Optimal Dynamical Decoupling Sequence for Ohmic Spectrum

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by  Yu Pan

Optimal Dynamical Decoupling Sequence for Ohmic Spectrum
Yu Pan1,2, Zai-Rong Xi1,∗ and Wei Cui1,2
1Key Laboratory of Systems and Control, Institute of Systems Science, Academy of Mathematics and Systems Science,
Chinese Academy of Sciences, Beijing 100190, People’s Republic of China and
2Graduate University of Chinese Academy of Sciences, Beijing 100039, People’s Republic of China
Weinvestigate theoptimal dynamicaldecoupling sequencefor aqubitcoupled toan ohmicenvi-
ronment. By analytically computing the derivativesof the decoherence function, the optimal pulse
locations are found to satisfy a set of non-linear equations which can be easily solved. These equa-
tionsincorporatestheenvironmentinformationsuchashigh-energy(UV)cutofffrequencyωc,giving
a complete description of thedecoupling process. The solutions explain previous experimental and
theoretical results of locally optimized dynamical decoupling (LODD) sequence in high-frequency-
0 dominated environment, which were obtained by purely numerical computation and experimental
1 feedback. As shown in numerical comparison, these solutions outperform the Uhrig dynamical
0 decoupling (UDD)sequenceby one or more orders of magnitudein theohmic case.
2
n PACSnumbers:
a
J
I. INTRODUCTION UDD for certain noise spectrum, especially for the one
8
with a high frequency part and sharp high-energy (UV)
1
Suppressing decoherence is one of the fundamental is- cutoff.

] sues in the field of quantum informationprocessing. De- However,inspiteofthegreatexperimentalsuccess,an-
h
p coherence, which has been caused by the environmental alytical results about the LODD sequence is insufficient.
- noise,plaguesalmostalltheimplementationsofquantum Until recently S. Pasini and G. S. Uhrig has made an
nt bit. Toeliminatetheunwantedcouplingbetweenaqubit analytical progress in optimizing the decoherence func-
a anditsenvironment,severalschemeshavebeenproposed tion for power law spectrum (PLODD) [13]. The power
u and tested. Among them a promising one is dynamical law spectrum ωα for α<1 without UV cutoff is consid-
q decoupling [1–4], which restores the qubit coherence by ered. They minimize the decoherence function through
[
applying delicately designed sequence of control pulses. expanding the function and separating, canceling diver-

1 For a qubit that can be modeled by a spin-1/2 par- gences from the relevant terms and solving variation
v ticle, the oldest dynamical decoupling sequence is peri- problems. Inspired by Pasini’s work, we try to analyze
0 odicdynamicaldecoupling(PDD).Originatedfrompulse the LODD problem with respect to the ohmic spectrum
6 sequences widely used in nuclear magnetic resonance S(ω)∼ω andasharpUVcutoff. Ohmicnoiseisthema-
9
(NMR) [5], the PDD sequence consists of periodic and jor decoherence source often found in a qubit’s environ-
2
. equidistant π pulses. To achieve better performance, ment,forexample,the semiconductingquantumdot[16]
1 there has been an extensive study in how to optimize and superconducting qubit [17]. Optimal performance
0
the pulselocations[6–15]. Oneimportantprogressisthe pulse sequence is found analytically which entirely dif-
0
1 powerful Uhrig DD (UDD) [10], which employs n pulses fers from the UDD sequence in such environment. We
v: located at tj according to the simple rules coahlml itchisspekcintrdumof)ofoprtismhaolrts.equence HLODD (LODD for
i
X δ =sin2(jπ/2(n+1)), We organize this paper as follows. In the second sec-
j
tion we propose the optimization problem of the deco-
r
a where δ =t /T and T is the totalevolutiontime. UDD herence function. In section III, we derive the analytical
j j
is first derived on spin-boson model and further proved equations for the optimal pulse sequence. In the follow-
tobe universalinthe sensethatitcanremovethequbit- ing section, we run a simulation to verify our results.
bath coupling to nth order in generic environment [11]. Conclusions are put in section V.
BeyondUDD,anotherlocallyoptimizeddynamicalde-
coupling (LODD) sequence has drawn great attention
[14]. LODD, along with its simplified version optimized
II. OPTIMIZATION OF THE DECOHERENCE
noise-filtration dynamic decoupling (OFDD) [15], gener- FUNCTION
ates the decoupling sequence by directly optimizing the
decoherence function using numeric methods as well as
Given a two-level quantum system, when the environ-
experimental feedback. It has been shown to be able to
mental noise behaves quantum-mechanically, we use the
suppress decoherence effect by orders of magnitude over
long-established spin-boson model with pure dephasing
1
H = ω b†b + σ λ (b†+b ). (1)
∗Electronicaddress: [email protected] i i i 2 z i i i
Xi Xi
2
Here we ignorethe qubit free evolution hamiltonian. On Then the integral I can be expressed as
n
the other hand, when the qubit is subjected to classical
noise, the system is modeled as [18, 19] In = lim In(x),
x→0+
1
H = [Ω+β(t)]σ , (2)
2 z n+1
I (x)= 2qi+qj(−1)i+jI (x), (6)
n ij
where the Ω is the qubit energy splitting and β(t) the iX,j=0
classicalrandomnoise. Lett bethe totalevolutiontime,
andn pulsesareappliedatt1 <t2 <...<tn insequence where the integrals Iij(x) are
with negligible pulse durations . We use the notation
δ = tj. This naturally leads to the definition of t = 0 zc e∆ijz
j t 0 I (x) = dz
ij
and tn+1 = 1. In either (1) or (2), the decay of coher- Zx z
ence under the dynamical decoupling sequence can be −∆ijzc e−z
described by the decoherence function [10, 18, 20, 21] = dz. (7)
e−2χ(t) with Z−∆ijx z
χ(t)= ∞ S(ω)|y (ωt)|2dω, (3) The limit x →0+ is carried out because Iij(0) does not
Z ω2 n existforarbitraryi,j. Makinguseoftheseriesrepresen-
0
tation of exponential function [22]
where S(ω) is environmental noise spectrum. The filter
∞ e−t
function yn(t) is given by E1(z) = dt
Z t
z
yn(t)=1+(−1)n+1eiωt+2 n (−1)jeiωtδj. (4) = −γ−lnz+ ∞ (−1)k+1zk,
k!k
Xj=1 Xk=1
Thus minimization of χ(t) with respect to δ gives the where γ is the Euler-Mascheroni constant and the sum
j
optimal decoupling sequence. convergesfor allthe complexz, Iij(x) canbe written as
We now consider the case when the noise spectrum is
I (x) = E (−∆ x)−E (−∆ z )
ohmic with a sharp cutoff at ω , i.e. S(ω)=S ωΘ(ω − ij 1 ij 1 ij c
c 0 c
ω). S is an irrelevantconstantfactorand Θ is unit step ∞ ∆k
0 = ln(z /x)+ ij(zk−xk). (8)
function. Thenminimizationof(3)turnstominimization c k!k c
of I with kX=1
n
Since we always have y (0)=0 which implies
zc |y (z)|2 n
n
I = dz, (5)
n
Z z n+1
0
|yn(0)|2 = 2qi+qj(−1)i+j =0, (9)
where zc = ωct. Since yn(0) = 0, the IR convergence iX,j=0
insures the integral converges to a finite value [13].
we can now proceed by taking the limit x→0+ in I
n
n+1 ∞ ∆k
III. DERIVATSIOEQNUOEFNOCPETIMAL PULSE In = xl→im0+iX,j=02qi+qj(−1)i+j[ln(zc/x)+Xk=1 k!ikj(zck−xk)]
n+1 ∞ ∆k
We follow the approach of Pasini and Uhrig [13] to = lim 2qi+qj(−1)i+j ij(zk−xk)
treat the integral (5). Here we use notation x→0+iX,j=0 Xk=1 k!k c
0 if j =0,n+1, n+1 ∞ ∆k
qj =(cid:26)1 if j ∈{1,2,...,n}, = 2qi+qj(−1)i+jk!ikjzck. (10)
iX,j=0Xk=1
and
TominimizeI ,UDDrequiresthefirstnderivativesofy
n n
vanish while OFDD simplifies the optimization process
∆ =i(δ −δ ),
ij i j by replacing S(ω) by a constant. Here we attempt to
minimize I directly to obtain optimal pulse sequence.
from which we get n
We notice that at the optimal pulse locations δ (j =
j
1,2,...,n), the gradient of I vanishes. So we impose
n+1 n
|yn(z)|2 = 2qi+qj(−1)i+jez∆ij. the following conditions ∂∂δImn = 0, for m from 1 to n.
iX,j=0 Although (10) are complex infinite series, we can still
3
explicitlycomputethederivativesof(10)aslongasthese ωc=1
1
derivatives converge. For arbitrary m we have
0.9
∂I ∂ n+1 ∞ ∆k 0.8
n = 2qi+qj(−1)i+j ijzk 0.7
∂δ ∂δ k!k c
m m iX,j=0kX=1 0.6 n=2
∂ n+1 ∞ ∆k δi0.5 n=5
= { 2qi+qm(−1)i+m imzk
∂δ k!k c 0.4
m Xi=0Xk=1 0.3
n+1 ∞ ∆k 0.2 OUDD
+ 2qm+qi(−1)m+i mizk} +HLODD
k!k c 0.1
Xi=0kX=1
0
= n+1 ∞ 2qm+qi(−1)m+izcki[∆k−1−(−1)k−1∆k−1]. 0 1 2 3i 4 5 6
k! mi mi ω=5
Xi=0Xk=1 1 c
(11)
0.9
0.8
The terms with k odd cancel, so the result can be sim-
0.7
plified as
n=2
0.6
n+1 ∞ z2k δi0.5 n=5
= 2qm+qi+1(−1)m+i c i2k(δm−δi)2k−1 0.4
(2k)!
Xi=0kX=1 0.3
n+1 1 ∞ z2k 0.2
= δ −δ 2qm+qi+1(−1)m+i (2ck)!i2k(δm−δi)2k 0.1
iX6=m m i kX=1 0
0 1 2 3 4 5 6
n+1 1 i
= 2qm+qi+1(−1)m+i{cos[(δm−δi)zc]−1}. ω=10
δ −δ c
iX6=m m i 1
(12) 0.9
0.8
Here we have used the expansion cos(z) = ∞ ((−2k1))k! z2k 00..67 n=2
kP=0
which convergeson the whole complex plane. From (12) δi0.5 n=5
weknowthatthederivativesofI indeedconvergetoafi-
n 0.4
nitevalue. Thustheoptimalpulselocations{δ ,δ ,...δ }
1 2 n 0.3
shall satisfy the following non-linear equations
0.2
0.1
n+1
1
2qm+qi+1(−1)m+i{cos[(δ −δ )z ]−1}=0. 0
m i c 0 1 2 3 4 5 6
iX6=mδm−δi i
(13)
Equations (13) are main results of this paper. The opti- FIG.1: ComparisonbetweenUDDandHLODDsequencefor
mal sequence obtained from (13) is quite different from differentUVcutofffrequencyωc. Pulsesequencesδiforn=2
the UDD sequence obeying and n=5 are plotted in one figureunderthe same ωc.
n+1
2qj(−1)jδp =0 IV. NUMERICAL RESULTS
j
Xj=1
Westartoursimulationbysolvingthenon-linearequa-
for p = {1,2,...n}. For the ohmic spectrum, our equa- tions (13) and evaluate the decoherence function with
tions incorporate the UV cutoff frequency ω , indicating these solutions. First, we set the total evolution time
c
that the solutions are specially tailored to combat this t = 1 without loss of generality. Then z = ω and we
c c
kind of noise. Although the UDD sequence is universal can concentrate on analyzing the influence of the cutoff
in suppressing decoherence,we believe that the HLODD frequency ω . Computing solutions to (13) for differ-
c
sequencewilloutperformtheUDDsequenceintheohmic ent ω , we find that the optimal pulse sequences behave
c
environment. In the next section, we use numeric meth- differently. We alsoevaluate the UDD sequencefor com-
ods to illustrate the performance of HLODD sequence. parison.
4
ωc=5 inspecting the points on the HLODD curve, we expect
105 the HLODD sequence suppresses decoherence in power
law n as UDD.
At last, we would like to explain the numerical results
100 in another way. If we fixed UV cutoff frequency ω at
c
the beginning, and compare the HLODD performance
for t=1, t=5, and t=10, the numerical results would
In10−5 be the same since zc did not change. So we can also
conclude that for the same number of pulses n, HLODD
will beat UDD with increasing total evolution time t.
10−10 *UDD V. CONCLUSIONS
+HLODD
In this paper we analytically find the optimal pulse
10−15 locations to decouple a qubit in an ohmic environment.
1 2 3 4 5 6 7 8 9 10 11
n Byderivingthe analyticalexpressionsforthe derivatives
of decoherence function, we obtain a set of non-linear
FIG.2: Semi-logplotofIn versuspulsenumbern. UDDand equations which the optimal pulse sequence must obey.
HLODDsequences are compared. These equations are completely different from UDD and
aremoreaccurate,becausethey incorporatethe effectof
UV cutoff frequency ω .
c
As shown in Fig. 1, deviation of the pulse locations δi Inournumericalsimulation,theanalyticalresultspro-
in HLODD sequence from their UDD counterparts in- vide an improvementover UDD sequence by an order or
creases with ωc. This agrees with our intuition since two of magnitude, which is consistent with previous re-
UDD focuses on suppressing decoherence by minimiz- sults inLODDandOFDD obtainedbypurely numerical
ing |yn(z)| in the neighborhood of yn(0), weakening its minimization and experimental feedback. We have to
ability to maintain small |yn(z)| on the other end of the mention that the pulse performance is influenced by the
spectrum. For large ωc, UDD sequence is no longer op- sharpUVcutofffrequencyωc greatly. ThelargertheUV
timal. In addition, we can see pulse number n plays an cutoff ω , the more HLODD deviates from UDD. Early
c
important role. By increasing n, UDD can narrow the work [13, 14, 18, 20] has pointed out that for soft large
difference fromHLODD. The difference between the two UV cutoff, UDD performs even worse and LODD is still
sequences when n = 2 is greatly reduced when we in- a better choice. However, the integral (3) for S(ω) with
crease n to 5, see Fig. 1. Especially for the case ωc =1, a soft cutoff is hard to analyze.
thedifferenceiscompletelyremoved. However,forlarger
Inconclusion,ourworkprovidesananalyticalsolution
ω this gap can’t be removed by increasing n.
c to optimal dynamical decoupling for ohmic case. Our
Next,todemonstratetheoptimaldecouplingabilityof
derivationis basedon ohmic spectrum, but we believe it
HLODD sequence, we compute In versus n while ωc is can be extended to super-ohmic case S(ω)∼ ωα(α > 1)
chosen to be 5. The results are depicted in Fig. 2, and
via slight modification.
again are compared with UDD. The obtained solutions
yield a significant improvement over UDD. For fixed n,
the HLODD suppresses decoherence better than UDD
Acknowledgments
byone ortwoordersofmagnitudewhichis inagreement
with the results in [14, 15], where LODD and OFDD
sequencesaretestedfor9Be+qubitsinapenningiontrap This work was supported by the National Natu-
and various spectrum. The qubit error rates are below ral Science Foundation of China (No. 60774099, No.
10−5 when n>5, and we see that HLODD is capable of 60821091), the Chinese Academy of Sciences (KJCX3-
suppressingthe errorratesfarbelowthe Fault-Tolerance SYW-S01), and by the CAS Special Grantfor Postgrad-
error threshold [23] by increasing n. Furthermore, by uate Research, Innovation and Practice.
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