Environmental noise reduction for holonomic quantum gates
Daniele Parodi,1,2 Maura Sassetti,1,3 Paolo Solinas,4 and Nino Zangh̀ı1,2
1 Dipartimento di Fisica,
Università di Genova, Genova, Italy
2 Istituto Nazionale di Fisica Nucleare (Sezione di Genova),
Genova, Italy 3 INFM-CNR Lamia
Via Dodecaneso 33, 16146 Genova, Italy
4 Laboratoire de Physique Théorique de la Matière Condensée,
Université Pierre et Marie Curie,
Place Jussieu, 75252 Paris Cedex 05, France
(Dated: October 27, 2018)
We study the performance of holonomic quantum gates, driven by lasers, under the effect of
a dissipative environment modeled as a thermal bath of oscillators. We show how to enhance the
performance of the gates by suitable choice of the loop in the manifold of the controllable parameters
of the laser. For a simplified, albeit realistic model, we find the surprising result that for a long time
evolution the performance of the gate (properly estimated in terms of average fidelity) increases. On
the basis of this result, we compare holonomic gates with the so-called Stimulated Raman adiabatic
passage (STIRAP) gates.
PACS numbers: 03.67.Lx
I. INTRODUCTION
The major challenge for quantum computation is posed
by the fact that generically quantum states are very del-
icate objects quite difficult to control with the required
accuracy—typically, by means of external driving fields,
e.g., a laser. The interaction with the many degrees of
freedom of the environment causes decoherence; more-
over, errors in processing the information may lead to a
wrong output state.
Among the approaches aiming at overcoming these dif-
ficulties are those for which the quantum gate depends
very weakly on the details of the dynamics, in particu-
lar, the holonomic quantum computation (HQC) [1] and
the so-called Stimulated Raman adiabatic passage (STI-
RAP) [2, 3, 4]. In the latter, the gate operator is obtained
acting on the phase difference of the driving lasers dur-
ing the evolution, while in the former the same goal is
achieved by exploiting the non-commutative analogue of
the Berry phase collected by a quantum state during a
cyclic evolution. Concrete proposals have been put for-
ward, for both Abelian [5, 6] and non-Abelian holonomies
[7, 8, 9, 10, 11, 12]. The main advantage of the HQC
is the robustness against noise deriving from a imperfect
control of the driving fields [13, 14, 15, 16, 17, 18, 19, 20].
In a recent paper [21] we have shown that the dis-
turbance of the environment on holonomic gates can be
suppressed and the performance of the gate optimized
for particular environments (purely superohmic thermal
bath). In the present paper we consider a different sort
of optimization, which is independent of the particular
nature of the environment.
By exploiting the full geometrical structure of HQC,
we show how the performance of a holonomic gate can
be enhanced by a suitable choice of the loop in the man-
ifold of the parameters of the external driving field: by
choosing the optimal loop which minimizes the “error”
(properly estimated in terms of average fidelity loss). Our
result is based on the observation that there are different
loops in the parameter manifold producing the same gate
and, since decoherence and dissipation crucially depend
on the dynamics, it is possible to drive the system over
trajectories which are less perturbed by the noise. For a
simplified, albeit realistic model, we find the surprising
result that the error decreases linearly as the gating time
increases. Thus the disturbance of the environment can
be drastically reduced. On the basis of this result, we
compare holonomic gates with the STIRAP gates.
In Sec. II the model is introduced and the explicit
expression of the error is derived. In Sec. III we find
the optimal loop, calculate the error, make a comparison
with other approaches, and briefly sketch how to treat a
different coupling with the environment.
II. MODEL
The physical model is given by three degenerate (or
quasidegenerate) states, |+〉, |−〉, and |0〉, optically con-
nected to another state |G〉. The system is driven by
lasers with different frequencies and polarizations, acting
selectively on the degenerate states. This model describes
various quantum systems interacting with a laser radia-
tion, ranging from semiconductor quantum dots, such as
excitons [12] and spin-degenerate electron states [3], to
trapped ions [8] or neutral atoms [7].
The (approximate) Hamiltonian modeling the effect of
the laser on the system is (for simplicity, ~ = 1) [8, 12]
H0(t) =
j=+,−,0
ǫ|j〉〈j|+(e−iǫtΩj(t)|j〉〈G|+H.c)
, (1)
where Ωj(t) are the timedependent Rabi frequencies de-
http://arxiv.org/abs/0704.0376v2
pending on controllable parameters, such as the phase
and intensity of the lasers, and ǫ is the energy of the
degenerate electron states. The Rabi frequencies are
modulated within the adiabatic time tad, (which coin-
cides with the gating time), to produce a loop in the pa-
rameter space and thereby realize the periodic condition
H0(tad) = H0(0).
The Hamiltonian (1) has four time dependent eigen-
states: two eigenstates |Ei(t)〉 , i = 1, 2, called bright
states, and two eigenstates |Ei(t)〉, i = 3, 4, called dark
states. The two dark states have degenerate eigen-
value ǫ and the two bright states have timedependent
energies λ±(t) = [ǫ ±
ǫ2 + 4Ω2(t)]/2 with Ω2(t) =
i=±,0 |Ωi(t)|
2 [22].
The evolution of the state is generated by
Ut = Te
dt′H0(t
′), (2)
where T is the time-ordered operator. In the adiabatic
approximation, the evolution of the state takes place in
the degenerate subspace generated by |+〉, |−〉, and |0〉.
This approximation allows to separate the dynamic con-
tribution and the geometric contribution from the evolu-
tion operator. Expanding Ut in the basis of instantaneous
eigenstates of H0(t) (the bright and dark states), in the
adiabatic approximation, we have
Ut ∼=
′)dt′ |Ej(t)〉〈Ej(t)| Ut, (3)
where
Ut = Te
dτV (τ), (4)
here V is the operator with matrix elements Vij(t) =
〈Ei(t)|∂t|Ej(t)〉. The unitary operator Ut plays the role
of timedependent holonomic operator and is the funda-
mental ingredient for realizing complex geometric trans-
formation whereas
′)dt′ |Ej(t)〉〈Ej(t)| is the
dynamic contribution.
Consider Ut for a closed loop, i.e., for t = tad,
U = Utad . (5)
If the initial state |ψ0〉 is a superposition of |+〉 and |−〉,
then U|ψ0〉 is still a superposition of the same vectors (in
general, with different coefficients)[12]. Thus the space
spanned by |+〉 and |−〉 can be regarded as the “logi-
cal space” on which the “logical operator” U acts as a
“quantum gate” operator. Note that for t < tad, Ut|ψ0〉
has, in general, also a component along |0〉. However,
as it is easy to show [22], at any instant t < tad, Ut|ψ0〉
can be expanded in the twodimensional space spanned
by the dark states |E3(t)〉 and |E4(t)〉. It is important
to observe that U depends only on global geometric fea-
tures of the path in the parameter manifold and not on
the details of the dynamical evolution [1, 12].
To construct a complete set of holonomic quantum
gates, it is sufficient to restrict the Rabi frequencies
Ωj(t) in such a way that the norm Ω of the vector
~Ω = [Ω0(t),Ω+(t),Ω−(t)] is time independent and the
vector lies on a real three dimensional sphere [8, 12].
We parametrize the evolution on this sphere as Ω+(t) =
sin θ(t) cos φ(t), Ω−(t) = sin θ(t) sin φ(t) and Ω0(t) =
cos θ(t) with fixed initial (and final) point in θ(0) = 0,
the north pole By straightforward calculation we obtain
the analytical expression for V (t) in eq. (4), V (t) =
iσy cos[θ(t)]φ̇(t), where σy is the usual Pauli matrix
written in the basis of dark states. Thus, the oper-
ator (4) becomes Ut = cos[a(t)] − iσy sin[a(t)], here
a(t) =
dτφ̇(τ) cos θ(τ). Accordingly, the logical op-
erator U (5) is
U = cos a− iσy sin a, (6)
where
a = a(tad) =
∫ tad
dτφ̇(τ) cos θ(τ) (7)
is the solid angle spanned on the sphere during the evolu-
tion. Note that the are many paths on the sphere which
generate the same logical operator U , and span the same
solid angle a.
In a previous work we have studied how interaction
with the environment disturbs the logical operator U [21].
The goal of the present paper is to analyze whether and
how such a disturbance can be minimized for a given
U . To this end, we model the environment as a thermal
bath of harmonic oscillators with linear coupling between
system and environment [23]. The total Hamiltonian is
H = H0(t) +
α + cαxαA), (8)
where A is the system interaction operator called, from
now on, noise operator.
We now consider the time evolution of the reduced
density matrix of the system, determined by the Hamil-
tonian (8). We rely on the standard methods of the “mas-
ter equation approach,” with the environment treated in
the Born approximation and assumed to be at each time
in its own thermal equilibrium state at temperature T .
This allows to include the effect of the environment in
the correlation function (kB = 1)
g(τ) =
cos(ωτ) − i sin(ωτ)
Here the spectral density is
J(ω) =
δ(ω − ωα), (10)
at the low frequencies regimes, is proportional to ωs, with
s ≥ 0, i.e., s = 1 describes a Ohmic environments, typ-
ical of baths of conduction electrons, s = 3 describes
a super-Ohmic environment, typical of baths of phonons
[21, 24]. The asymptotic decay of the real part of g(τ) de-
fines the characteristic memory time of the environment.
Denoting with ρ̃(t) the time evolution of the reduced den-
sity matrix of the system in the interaction picture, e.g.,
ρ̃(t) = U
t ρUt, one has [24]
ρ̃(tad) = ρ(0) +
dτ{g(τ )[ÃÃ
ρ̃(t− τ )− Ã
ρ̃(t− τ )Ã]
+ g(−τ )[ρ̃(t− τ )Ã
Ã− Ãρ̃(t− τ )Ã
]. (11)
Here Ã and Ã′ stand for Ã(t) and Ã(t−τ), with the tilde
denoting the time evolution in the interaction picture.
In quantum information the quality of a gate is usually
evaluated by the fidelity F , which measures the closeness
between the unperturbed state and the final state,
F = 〈ψ0(0)|U
†ρ(tad)U|ψ0(0)〉, (12)
where |ψ0(0)〉 is the initial state, and ρ(tad) = U ρ̃(tad)U
is the reduced density matrix in the Schrödinger picture
starting from the initial condition ρ(0) = |ψ0(0)〉〈ψ0(0)|.
The average error is defined as the average fidelity loss,
i.e.,
δ =< 1−F >= 1− < 〈ψ0(0)|ρ̃(tad)|ψ0(0)〉 >, (13)
where < · · · > denotes averaging with respect to the
uniform distribution over the initial state |ψ0(0)〉.
The right-handside of Eq. (13) can be computed by
the following steps:
(1) solving Eq. (11) in strictly second order approxima-
tion; this approximation corresponds to replace ρ̃(t − τ)
with ρ(0);
(2) using the adiabatic approximation U(t − τ, t) ≈
exp(iτH0(t));
(3) expanding the scalar product in Eq. (13) with respect
to a complete orthonormal basis {|ϕn(t)〉}, n = 1, 2, 3,
orthogonal to |ψ0(t)〉. In this way, one obtains
∫ tad
dt G(t)|〈ψ0(t)|A|ϕn(t)〉|
, (14)
where
G(t) =
Re[g(τ)] cos(ω0nt) + Im[g(τ)] sin(ω0nt))
Here, ω0n = ω0−ωn are the energy differences associated
to the transition ψ0 ↔ φn, with ω0 = ǫ, ω1 = λ+, ω2 =
λ−, and ω3 = ǫ.
The interaction between system and environment is ex-
pressed by the noise operator A in Eq. (8). We shall now
make the assumption that A = diag{0, 0, 0, 1} in the |G〉,
|±〉, and |0〉 basis. In this case the transition between de-
generate states are forbidden, however the noise breaks
their degeneracy, shifting one of them. In spite of its sim-
ple form, this A is nevertheless a realistic noise operator
for physical semiconductor systems [4].
III. MINIMIZING THE ERROR
The problem can be stated in the following way: given
the noise operator A and the logical operator U , find a
path on the parameter space (the surface of the sphere,
described above) which minimizes the error δ.
The total error δ, given by Eq. (14), can be decom-
posed as
δ = δtr + δpd, (16)
where the transition error, δtr, is the contribution to the
sum of the nondegenerate states (ω0n 6= 0) and the pure
dephasing error δpd is the contribution of the degenerate
states (ω0n = 0). Thus
δpd =
∫ tad
sin(ωt)
sin2 2a(t)
sin4 θ(t) (17)
δtr =
n=+,−
1 + [(λn − ǫ)/Ω]2
∫ tad
sin2 2θ(t)dt,
where
Γ0n = J(|ω0n|)
|ω0n|
− sgn(ω0n)
correspond to the transition rates calculated by standard
Fermi golden rules, supposing, as usual, G(t) ≈ G(∞) for
g(τ) strongly peaked around τ = 0. In the following we
define for simplicity
n=+,−
1 + [(λn − ǫ)/Ω]2
Since we are interested at long time evolution, we start
discussing the transition error which dominates in this
regime [4, 25].
A. Transition rate
As explained in Sec. II, the holonomic paths are closed
curves on the surface of the sphere which start from the
north pole. It turns out that the curve minimizing δtr
can be found among the loops which are composed by a
simple sequence of three paths (see the Appendix): evo-
lution along a meridian (φ = const), evolution along a
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FIG. 1: The error δtr versus θM for two different a values:
a = π/2 (dashed line) and a = π/4 (full line) correspond to
NOT and Hadamard gate, respectively.
parallel (θ = const) and a final evolution along a merid-
ian to come back to the north pole.
The error δtr in (18), depends on a given by Eq. (7),
θM (the maximum angle spanned during the evolution
along the meridian), ∆φ (the angle spanned along the
parallel), and angular velocity v. We allow ∆φ ≥ 2π
which corresponds to cover more than one loop along
the parallel. The velocity along the parallel is v(t) =
φ̇(t) sin θ and that along the meridian is v(t) = θ̇(t). In
the following we assume that v is constant, and it cannot
exceed the maximal value of vmax, fixed by adiabatic
condition vmax ≪ Ω.
The parameters a, θM , and ∆φ are connected by the
relation a = ∆φ(1 − cos θM ). The error δtr is then
δtr = δ
tr + δ
tr, (20)
where
δMtr =
sin 4θM
is the contribution along the meridian and
δPtr = K
sin θM sin
2 2θM
1− cos θM
is the contribution along the parallel.
In Fig. 1 δtr is plotted for a = π/2 and a = π/4 (corre-
sponding to NOT and Hadamard gate, respectively) as a
function of θM . One can see that δtr has a local minimum
for θM = π/2 and a global minimum for θM = 0 where
the error vanishes. This suggests that the best choice is
to take θM as small as possible.
It is interesting to consider the dependence of δtr also
on the evolution time tad. For simplicity, we set the ve-
locity v = vmax. In this case, changing θM (and then
∆φ) corresponds to a change in the evolution time. We
obtain
θM = arccos
, (23)
5 10 15 20 25 30
vmaxtad
FIG. 2: The error δtr versus vmaxtad for two different a val-
ues: a = π/2 (dashed line) and a = π/4 (full line) correspond
to NOT and Hadamard gate, respectively. The dotted-dashed
line shows the value of the error at θ = π/2. The circles show
the critical value of vmaxtad above which the best loop is the
one with the minimal θM .
where
(vmaxtad)
2 + a2
. (24)
Using these relations, δMtr and δ
tr, given by (21) and
(22) become functions of tad, vmax, and a. Note that
m measures the space covered along the parallel, in fact
∆φ = 2πm.
In Fig. 2 we see the behavior of δtr as a function
of vmaxtad. The first minimum for both curves corre-
sponds to θM = π/2, then the curves for long tad de-
crease asymptotically to zero corresponding to the region
in which θM → 0. In this regime we have δtr ∝ 1/tad
which is drastically different from the results obtained
with other methods where δtr ∝ tad, (see Refs [4, 25]
and below Sec. III C). It should be observed that this
surprising results is a merit of holonomic approach which
allows to choose the loop in the parameter space, with-
out changing the logical operation as long as it subtends
the same solid angle. Observe that small θM and long
tad mean large value of m, i.e., multiple loops around the
north pole.
Figure 2 shows that, for a given gate, there is a criti-
cal value kc of vmaxtad which discriminate between the
choice of θM (e.g., k = 6 for the Hadamard gate and
k = 25 for the NOT gate). For vmaxtad < kc the best
choice for the loop is θM = π/2; For vmaxtad > kc the
best choice is the value of θM determined by eq. (23) and
(24).
Note that the region vmaxtad > kc is accessible with
physical realistic parameters [12]. For example, if we
choose the laser intensity Ω = 20 meV and vmax = Ω/50
(for which values the nonadiabatic transitions are forbid-
den), the critical parameter corresponds to the critical
time of 15 ps for the Hadamard gate and 42 ps for the
NOT gate.
B. Pure Dephasing
Until now we have ignored the pure dephasing effect
because we have assumed that it is negligible in com-
parison with the transition error for long evolution time.
Now, we check that the pure dephasing error contribu-
tion can indeed be neglected. We can write the pure
dephasing error using Eq. (17) and splitting to parallel
and meridian part as
δPpd =
∫ tad
Q[a(t)] sin ωt sin4 θM (25)
δMpd =
Q[a(t)]
sin ωt
sin4(vmaxt) + sin
vmaxt
,(26)
where Q[a(t)] = 1 + 1/2 sin2[2a(t)].
To estimate δpd we assume that tad is longer with re-
spect to the characteristic time of the bath. Remember-
ing that J(ω) ∝ ωs, the pure dephasing error behavior
along the parallel part at the temperature T is
δPpd ∝
, T ≪ 1/tad
, T ≫ 1/tad
while the along meridian is
δMpd ∝
. (28)
Then, we can conclude that the pure dephasing can al-
ways be neglected for long time evolution because it de-
creases faster than the transition error.
C. Comparison between HQC and STIRAP
We make a comparison between holonomic quantum
computation (HQC) and the STIRAP procedure which is
an analogous approach to process quantum information.
The STIRAP procedure ([2, 4]) is, in its basic points,
very similar to the holonomic information manipulation.
The level spectrum, the information encoding, the evolu-
tion produced by adiabatic evolving laser are exactly the
same. The fundamental difference is that in STIRAP
the dynamical evolution is fixed (we must pass through
a precise sequence of states) and then the correspond-
ing loop in the parameter space is fixed. In particular,
we go from the north pole to the south pole and back
to the north pole along meridians. Since the loop, as
in our model, is a sequence of meridian-parallel-meridian
path, we can calculate the error and make a direct com-
parison. In this case, the transition error results propor-
tional to δtr ∝ tad and grows linearly in time while for
HQC δtr ∝ 1/tad. Therefore, the HQC is fundamentally
the favorite for long application times with respect to the
STIRAP ones.
Moreover, we can show that the freedom in the choice
of the loop allows us to construct HQC which perform
better than the best STIRAP gates. In Ref. [4] the mini-
mum error (not depending on the evolution time) for STI-
RAP was obtained reaching a compromise between the
necessity to minimize the transition, pure dephasing error
and the constraint of adiabatic evolution. With realistic
physical parameters [21] (J(ω) = kω3e(−ω/ωc)
, Ω = 10
meV, ǫ = 1eV, vmax = Ω/50, k = 10
−2(meV)−2, ωc = 0.5
meV and for low temperature), the total minimum error
in Ref. [4] is δstirap = 10
−3. With the same parame-
ters, we still have the possibility to increase the evolution
time in order to reduce the environmental error. How-
ever, for evolution time tad = 50 ps we obtain a total
error δ = 1.5× 10−4 for the NOT gate and δ = 4× 10−5
for the Hadamard gate, respectively. As can be seen, the
logical gate performance is greatly increased.
D. More general noise
Until now we have discussed the possibility to minimize
the environmental error by choosing a particular loop in
the parameter sphere but the structure of the error func-
tional clearly depends on the system-environment inter-
action. Then one might wonder if the same approach can
be used for a different noise environment.
For this reason, we now briefly analyze the case of noise
matrix in the form A = diag{0, 1, 0,−1}. Again, for long
evolution we can neglect the contribution of the pure de-
phasing and focus on the transition error. In this case the
interesting part of the error functional takes the form
δtr = K[(
sin 2θ cos 2θ)2 + (sin θ sin 2φ)2]. (29)
Even if the analysis in this case is much more com-
plicated, it can be seen that δtr has an absolute mini-
mum for θM = 0. The long time behavior is the same
(δtr ∝ 1/tad) such that the results are qualitatively anal-
ogous to the above ones: for small θM loops (or long
evolution at fixed velocity) the holonomic quantum gate
presents a decreasing error. Then even in this case it is
possible to minimize the environmental error.
IV. CONCLUSIONS
In summary, we have analyzed the performance of
holonomic quantum gates in the presence of environmen-
tal noise by focusing on the possibility to have small
errors choosing different loops in the parameter mani-
fold. Due to the geometric dependence, we can imple-
ment the same logical gate with different loops. Since
different loops correspond to different dynamical evolu-
tions, we have used this freedom to construct an evolu-
tion through “protected” or “weakly influenced” states
leading to good holonomic quantum gates performances.
This allows to select (once that the physical parameter
are fixed) the best loop which minimizes the environ-
mental effect. (Note that this optimization procedure
is rather independent of the details of the simple model
we have considered and arguably, it could be extended
to more complicated systems without any substantial
modification.) We have shown that for long time evo-
lutions the noise decreases as 1/tad while in the other
cases it increases linearly with adiabatic time. We also
have shown that the same features can be found with
different kinds of noise suggesting the possibility to find
a way to minimize the environmental effect in the pres-
ence of any noise. These results open a new possibility
for implementation of holonomic quantum gates to build
quantum computation because they seem robust against
both control error and environmental noise.
Acknowledgment
The autors thank E. De Vito for useful discussions.
One of the authors (P. S.) acknowledges support from
INFN. Financial support by the italian MIUR via
PRIN05 and INFN is acknowledged.
APPENDIX A: MINIMIZING THEOREM
Let us consider the family Cn composed of the closed
curves generated by a sequence of n paths along a parallel
(θ = const) alternated with paths along a meridian (φ =
const). We call Cn a generic curve in this family. For
example, the family C1 contains all the closed curves com-
posed by the sequence of path meridian-parallel-meridian
while the family C2 contains the curves meridian-parallel-
meridian-parallel-meridian.
We argue that the closed curve minimizing the error
in Eq. (18) can be found in the C1 family. First, we
show that any closed curve in C2 spanning a solid angle
a on the sphere can be replaced by a closed curve in C1
spanning the same angle and producing a smaller error.
In analogous way any closed curve in C3 can be replaced
by a closed curve in C2 with smaller error and so on. By
induction we obtain that any closed curve in Cn can be
replaced by a curve in C1 spanning the same solid angle
but producing smaller error. Since the curve belonging
to Cn can approximate any closed curve on the sphere,
the best curve can be found in C1.
The crucial point is to show that any curve in C2 can be
replaced by a curve in C1. Let us consider a generic curve
C2 in C2 spanning a solid angle a: composed by a seg-
ment of a meridian (with θ going from 0 to θ1), a parallel
(spanning a ∆φ1 angle), meridian (with θ : θ1 → θ2), a
parallel (spanning a ∆φ2 angle), and finally a segment
to the north pole along a meridian. Let us consider two
closed curves C11 and C
1 in C1 subtending the same solid
angle a with, respectively, θ1 and θ2 as maximum angle
spanned during the evolution along the meridian. First
we analyze (20) along the meridian. Without losing gen-
erality, we can take θ1 < θ2; it is clear from Eq. (21) that
the value of δtr along the meridian for C
1 is smaller that
for C21 : δ
. We note from the Eq. (21), suitable
extended to C2, that the two paths along the meridians
depends only on θ2 and then produce the same error of
C21 ,
< δMC2
= δMC2 . (A1)
The difference between the contribution along the par-
allel is
δPC2−δ
= ∆φ1
sin θ1 sin
2 2θ1−
1− cos θ1
1− cos θ2
sin θ2 sin
2 2θ2
δPC2−δ
= ∆φ2
sin θ2 sin
2 2θ2−
1− cos θ2
1− cos θ1
sin θ1 sin
2 2θ1
Analysis of the positivity of the quantities given by
Eqs. (A2) and (A3) shows that δPC2 cannot be at the
same time smaller than δP
and δP
. In fact, there are
two possibilities: If δPC2 > δ
, from Eq. (A1) and (A3),
δC2 = δ
+ δPC2 > δ
+ δPC1
= δC1
, (A4)
and the best closed curve is C11 . If δ
, from Eqs.
(A1) and (A2),
δC2 = δ
+ δPC2 > δ
+ δPC2
= δC2
, (A5)
and the best closed curve is C21 .
In the same way it can be shown that any closed curve
in C3 can be replaced by a closed curve in C2 with smaller
error.
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  We study the performance of holonomic quantum gates, driven by lasers, under
the effect of a dissipative environment modeled as a thermal bath of
oscillators. We show how to enhance the performance of the gates by suitable
choice of the loop in the manifold of the controllable parameters of the laser.
For a simplified, albeit realistic model, we find the surprising result that
for a long time evolution the performance of the gate (properly estimated in
terms of average fidelity) increases. On the basis of this result, we compare
holonomic gates with the so-called stimulated Raman adiabatic passage (STIRAP)
gates.

Environmental noise reduction for holonomic quantum gates
Daniele Parodi,1,2 Maura Sassetti,1,3 Paolo Solinas,4 and Nino Zangh̀ı1,2
1 Dipartimento di Fisica,
Università di Genova, Genova, Italy
2 Istituto Nazionale di Fisica Nucleare (Sezione di Genova),
Genova, Italy 3 INFM-CNR Lamia
Via Dodecaneso 33, 16146 Genova, Italy
4 Laboratoire de Physique Théorique de la Matière Condensée,
Université Pierre et Marie Curie,
Place Jussieu, 75252 Paris Cedex 05, France
(Dated: October 27, 2018)
We study the performance of holonomic quantum gates, driven by lasers, under the effect of
a dissipative environment modeled as a thermal bath of oscillators. We show how to enhance the
performance of the gates by suitable choice of the loop in the manifold of the controllable parameters
of the laser. For a simplified, albeit realistic model, we find the surprising result that for a long time
evolution the performance of the gate (properly estimated in terms of average fidelity) increases. On
the basis of this result, we compare holonomic gates with the so-called Stimulated Raman adiabatic
passage (STIRAP) gates.
PACS numbers: 03.67.Lx
I. INTRODUCTION
The major challenge for quantum computation is posed
by the fact that generically quantum states are very del-
icate objects quite difficult to control with the required
accuracy—typically, by means of external driving fields,
e.g., a laser. The interaction with the many degrees of
freedom of the environment causes decoherence; more-
over, errors in processing the information may lead to a
wrong output state.
Among the approaches aiming at overcoming these dif-
ficulties are those for which the quantum gate depends
very weakly on the details of the dynamics, in particu-
lar, the holonomic quantum computation (HQC) [1] and
the so-called Stimulated Raman adiabatic passage (STI-
RAP) [2, 3, 4]. In the latter, the gate operator is obtained
acting on the phase difference of the driving lasers dur-
ing the evolution, while in the former the same goal is
achieved by exploiting the non-commutative analogue of
the Berry phase collected by a quantum state during a
cyclic evolution. Concrete proposals have been put for-
ward, for both Abelian [5, 6] and non-Abelian holonomies
[7, 8, 9, 10, 11, 12]. The main advantage of the HQC
is the robustness against noise deriving from a imperfect
control of the driving fields [13, 14, 15, 16, 17, 18, 19, 20].
In a recent paper [21] we have shown that the dis-
turbance of the environment on holonomic gates can be
suppressed and the performance of the gate optimized
for particular environments (purely superohmic thermal
bath). In the present paper we consider a different sort
of optimization, which is independent of the particular
nature of the environment.
By exploiting the full geometrical structure of HQC,
we show how the performance of a holonomic gate can
be enhanced by a suitable choice of the loop in the man-
ifold of the parameters of the external driving field: by
choosing the optimal loop which minimizes the “error”
(properly estimated in terms of average fidelity loss). Our
result is based on the observation that there are different
loops in the parameter manifold producing the same gate
and, since decoherence and dissipation crucially depend
on the dynamics, it is possible to drive the system over
trajectories which are less perturbed by the noise. For a
simplified, albeit realistic model, we find the surprising
result that the error decreases linearly as the gating time
increases. Thus the disturbance of the environment can
be drastically reduced. On the basis of this result, we
compare holonomic gates with the STIRAP gates.
In Sec. II the model is introduced and the explicit
expression of the error is derived. In Sec. III we find
the optimal loop, calculate the error, make a comparison
with other approaches, and briefly sketch how to treat a
different coupling with the environment.
II. MODEL
The physical model is given by three degenerate (or
quasidegenerate) states, |+〉, |−〉, and |0〉, optically con-
nected to another state |G〉. The system is driven by
lasers with different frequencies and polarizations, acting
selectively on the degenerate states. This model describes
various quantum systems interacting with a laser radia-
tion, ranging from semiconductor quantum dots, such as
excitons [12] and spin-degenerate electron states [3], to
trapped ions [8] or neutral atoms [7].
The (approximate) Hamiltonian modeling the effect of
the laser on the system is (for simplicity, ~ = 1) [8, 12]
H0(t) =
j=+,−,0
ǫ|j〉〈j|+(e−iǫtΩj(t)|j〉〈G|+H.c)
, (1)
where Ωj(t) are the timedependent Rabi frequencies de-
http://arxiv.org/abs/0704.0376v2
pending on controllable parameters, such as the phase
and intensity of the lasers, and ǫ is the energy of the
degenerate electron states. The Rabi frequencies are
modulated within the adiabatic time tad, (which coin-
cides with the gating time), to produce a loop in the pa-
rameter space and thereby realize the periodic condition
H0(tad) = H0(0).
The Hamiltonian (1) has four time dependent eigen-
states: two eigenstates |Ei(t)〉 , i = 1, 2, called bright
states, and two eigenstates |Ei(t)〉, i = 3, 4, called dark
states. The two dark states have degenerate eigen-
value ǫ and the two bright states have timedependent
energies λ±(t) = [ǫ ±
ǫ2 + 4Ω2(t)]/2 with Ω2(t) =
i=±,0 |Ωi(t)|
2 [22].
The evolution of the state is generated by
Ut = Te
dt′H0(t
′), (2)
where T is the time-ordered operator. In the adiabatic
approximation, the evolution of the state takes place in
the degenerate subspace generated by |+〉, |−〉, and |0〉.
This approximation allows to separate the dynamic con-
tribution and the geometric contribution from the evolu-
tion operator. Expanding Ut in the basis of instantaneous
eigenstates of H0(t) (the bright and dark states), in the
adiabatic approximation, we have
Ut ∼=
′)dt′ |Ej(t)〉〈Ej(t)| Ut, (3)
where
Ut = Te
dτV (τ), (4)
here V is the operator with matrix elements Vij(t) =
〈Ei(t)|∂t|Ej(t)〉. The unitary operator Ut plays the role
of timedependent holonomic operator and is the funda-
mental ingredient for realizing complex geometric trans-
formation whereas
′)dt′ |Ej(t)〉〈Ej(t)| is the
dynamic contribution.
Consider Ut for a closed loop, i.e., for t = tad,
U = Utad . (5)
If the initial state |ψ0〉 is a superposition of |+〉 and |−〉,
then U|ψ0〉 is still a superposition of the same vectors (in
general, with different coefficients)[12]. Thus the space
spanned by |+〉 and |−〉 can be regarded as the “logi-
cal space” on which the “logical operator” U acts as a
“quantum gate” operator. Note that for t < tad, Ut|ψ0〉
has, in general, also a component along |0〉. However,
as it is easy to show [22], at any instant t < tad, Ut|ψ0〉
can be expanded in the twodimensional space spanned
by the dark states |E3(t)〉 and |E4(t)〉. It is important
to observe that U depends only on global geometric fea-
tures of the path in the parameter manifold and not on
the details of the dynamical evolution [1, 12].
To construct a complete set of holonomic quantum
gates, it is sufficient to restrict the Rabi frequencies
Ωj(t) in such a way that the norm Ω of the vector
~Ω = [Ω0(t),Ω+(t),Ω−(t)] is time independent and the
vector lies on a real three dimensional sphere [8, 12].
We parametrize the evolution on this sphere as Ω+(t) =
sin θ(t) cos φ(t), Ω−(t) = sin θ(t) sin φ(t) and Ω0(t) =
cos θ(t) with fixed initial (and final) point in θ(0) = 0,
the north pole By straightforward calculation we obtain
the analytical expression for V (t) in eq. (4), V (t) =
iσy cos[θ(t)]φ̇(t), where σy is the usual Pauli matrix
written in the basis of dark states. Thus, the oper-
ator (4) becomes Ut = cos[a(t)] − iσy sin[a(t)], here
a(t) =
dτφ̇(τ) cos θ(τ). Accordingly, the logical op-
erator U (5) is
U = cos a− iσy sin a, (6)
where
a = a(tad) =
∫ tad
dτφ̇(τ) cos θ(τ) (7)
is the solid angle spanned on the sphere during the evolu-
tion. Note that the are many paths on the sphere which
generate the same logical operator U , and span the same
solid angle a.
In a previous work we have studied how interaction
with the environment disturbs the logical operator U [21].
The goal of the present paper is to analyze whether and
how such a disturbance can be minimized for a given
U . To this end, we model the environment as a thermal
bath of harmonic oscillators with linear coupling between
system and environment [23]. The total Hamiltonian is
H = H0(t) +
α + cαxαA), (8)
where A is the system interaction operator called, from
now on, noise operator.
We now consider the time evolution of the reduced
density matrix of the system, determined by the Hamil-
tonian (8). We rely on the standard methods of the “mas-
ter equation approach,” with the environment treated in
the Born approximation and assumed to be at each time
in its own thermal equilibrium state at temperature T .
This allows to include the effect of the environment in
the correlation function (kB = 1)
g(τ) =
cos(ωτ) − i sin(ωτ)
Here the spectral density is
J(ω) =
δ(ω − ωα), (10)
at the low frequencies regimes, is proportional to ωs, with
s ≥ 0, i.e., s = 1 describes a Ohmic environments, typ-
ical of baths of conduction electrons, s = 3 describes
a super-Ohmic environment, typical of baths of phonons
[21, 24]. The asymptotic decay of the real part of g(τ) de-
fines the characteristic memory time of the environment.
Denoting with ρ̃(t) the time evolution of the reduced den-
sity matrix of the system in the interaction picture, e.g.,
ρ̃(t) = U
t ρUt, one has [24]
ρ̃(tad) = ρ(0) +
dτ{g(τ )[ÃÃ
ρ̃(t− τ )− Ã
ρ̃(t− τ )Ã]
+ g(−τ )[ρ̃(t− τ )Ã
Ã− Ãρ̃(t− τ )Ã
]. (11)
Here Ã and Ã′ stand for Ã(t) and Ã(t−τ), with the tilde
denoting the time evolution in the interaction picture.
In quantum information the quality of a gate is usually
evaluated by the fidelity F , which measures the closeness
between the unperturbed state and the final state,
F = 〈ψ0(0)|U
†ρ(tad)U|ψ0(0)〉, (12)
where |ψ0(0)〉 is the initial state, and ρ(tad) = U ρ̃(tad)U
is the reduced density matrix in the Schrödinger picture
starting from the initial condition ρ(0) = |ψ0(0)〉〈ψ0(0)|.
The average error is defined as the average fidelity loss,
i.e.,
δ =< 1−F >= 1− < 〈ψ0(0)|ρ̃(tad)|ψ0(0)〉 >, (13)
where < · · · > denotes averaging with respect to the
uniform distribution over the initial state |ψ0(0)〉.
The right-handside of Eq. (13) can be computed by
the following steps:
(1) solving Eq. (11) in strictly second order approxima-
tion; this approximation corresponds to replace ρ̃(t − τ)
with ρ(0);
(2) using the adiabatic approximation U(t − τ, t) ≈
exp(iτH0(t));
(3) expanding the scalar product in Eq. (13) with respect
to a complete orthonormal basis {|ϕn(t)〉}, n = 1, 2, 3,
orthogonal to |ψ0(t)〉. In this way, one obtains
∫ tad
dt G(t)|〈ψ0(t)|A|ϕn(t)〉|
, (14)
where
G(t) =
Re[g(τ)] cos(ω0nt) + Im[g(τ)] sin(ω0nt))
Here, ω0n = ω0−ωn are the energy differences associated
to the transition ψ0 ↔ φn, with ω0 = ǫ, ω1 = λ+, ω2 =
λ−, and ω3 = ǫ.
The interaction between system and environment is ex-
pressed by the noise operator A in Eq. (8). We shall now
make the assumption that A = diag{0, 0, 0, 1} in the |G〉,
|±〉, and |0〉 basis. In this case the transition between de-
generate states are forbidden, however the noise breaks
their degeneracy, shifting one of them. In spite of its sim-
ple form, this A is nevertheless a realistic noise operator
for physical semiconductor systems [4].
III. MINIMIZING THE ERROR
The problem can be stated in the following way: given
the noise operator A and the logical operator U , find a
path on the parameter space (the surface of the sphere,
described above) which minimizes the error δ.
The total error δ, given by Eq. (14), can be decom-
posed as
δ = δtr + δpd, (16)
where the transition error, δtr, is the contribution to the
sum of the nondegenerate states (ω0n 6= 0) and the pure
dephasing error δpd is the contribution of the degenerate
states (ω0n = 0). Thus
δpd =
∫ tad
sin(ωt)
sin2 2a(t)
sin4 θ(t) (17)
δtr =
n=+,−
1 + [(λn − ǫ)/Ω]2
∫ tad
sin2 2θ(t)dt,
where
Γ0n = J(|ω0n|)
|ω0n|
− sgn(ω0n)
correspond to the transition rates calculated by standard
Fermi golden rules, supposing, as usual, G(t) ≈ G(∞) for
g(τ) strongly peaked around τ = 0. In the following we
define for simplicity
n=+,−
1 + [(λn − ǫ)/Ω]2
Since we are interested at long time evolution, we start
discussing the transition error which dominates in this
regime [4, 25].
A. Transition rate
As explained in Sec. II, the holonomic paths are closed
curves on the surface of the sphere which start from the
north pole. It turns out that the curve minimizing δtr
can be found among the loops which are composed by a
simple sequence of three paths (see the Appendix): evo-
lution along a meridian (φ = const), evolution along a
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FIG. 1: The error δtr versus θM for two different a values:
a = π/2 (dashed line) and a = π/4 (full line) correspond to
NOT and Hadamard gate, respectively.
parallel (θ = const) and a final evolution along a merid-
ian to come back to the north pole.
The error δtr in (18), depends on a given by Eq. (7),
θM (the maximum angle spanned during the evolution
along the meridian), ∆φ (the angle spanned along the
parallel), and angular velocity v. We allow ∆φ ≥ 2π
which corresponds to cover more than one loop along
the parallel. The velocity along the parallel is v(t) =
φ̇(t) sin θ and that along the meridian is v(t) = θ̇(t). In
the following we assume that v is constant, and it cannot
exceed the maximal value of vmax, fixed by adiabatic
condition vmax ≪ Ω.
The parameters a, θM , and ∆φ are connected by the
relation a = ∆φ(1 − cos θM ). The error δtr is then
δtr = δ
tr + δ
tr, (20)
where
δMtr =
sin 4θM
is the contribution along the meridian and
δPtr = K
sin θM sin
2 2θM
1− cos θM
is the contribution along the parallel.
In Fig. 1 δtr is plotted for a = π/2 and a = π/4 (corre-
sponding to NOT and Hadamard gate, respectively) as a
function of θM . One can see that δtr has a local minimum
for θM = π/2 and a global minimum for θM = 0 where
the error vanishes. This suggests that the best choice is
to take θM as small as possible.
It is interesting to consider the dependence of δtr also
on the evolution time tad. For simplicity, we set the ve-
locity v = vmax. In this case, changing θM (and then
∆φ) corresponds to a change in the evolution time. We
obtain
θM = arccos
, (23)
5 10 15 20 25 30
vmaxtad
FIG. 2: The error δtr versus vmaxtad for two different a val-
ues: a = π/2 (dashed line) and a = π/4 (full line) correspond
to NOT and Hadamard gate, respectively. The dotted-dashed
line shows the value of the error at θ = π/2. The circles show
the critical value of vmaxtad above which the best loop is the
one with the minimal θM .
where
(vmaxtad)
2 + a2
. (24)
Using these relations, δMtr and δ
tr, given by (21) and
(22) become functions of tad, vmax, and a. Note that
m measures the space covered along the parallel, in fact
∆φ = 2πm.
In Fig. 2 we see the behavior of δtr as a function
of vmaxtad. The first minimum for both curves corre-
sponds to θM = π/2, then the curves for long tad de-
crease asymptotically to zero corresponding to the region
in which θM → 0. In this regime we have δtr ∝ 1/tad
which is drastically different from the results obtained
with other methods where δtr ∝ tad, (see Refs [4, 25]
and below Sec. III C). It should be observed that this
surprising results is a merit of holonomic approach which
allows to choose the loop in the parameter space, with-
out changing the logical operation as long as it subtends
the same solid angle. Observe that small θM and long
tad mean large value of m, i.e., multiple loops around the
north pole.
Figure 2 shows that, for a given gate, there is a criti-
cal value kc of vmaxtad which discriminate between the
choice of θM (e.g., k = 6 for the Hadamard gate and
k = 25 for the NOT gate). For vmaxtad < kc the best
choice for the loop is θM = π/2; For vmaxtad > kc the
best choice is the value of θM determined by eq. (23) and
(24).
Note that the region vmaxtad > kc is accessible with
physical realistic parameters [12]. For example, if we
choose the laser intensity Ω = 20 meV and vmax = Ω/50
(for which values the nonadiabatic transitions are forbid-
den), the critical parameter corresponds to the critical
time of 15 ps for the Hadamard gate and 42 ps for the
NOT gate.
B. Pure Dephasing
Until now we have ignored the pure dephasing effect
because we have assumed that it is negligible in com-
parison with the transition error for long evolution time.
Now, we check that the pure dephasing error contribu-
tion can indeed be neglected. We can write the pure
dephasing error using Eq. (17) and splitting to parallel
and meridian part as
δPpd =
∫ tad
Q[a(t)] sin ωt sin4 θM (25)
δMpd =
Q[a(t)]
sin ωt
sin4(vmaxt) + sin
vmaxt
,(26)
where Q[a(t)] = 1 + 1/2 sin2[2a(t)].
To estimate δpd we assume that tad is longer with re-
spect to the characteristic time of the bath. Remember-
ing that J(ω) ∝ ωs, the pure dephasing error behavior
along the parallel part at the temperature T is
δPpd ∝
, T ≪ 1/tad
, T ≫ 1/tad
while the along meridian is
δMpd ∝
. (28)
Then, we can conclude that the pure dephasing can al-
ways be neglected for long time evolution because it de-
creases faster than the transition error.
C. Comparison between HQC and STIRAP
We make a comparison between holonomic quantum
computation (HQC) and the STIRAP procedure which is
an analogous approach to process quantum information.
The STIRAP procedure ([2, 4]) is, in its basic points,
very similar to the holonomic information manipulation.
The level spectrum, the information encoding, the evolu-
tion produced by adiabatic evolving laser are exactly the
same. The fundamental difference is that in STIRAP
the dynamical evolution is fixed (we must pass through
a precise sequence of states) and then the correspond-
ing loop in the parameter space is fixed. In particular,
we go from the north pole to the south pole and back
to the north pole along meridians. Since the loop, as
in our model, is a sequence of meridian-parallel-meridian
path, we can calculate the error and make a direct com-
parison. In this case, the transition error results propor-
tional to δtr ∝ tad and grows linearly in time while for
HQC δtr ∝ 1/tad. Therefore, the HQC is fundamentally
the favorite for long application times with respect to the
STIRAP ones.
Moreover, we can show that the freedom in the choice
of the loop allows us to construct HQC which perform
better than the best STIRAP gates. In Ref. [4] the mini-
mum error (not depending on the evolution time) for STI-
RAP was obtained reaching a compromise between the
necessity to minimize the transition, pure dephasing error
and the constraint of adiabatic evolution. With realistic
physical parameters [21] (J(ω) = kω3e(−ω/ωc)
, Ω = 10
meV, ǫ = 1eV, vmax = Ω/50, k = 10
−2(meV)−2, ωc = 0.5
meV and for low temperature), the total minimum error
in Ref. [4] is δstirap = 10
−3. With the same parame-
ters, we still have the possibility to increase the evolution
time in order to reduce the environmental error. How-
ever, for evolution time tad = 50 ps we obtain a total
error δ = 1.5× 10−4 for the NOT gate and δ = 4× 10−5
for the Hadamard gate, respectively. As can be seen, the
logical gate performance is greatly increased.
D. More general noise
Until now we have discussed the possibility to minimize
the environmental error by choosing a particular loop in
the parameter sphere but the structure of the error func-
tional clearly depends on the system-environment inter-
action. Then one might wonder if the same approach can
be used for a different noise environment.
For this reason, we now briefly analyze the case of noise
matrix in the form A = diag{0, 1, 0,−1}. Again, for long
evolution we can neglect the contribution of the pure de-
phasing and focus on the transition error. In this case the
interesting part of the error functional takes the form
δtr = K[(
sin 2θ cos 2θ)2 + (sin θ sin 2φ)2]. (29)
Even if the analysis in this case is much more com-
plicated, it can be seen that δtr has an absolute mini-
mum for θM = 0. The long time behavior is the same
(δtr ∝ 1/tad) such that the results are qualitatively anal-
ogous to the above ones: for small θM loops (or long
evolution at fixed velocity) the holonomic quantum gate
presents a decreasing error. Then even in this case it is
possible to minimize the environmental error.
IV. CONCLUSIONS
In summary, we have analyzed the performance of
holonomic quantum gates in the presence of environmen-
tal noise by focusing on the possibility to have small
errors choosing different loops in the parameter mani-
fold. Due to the geometric dependence, we can imple-
ment the same logical gate with different loops. Since
different loops correspond to different dynamical evolu-
tions, we have used this freedom to construct an evolu-
tion through “protected” or “weakly influenced” states
leading to good holonomic quantum gates performances.
This allows to select (once that the physical parameter
are fixed) the best loop which minimizes the environ-
mental effect. (Note that this optimization procedure
is rather independent of the details of the simple model
we have considered and arguably, it could be extended
to more complicated systems without any substantial
modification.) We have shown that for long time evo-
lutions the noise decreases as 1/tad while in the other
cases it increases linearly with adiabatic time. We also
have shown that the same features can be found with
different kinds of noise suggesting the possibility to find
a way to minimize the environmental effect in the pres-
ence of any noise. These results open a new possibility
for implementation of holonomic quantum gates to build
quantum computation because they seem robust against
both control error and environmental noise.
Acknowledgment
The autors thank E. De Vito for useful discussions.
One of the authors (P. S.) acknowledges support from
INFN. Financial support by the italian MIUR via
PRIN05 and INFN is acknowledged.
APPENDIX A: MINIMIZING THEOREM
Let us consider the family Cn composed of the closed
curves generated by a sequence of n paths along a parallel
(θ = const) alternated with paths along a meridian (φ =
const). We call Cn a generic curve in this family. For
example, the family C1 contains all the closed curves com-
posed by the sequence of path meridian-parallel-meridian
while the family C2 contains the curves meridian-parallel-
meridian-parallel-meridian.
We argue that the closed curve minimizing the error
in Eq. (18) can be found in the C1 family. First, we
show that any closed curve in C2 spanning a solid angle
a on the sphere can be replaced by a closed curve in C1
spanning the same angle and producing a smaller error.
In analogous way any closed curve in C3 can be replaced
by a closed curve in C2 with smaller error and so on. By
induction we obtain that any closed curve in Cn can be
replaced by a curve in C1 spanning the same solid angle
but producing smaller error. Since the curve belonging
to Cn can approximate any closed curve on the sphere,
the best curve can be found in C1.
The crucial point is to show that any curve in C2 can be
replaced by a curve in C1. Let us consider a generic curve
C2 in C2 spanning a solid angle a: composed by a seg-
ment of a meridian (with θ going from 0 to θ1), a parallel
(spanning a ∆φ1 angle), meridian (with θ : θ1 → θ2), a
parallel (spanning a ∆φ2 angle), and finally a segment
to the north pole along a meridian. Let us consider two
closed curves C11 and C
1 in C1 subtending the same solid
angle a with, respectively, θ1 and θ2 as maximum angle
spanned during the evolution along the meridian. First
we analyze (20) along the meridian. Without losing gen-
erality, we can take θ1 < θ2; it is clear from Eq. (21) that
the value of δtr along the meridian for C
1 is smaller that
for C21 : δ
. We note from the Eq. (21), suitable
extended to C2, that the two paths along the meridians
depends only on θ2 and then produce the same error of
C21 ,
< δMC2
= δMC2 . (A1)
The difference between the contribution along the par-
allel is
δPC2−δ
= ∆φ1
sin θ1 sin
2 2θ1−
1− cos θ1
1− cos θ2
sin θ2 sin
2 2θ2
δPC2−δ
= ∆φ2
sin θ2 sin
2 2θ2−
1− cos θ2
1− cos θ1
sin θ1 sin
2 2θ1
Analysis of the positivity of the quantities given by
Eqs. (A2) and (A3) shows that δPC2 cannot be at the
same time smaller than δP
and δP
. In fact, there are
two possibilities: If δPC2 > δ
, from Eq. (A1) and (A3),
δC2 = δ
+ δPC2 > δ
+ δPC1
= δC1
, (A4)
and the best closed curve is C11 . If δ
, from Eqs.
(A1) and (A2),
δC2 = δ
+ δPC2 > δ
+ δPC2
= δC2
, (A5)
and the best closed curve is C21 .
In the same way it can be shown that any closed curve
in C3 can be replaced by a closed curve in C2 with smaller
error.
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