 
or-
responds formally to Eq. (6) with M − 1 non-vanishing
RTN �u
tuators. Thus we have a
hieved an exponential
improvement in the e�
ien
y of the noise approximation.
Alternatively, we 
an 
hoose the eigenvalues of Γ freely
and obtain a valid matrix V with numeri
al optimization,
0 0.2 0.4 0.6 0.8 1 1.2 1.4
[(2πf)/γ0]
FIG. 1: Logarithm of the power spe
tral density for �ve
independent RTN �u
tuators (dash-dotted line), a multi-state
Markovian sour
e 
orresponding to 31 RTN �u
tuators (solid
line), and an ideal 1/f noise (dotted line). The transition
rates of the RTN �u
tuators are in both 
ases distributed
uniformly on the interval [γ0, 30γ0].
whi
h may result in even more faithful approximation of
1/fα noise.
Figure 1 
ompares the approximation of the spe
-
tral density of 1/f noise generated by independent RTN
sour
es and by a multi-state Markovian sour
e. For the
RTN approa
h, we 
hoose 5 independent noise sour
es,
for whi
h the transition rates γk are uniformly distributed
in the range [γmin, γmax] = [γ0, 30γ0], and the strengths
are given by ∆k = 1/
γk. This yields a �u
tuator
with 32 distin
t noise states. For the multi-state �u
-
tuator, we 
hoose a 32-state noise sour
e, for whi
h
the nonzero eigenvalues λk of its transition rate ma-
trix Γ are distributed uniformly on [−60γ0,−2γ0], and
χk = 1/
−λk/2. Thus the 
ondition in Eq. (9) is satis-
�ed for both of the approa
hes and the multi-state noise
sour
e has an auto
orrelation fun
tion and power spe
-
tral density whi
h are equal to those for a 
ertain ensem-
ble of 31 RTN �u
tuators. We employ representations of
similar 
omputational 
omplexity here in order to be able
to assess the relative a

ura
y for a given 
omputational
e�ort.
Figure 1 shows that an ensemble of �ve RTN pro
esses
is not an a

urate model for 1/f noise, whereas a single
32-state Markovian noise sour
e is quite a

urate, espe-

ially in the range 3γ0 . ω . 16γ0. The poor quality
of the approximation with �ve RTN �u
tuators is due to
the small number of independent noise sour
es employed
here, whereas the 32-state Markovian �u
tuator 
ontains
more parameters and thereby introdu
es more �exibility
in the noise approximation. The frequen
y range over
whi
h the approximation is a

urate is relatively short
if one 
onsiders that the 1/f noise dete
ted in experi-
mental appli
ations often extends over several frequen
y
de
ades. The width of this frequen
y range 
an of 
ourse
be in
reased by in
reasing the width of the region from
whi
h the eigenvalues of the matrix Γ are 
hosen. In
this 
ase, however, the number of dis
rete levels in the
Markovian sour
e must also be in
reased to preserve the
desired a

ura
y. For the main purpose of demonstrating
the feasibility of the numeri
al optimization algorithm,
in the rest of this paper we will 
ontinue to approximate
1/fα noise by a single Markovian noise sour
e with 32
levels.
III. QUBIT DYNAMICS AND CONTROL
In Ref. [24℄, the temporal evolution of the qubit density
matrix was 
al
ulated by averaging over 104�105 unitary
quantum traje
tories, ea
h 
orresponding to a sample
noise path. To ensure a

ura
y, a large number of uni-
tary traje
tories are required, whi
h results in extensive

omputational e�ort. In Ref. [29℄, exa
t deterministi
master equations des
ribing the average temporal evo-
lution of quantum systems under Markovian noise were
derived.
Following Ref. [29℄, we introdu
e a 
onditional density
operator ρk(t) whi
h 
orresponds to the density operator
of the system averaged over all the noise sample paths
o

upying the kth state at the time instant t. The 
on-
ditional density operators are normalized su
h that the
tra
e of the operator ρk(t) yields the probability of the
kth noise state as Pk(t) = Tr [ρk(t)]. The total average
density operator 
an be expressed as
ρ(t) =
ρk(t). (16)
The dynami
s of ρk is obtained from the 
oupled master
equations [29℄
∂tρk(t) =
[Hk(t), ρk(t)] +
Γkjρj(t), (17)
whereHk(t) is the Hamiltonian of the system 
orrespond-
ing to the kth noise state, and Γkj the transition rate
from the jth state to the kth state, as de�ned in Se
. II.
Spe
i�
ally, in our one-qubit 
ase,
Hk(t) =
a(t)σx +
bkσz, (18)
where bk is the noise amplitude of the state k. We shall
use Ea {ρ} to denote the state ρ evolved under the in�u-
en
e of noise and the 
ontrol sequen
e a.
The �delity fun
tion quantifying the overlap between
the desired state ρf and the a
tual a
hieved �nal state is
de�ned as
φ(ρf , Ea {ρ0}) = Tr
Ea {ρ0}
, (19)
where ρ0 is the initial state of the system. To measure
how 
lose the evolution Ea is to the intended quantum
gate operation U , we 
al
ulate the average of the �delity
φ(Uρ0U
†, Ea {ρ0}) over all pure initial states ρ0, and ob-
tain the gate �delity fun
tion [24℄
Φ(U) =
k=x,y,z
†Ea {σk}
. (20)
We aim to �nd the optimal 
ontrol pulses whi
h max-
imize the �delity of the a
hieved quantum operation,
and hen
e apply a typi
al gradient based optimization
algorithm su
h as the gradient as
ent pulse engineering
(GRAPE) method developed in Ref. [31℄. If the 
ontin-
uous pulse pro�les are approximated by pie
ewise 
on-
stant fun
tions, the gradient of the �delity fun
tion with
respe
t to these 
onstant pulse values and durations 
an
be 
al
ulated by the 
hain rule. This gradient is further
used as a proportional adjustment to update the 
ontrol
pulse pro�le. The optimization pro
edure is terminated
when 
ertain desired a

ura
y is a
hieved. Note that
due to the non-
onvex nature of the problem, the gra-
dient based algorithm will only yield a lo
ally optimal
solution. We further employ a multitude of initial 
on-
ditions to �nd a 
ontrol pulse whi
h a
hieves the highest
�delity.
IV. QUANTUM MEMORY
In this se
tion, we fo
us on the implementation of
quantummemory, i.e., the identity operator. For the pur-
pose of 
omparison with the optimized pulse sequen
es,
we introdu
e four other kinds of 
ontrol s
hemes whi
h
generate the identity operator.
The �rst referen
e sequen
e is simply not to apply any
external 
ontrol pulse, i.e., a(t) = 0. This pulse has
no 
ompensation for de
oheren
e or error. The se
ond
referen
e sequen
e is a 
onstant 2π pulse given by
a2π(t) = amax, for t ∈ [0, 2π~/amax]. (21)
The third referen
e sequen
e is the 
omposite pulse se-
quen
e known as 
ompensation for o�-resonan
e with a
pulse sequen
e (CORPSE), whi
h was originally designed
to 
orre
t systemati
 errors in the implementation of one-
qubit quantum operations and to provide high order 
on-
trol proto
ols for systemati
 qubit bias, i.e., for the noise

orrelation time τc → ∞ [32, 33℄. For the identity oper-
ation, the CORPSE pulse sequen
e 
an be obtained as
aSC2π(t) =
amax, for 0 < t
′ < π
−amax, for π ≤ t′ ≤ 3π
amax, for 3π < t
′ < 4π,
where the dimensionless time t′ is de�ned as t′ = amaxt/~.
In the absen
e of noise, the CORPSE sequen
e gen-
erates the identity operator exa
tly although it requires
twi
e as long operation time as a 2π pulse, the se
ond
referen
e pulse above. In the presen
e of small system-
ati
 errors, the CORPSE sequen
e is mu
h more a

urate
than the 2π pulse. For example, 
onsider a state trans-
formation from the north pole ba
k to itself on the Blo
h
sphere. For η(t) ≡ ∆ in Eq. (1), the �delities de�ned in
Eq. (19) 
an be derived to be
φ2π = 1−
, (23)
SC2π = 1− 4π2
. (24)
We observe that the error in the �delity of the 2π pulse
is fourth order in the relative noise strength ∆/amax,
whereas for the CORPSE pulse sequen
e it is eighth or-
der. Thus the CORPSE sequen
e is mu
h more a

urate
than a 2π pulse in 
orre
ting the e�e
ts of systemati
errors on quantum memory.
The fourth standard pulse sequen
e whi
h we take as a
referen
e is the Carr-Pur
ell-Meiboom-Gill (CPMG) [34℄
sequen
e whi
h is designed to preserve qubit 
oheren
e.
In our 
ontext, this sequen
e 
onsists of a π/2 pulse fol-
lowed by multiple π pulses at intervals tp, followed by a
�nal π/2 pulse to bring the system ba
k to the original
state. This pulse sequen
e is designed for T2 measure-
ments on spins, starting from the |0〉 state. Thus one
does not expe
t a CPMG pulse sequen
e to perform as
well if the initial state is averaged over the Blo
h sphere
as is done to 
ompute a gate �delity.
We �rst present the �delities obtained for the iden-
tity operator using the various 
ontrol pulse options in
the presen
e of 1/f noise. The noise is generated here
by the single Markovian noise sour
e dis
ussed in Se
. II,
with transition rates distributed uniformly over the inter-
val [1/τc, 30/τc] . In Fig. 2, the �delities obtained from
optimized 
ontrol pulses, 2π pulse, CORPSE, CPMG,
and zero pulse sequen
es are plotted as fun
tions of the

hara
teristi
 
orrelation time τc of the approximate 1/f
noise. Here, CPMG1 and CPMG2 refer to two CPMG
types of pulses with the intervals between π pulses be-
ing π and 2π, respe
tively. The total duration for these
pulses are all 12π~/a
. The optimal 
ontrol pulse is de-
signed for 6π, and therefore we repeat it twi
e. Similarly,
we repeat the 2π pulse 6 times, the CORPSE sequen
e
3 times, the CPMG1 sequen
e 3 times, and the CPMG2
sequen
e twi
e. The optimal 
ontrol pulse yields 
learly
the highest �delity among all these pulses, whereas the
zero pulse sequen
e has the worst performan
e as there
are no 
orre
tion me
hanisms. Note that due to motional
narrowing, all 
urves approa
h unit �delity in the limit
τc → 0.
The memory a

ess rate is an important spe
i�
ation
in modern 
omputer te
hnology [35℄. In our 
ontext, it

orresponds to the total duration of the 
ontrol pulses.
Figure 3 shows the �delity as a fun
tion of the dura-
tion for the numeri
ally optimized 
ontrol pulses. Equa-
tion (1) implies that in the absen
e of noise, the quantum
system will generate an identity operator for a = amax
0
 5
 10
 15
 20
 25
 30
0.55
0.65
0.75
0.85
0.95
PSfrag repla
ements
τc/(~/amax)
FIG. 2: Fidelity of the quantum memory as a fun
tion of
the 
hara
teristi
 
orrelation time τc for optimized 
ontrol
pulses (bla
k solid), a 2π pulse (bla
k dash-dotted), CORPSE
pulse sequen
e (bla
k dotted), CPMG1 pulse sequen
e (bla
k
dashed), CPMG2 pulse sequen
e (gray solid), zero pulse se-
quen
e (gray dash-dotted). The operation time is 
hosen to be
12π~/a
. The noise is produ
ed by a single 32-state Marko-
vian sour
e with the average strength 〈|η|〉 = 0.125 × amax

orresponding to 31 RTN �u
tuators with the transition
rates uniformly distributed over the region [1/τ
, 30/τ
] and
strengths 
hosen as des
ribed in Se
. II.
0 1 2 3 4 5 6 7 8 9 10
PSfrag repla
ements
T/(π~/amax)
FIG. 3: Fidelity of the quantum memory as a fun
tion of the
operation time for 
ontrol pulses optimized at ea
h point. The
noise is produ
ed by a similar multi-state Markovian sour
e
as in Fig. 2, with τc = 3~/amax.
and the duration T = 2nπ/amax. In Fig. 3, we observe
that, despite an overall de
rease, there are peaks in the
�delity near these operation times. Thus we 
an regard
2nπ/amax as the natural periods for quantum memory,
and we always 
hoose the total duration of 
ontrol pulses

orrespondingly.
Here, we study the relation between the optimized �-
delities a
hieved above and the average noise strength
〈|η|〉, for a �xed value of the 
hara
teristi
 
orrelation
0
 0.1
 0.2
 0.3
 0.4
 0.5
 0.6
 0.7
PSfrag repla
ements
〈|η|〉
FIG. 4: Fidelity of the quantum memory as a fun
tion of the
average absolute noise strength for optimized 
ontrol pulses
(bla
k solid), 2π pulse (bla
k dash-dotted), CORPSE (bla
k
dotted), CPMG1 (bla
k dashed), CPMG2 (gray solid), and
zero (gray dash-dotted). The operation time is 
hosen to be
12π~/a
. Ex
ept for its strength, the noise is produ
ed by
a similar multi-state Markovian sour
e as in Fig. 2 with the

orrelation time τ
= 30~/a
time τc = 30~/amax. Figure 4 shows the �delity as a
fun
tion of the noise strength for the optimized 
ontrol
pulses, 2π pulse, the CORPSE, CPMG1, CPMG2, and
zero pulse sequen
es. At small values of 〈|η|〉 again, the
optimized 
ontrol pulses 
onsistently a
hieve higher �-
delities than all referen
e pulses. However, we note that
if the noise strength ex
eeds ∼0.4, the optimized pulse
sequen
e redu
es to the zero pulse sequen
e, i.e., any
nonzero pulse sequen
e will a
tually deteriorate the �-
delity performan
e.
The dis
ussion above is based on the spe
i�
 noise den-
sity spe
trum 1/fα with α = 1. Figure 5 shows the
�delities of quantum memory for four optimized 
ontrol
pulses, ea
h of whi
h is obtained for a di�erent value of α.
The noise is produ
ed here by a single multi-state Marko-
vian sour
e with average strength 〈|η|〉 = 0.125 × amax,
and the total duration for all 
ontrol pulses are �xed
to 6π. A systemati
 s
aling of the 
orrelation time axis
with respe
t to α is 
learly visible in Fig. 5. This phe-
nomenon is explained by the fa
t that the 
on
entra-
tion of the power spe
trum of 1/fα to high frequen
ies,
i.e., long 
orrelation times, in
reases with α. Hen
e, the

urves s
ale down in τ
with in
reasing α.
V. NOT GATE
In this se
tion, we fo
us on the generation of high-
�delityNOT gates, i.e., the σx operator, under 1/f noise.
As in the 
ase of quantum memory, we 
ompare the nu-
meri
ally optimized results with referen
e pulses. In this
0
 5
 10
 15
 20
 25
 30
0.95
0.955
0.96
0.965
0.97
0.975
0.98
0.985
0.99
0.995
PSfrag repla
ements
τc/(~/amax)
FIG. 5: Fidelity of the quantum memory a
hieved with op-
timized 
ontrol pulses as a fun
tion of the 
hara
teristi
 
or-
relation time τc for 1/f
noise with α = 1 (solid ), α = 1.25
(dotted), α = 1.5 (dash-dotted), and α = 1.75 (dashed). The
operation time is 
hosen to be 6π~/a
. The noise is pro-
du
ed by a similar multi-state Markovian sour
e as in Fig. 2
with variable values of the power α.

ase, our �rst referen
e pulse is the π pulse given by
aπ(t) = amax, for t ∈ [0, π~/amax], (25)
whi
h in the absen
e of noise is the most e�
ient way of
a
hieving a NOT gate. In addition, we will use the two

omposite pulse sequen
es CORPSE and short CORPSE
[32, 33℄ whi
h assume here the form
aCπ(t) =
amax, for 0 < t
′ < π/3
−amax, for π/3 ≤ t′ ≤ 2π
amax, for 2π < t
′ < 13π/3,
aSCπ(t) =
−amax, for 0 < t′ < π/3
amax, for π/3 ≤ t′ ≤ 2π
−amax, for 2π < t′ < 7π/3,
respe
tively. Both of these pulse sequen
es 
orre
t for
systemati
 error, CORPSE being more e�
ient. How-
ever, the operation time of short CORPSE is mu
h
shorter than that of CORPSE, and hen
e it 
an yield
higher �delities in the presen
e of noise.
Figure 6 shows the NOT gate �delities obtained by
the referen
e and optimized pulses in the presen
e of the
same 1/f noise as employed in the analysis of quantum
memory in Se
. IV. We observe that for long enough

orrelation times, the 
omposite pulse sequen
es provide
good error 
orre
tion. Furthermore, as observed earlier
for RTN [24℄, for intermediate 
orrelation times, short
CORPSE a
hieves the highest �delity among the refer-
en
e pulses. Figure 7 presents the pulse sequen
es ob-
tained from the numeri
al optimizations for three dif-
ferent values of the noise 
orrelation time τc. For the
optimized pulse sequen
e, we �nd a transition from an
approximately 
onstant pulse to a short CORPSE -like
pulse sequen
e at 
hara
teristi
 
orrelation time τ
50~/a
. This 
hange in optimal pulse sequen
e is
responsible for the apparent dis
ontinuity in the �rst
derivative of the �delity 
urve in Fig. 6.
0
 50
 100
 150
 200
0.955
0.96
0.965
0.97
0.975
0.98
0.985
0.99
0.995
τc/(~/amax)
FIG. 6: NOT gate �delities as fun
tions of the 
hara
teris-
ti
 noise 
orrelation time τc for a π pulse (dotted), CORPSE
(dash-dotted), short CORPSE (dashed), and gradient opti-
mized pulse sequen
e (solid). The 1/f noise is generated as
in Fig. 2.
These results for the generation of NOT gates under
1/f noise are qualitatively quite similar to the previous
results presented in Refs. [24, 29℄ for a single RTN. This
similarity is due to the fa
t that 1/f noise 
an be re-
garded as arising from a sum of independent RTN �u
-
tuators, ea
h of whi
h having a similar �delity depen-
den
e on their 
orrelation times. Note that the s
ale for
the referen
e 
orrelation time τc of the �delity obtained
in presen
e of 1/f noise in Fig. 6 is somewhat di�erent
from the 
orresponding s
ale for the 
orrelation time of
a single RTN sour
e, sin
e the 1/f noise involves an en-
semble of RTN �u
tuators with a range of 
orrelation
times.
VI. CONCLUSIONS
We have studied a single qubit under the in�uen
e of
1/fα noise for 2 > α > 0 and investigated how de
o-
heren
e due to this noise 
an be suppressed in the im-
plementation of single qubit operations. We presented
an e�
ient way to approximate the noise with a dis
rete
multi-state Markovian �u
tuator. Due to this �nding,
the average temporal evolution of the qubit density ma-
trix under 1/fα noise 
an be e�
iently determined from
a deterministi
 master equation.
Employing these exa
t deterministi
 master equations
des
ribing the temporal evolution of the qubit density
operator under Markovian noise, we applied a gradient
FIG. 7: Optimized pulse sequen
es yielding the highest gate
�delities for 
orrelation times (a) 45~/a
, (b) 100~/a
and (
) 150~/a

orresponding to Fig. 6.
based optimization pro
edure to sear
h for optimal 
on-
trol pulses implementing quantum operations. In par-
ti
ular, we studied the physi
al appli
ation of quantum
memory, i.e., the identity operator, whi
h is a funda-
mental 
on
ept in the realization of a quantum 
om-
puter. The optimized 
ontrol pulses signi�
antly im-
proved the �delity over several referen
e sequen
es su
h
as 2π, CORPSE, CPMG, and zero pulses. We observe
peaks on �delity 
urves 
orresponding to integer multi-
ples of 2π~/a
in the total durations of 
ontrol pulses,
where a
is the maximum magnitude of the external

ontrol �eld. We also studied the performan
e of opti-
mal 
ontrol pulses under 1/fα noise for several di�erent
values of 2 > α ≥ 1, and found a monotoni
 behavior in
the noise frequen
y as a fun
tion of α, i.e., the �delity

urves are s
aled down in the 
orrelation time for in
reas-
ing α. We also investigated how the �delities degraded as
the noise strength in
reases. For the generation of high-
�delity NOT gates, we obtained results showing qualita-
tively similar behavior to the previous results presented
in Refs. [24, 29℄ for a single RTN sour
e. In parti
u-
lar, just as for a single noise sour
e, in the presen
e of
1/fα noise we observed a transition in the optimal 
ontrol
pulse sequen
e from a 
onstant pulse to a CORPSE-like
sequen
e as the noise 
hara
teristi
 
orrelation time τc is
in
reased.
This approa
h of 
oupled master equations indexed by
noise states of the environment, together with an opti-
mization te
hnique for pulse design 
an be readily gen-
eralized to multiple qubits evolving in the presen
e of
1/fα noise and other Markovian noise sour
es. Further-
more, it 
an be used to develop realisti
 pulse sequen
es
for mitigation of nu
lear spin and surfa
e magneti
 noise
a
ting on donor spins implanted in sili
on [11℄, as well
as for suppression of ba
kground 
harge noise a
ting on
super
ondu
ting qubits [4℄. In future, we will study the
implementation of multi-qubit gates, e.g., the 
ontrolled
NOT gate, in noisy systems and the swapping of quan-
tum information from a noisy qubit to long term quan-
tum memory. We will also 
onsider more realisti
 noise
with 1/fα spe
trum over many frequen
y de
ades.
A
knowledgments
This work was supported by the A
ademy of Fin-
land, the National Se
urity Agen
y (NSA) under
MOD713106A and by the NSF ITR program under grant
number EIA-0205641. M. M. and V. B. a
knowledge the
Finnish Cultural Foundation, M. M. the Väisälä founda-
tion and Magnus Ehrnrooth Foundation for the �nan
ial
support. We thank J. Clarke for insightful dis
ussions.
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