#  Limitations on the maximal level of entanglement of two singlet-triplet
qubits in GaAs quantum dots

Igor Bragar  0000-0003-1321-2371 Institute of Physics, Polish Academy of
Sciences, al. Lotników 32/46, PL 02-668 Warszawa, Poland  Łukasz Cywiński
0000-0002-0162-7943<EMAIL_ADDRESS>Institute of Physics, Polish Academy of
Sciences, al. Lotników 32/46, PL 02-668 Warszawa, Poland

###### Abstract

We analyze in detail a procedure of entangling of two singlet-triplet
($S$-$T_{0}$) qubits operated in a regime when energy associated with the
magnetic field gradient, $\Delta B_{z}$, is an order of magnitude smaller than
the exchange energy, $J$, between singlet and triplet states [Shulman M. et
al., Science 336, 202 (2012)]. We have studied theoretically a single
$S$-$T_{0}$ qubit in free induction decay and spin echo experiments. We have
obtained analytical expressions for time dependence of components of its Bloch
vector for quasistatical fluctuations of $\Delta B_{z}$ and quasistatical or
dynamical $1/f^{\beta}$-type fluctuations of $J$. We have then considered the
impact of fluctuations of these parameters on the efficiency of the entangling
procedure which uses an Ising-type coupling between two $S$-$T_{0}$ qubits.
Particularly, we have obtained an analytical expression for evolution of two
qubits affected by $1/f^{\beta}$-type fluctuations of $J$. This expression
indicates the maximal level of entanglement that can be generated by
performing the entangling procedure. Our results deliver also an evidence that
in the above-mentioned experiment $S$-$T_{0}$ qubits were affected by
uncorrelated $1/f^{\beta}$ charge noises.

##  I Introduction

Spin qubits based on gated quantum dots (QDs) [1, 2, 3] can be initialized,
coherently controlled, and read out. Qubits based on a spin of a single
electron [4] localized in a QD can be controlled with electron-spin resonance
techniques [5, 6, 7], while two-qubit gates can be performed with the help of
exchange interaction [4, 8, 9], which is controlled electrically [10, 11, 12,
13, 14, 15, 16]. Implementation of single-spin control with ac electric [6, 7,
17] and magnetic [5, 11] fields is nevertheless experimentally challenging,
especially in GaAs based QDs, in which interaction with nuclei [18, 19, 20,
21] leads to significant broadening of electron spin resonance lines (this is
in contrast to experimental situation in Si-based single-electron QDs [22, 23,
24], for which nuclear noise can be removed by isotopic purification [24,
25]). Research on all-electrical control (possibly without ac fields) is an
active field. Such a control is possible for spin qubits based on a few
electrons localized in multiple quantum dots [2]. In a double quantum dot
(DQD) containing two electrons one can easily initialize and partially control
a qubit, whose logical states correspond to singlet $S$ and unpolarized
triplet $T_{0}$ formed by the two electrons [10] (note that one can as well
use two QDs with larger odd number of electrons per dot [26]), and all-
electrical ac control is also possible in a variety of other designs based on
two [27], three [28, 29, 30] and four [31] electrons. We focus here on the
DQD-based two-electron singlet-triplet ($S$-$T_{0}$) qubit, for which full
electrical control over the state of the qubit is possible when the spin
splittings of electrons localized in the two dots are distinct. This can be
achieved with creation of a gradient of nuclear spin polarization [32, 33] in
nuclear spin rich material such as GaAs or with the help of micromagnets
creating a gradient of magnetic field [11].

Creation of entanglement [34, 35] of two spin-based qubits is the next
necessary step in development of a QD spin qubit platform for quantum
information processing. For two single-spin qubits, exchange interaction leads
to creation of entangled two-qubit states [11, 36, 15, 12], while in the case
of $S$-$T_{0}$ qubits it is the electric dipole-dipole (capacitive)
interaction [37] that has been most commonly used for interqubit coupling [38,
39] (although exchange coupling could be used too [40, 41, 42]). Our focus in
the present paper is on creation of entanglement and its evolution due to
capacitive interqubit interaction, as it was demonstrated experimentally for
two $S$-$T_{0}$ qubits in Ref. [39].

Interaction of qubits with their environments that fluctuate in an
uncontrolled manner leads to decoherence [43] of their quantum states. In this
process, the entanglement, which requires existence of a coherent
superposition of at least two product states of the two qubits, is also
destroyed [35]. For single electron spins in QDs, the dominant cause of
entanglement decay is their hyperfine interaction with the nuclear baths [44,
45]. Decoherence of $S$-$T_{0}$ qubits, on the other hand, can be dominated by
the nuclear bath at low singlet-triplet splitting [46], but for large
splittings it is mostly caused by charge noise [47, 48, 49], with the nuclear-
induced decay [50, 51] possibly playing a role when the fluctuations of
$S$-$T_{0}$ splitting are suppressed [52]. Influence of quasistatic charge and
nuclear noises on the simplest protocol of generation of entanglement of two
capacitively coupled $S$-$T_{0}$ qubits was considered in Ref. [53]. In this
paper, we consider a more involved protocol from Ref. [39] under the influence
of dynamical charge noise.

In the experiment [39], the two $S$-$T_{0}$ qubits were initialized in a
separable state, and subsequently they were evolving in the presence of a
finite singlet-triplet splitting $J$. With both qubits having nonzero $J$,
dipolar interaction between them leads to the appearance of an Ising-type
interaction, which in the decoherence-free case would lead to periodic
creation of maximal two-qubit entanglement. In the experiment, only one period
of such entanglement oscillation (with entanglement reaching only a fraction
of its maximal possible value) is visible. Furthermore, for the entanglement
to be nonzero the two qubits have to be subjected to a spin echo refocusing
pulse that removes the influence of the slowest environmental fluctuations on
their evolution. The strong decoherence is significantly affecting the
entanglement generation and evolution. The goal of this paper is to understand
processes originated from nuclear polarization fluctuations and charge noise
affecting the singlet-triplet splittings which limit the ability to generate
entangled two-qubit states for experimental protocol performed in Ref. [39].
The main conclusion of our work is that the spin echo protocol removes most of
the influence of the nuclear polarization fluctuations, and the observed
decoherence is caused by charge noise. The experimental data from Ref. [39]
are consistent with the assumption that each qubit is subjected to
$1/f^{\beta}$ charge noise with $\beta\\!\lesssim\\!1$ (as seen for a single
$S$-$T_{0}$ qubit [49]), and the noises affecting the two qubits are
independent. We also discuss the qualitative difference in the character of
the dominant decoherence process between the cases of small
$\beta\\!\lesssim\\!1$ and large $\beta\\!>\\!1$ – in the former case the
imperfectly echoed-out single-qubit noise has dominant influence, while in the
latter case of very low frequency noise the fluctuations of two-qubit coupling
are the main reason for imperfect entanglement.

The paper is organized as follows. In Sec. II we give an overview of the
physics of a single $S$-$T_{0}$ qubit, and we discuss the influence of
fluctuating external magnetic or electric fields on the decoherence of such
qubit seen in the free induction decay (FID) as well as spin echo (SE)
experiments. In Sec. III we recall a procedure that has been designed for
entangling two $S$-$T_{0}$ qubits [39]. We briefly discuss ways of
entanglement quantification that have been applied to the system of two qubits
in Sec. IV. In Sec. V we then analyze the influence of the above-mentioned
factors that lead to decoherence on the efficiency of the procedure of
entangling of two $S$-$T_{0}$ qubits. We show here that a dynamically
fluctuating electric field affecting qubits’ exchange splittings may limit the
maximal level of two-qubit state entanglement by destroying two-qubit
entangling gate or simply by dephasing of single qubits. Finally, Sec. VI
closes the paper with a discussion of conclusions on the nature of charge
noise in the system studied in Ref. [39] that one can draw by comparing the
results of our calculations to the observations described there. In the
appendices we present explicit expressions that describe the averaged values
of $S$-$T_{0}$ qubit components as functions of a duration of free induction
decay and spin echo experiments as well as attenuation factors originated from
$1/f^{\beta}$ dynamical noise of exchange splittings of the qubits.

##  II The physics of a Singlet-Triplet qubit

###  II.1 The Hamiltonian and control over the qubit

In a singlet-triplet qubit, the quantum state is stored in the joint spin
state of two electrons in a DQD, with one electron localized in each of the
two dots, the left (L) and the right (R) one [37, 10, 33, 49]. The logical
states of the qubit are the singlet
$|S\rangle~{}~{}=~{}\frac{1}{\sqrt{2}}(\left|\uparrow_{L}\downarrow_{R}\right\rangle-\left|\downarrow_{L}\uparrow_{R}\right\rangle)$
and the spin unpolarized triplet
$|T_{0}\rangle~{}=~{}\frac{1}{\sqrt{2}}(\left|\uparrow_{L}\downarrow_{R}\right\rangle+\left|\downarrow_{L}\uparrow_{R}\right\rangle)$.
The remaining two states, spin polarized triplets,
$\left|\uparrow_{L}\uparrow_{R}\right\rangle$ and
$\left|\downarrow_{L}\downarrow_{R}\right\rangle$, are split off by the
constant magnetic field applied in the plane of the nanostructure. In the
paper, we adopt the convention that the Bloch sphere of $S$-$T_{0}$ qubit is
defined in such a way that state $|S\rangle$ ($|T_{0}\rangle$) coincides with
north (south) pole of the sphere and the axis connecting these two points is
the $z$ axis.

It was demonstrated in Ref. [10] that it is possible to reliably initialize
the two electron spins in singlet state $|S\rangle$, perform rotations around
$z$ axis of the Bloch sphere, as well as a read-out in a form of projective
measurements onto $|S\rangle$. All these operations can be realized utilizing
the fast control of the exchange interaction that is achieved by applying
proper voltage pulses to the metallic gates on the surface of the sample. The
states $|S\rangle$ and $|T_{0}\rangle$ are naturally split by energy $J$ due
to the exchange interaction between electrons in the DQD [8, 9]. This energy
difference can be influenced by controlling either the energy difference
between the ground single-electron states of the two dots (the so-called
detuning $\epsilon$) [10, 54], or the height of the inter-dot barrier [4, 14].
According to Ref. [39], the value of $J$ can be varied from much less than
$1~{}\mu$eV to a few $\mu$eV on a time scale of a nanosecond. We have then the
following time-dependent and externally controlled term in the Hamiltonian of
the qubit (in units of $\hbar$):

$\hat{H}_{J}(t)=J(t)\frac{\hat{\sigma}_{z}}{2}\,\,,$ (1)

where $\hat{\sigma}_{z}=|S\rangle\langle S|-|T_{0}\rangle\langle T_{0}|$.

Rotations about the $x$ axis of the Bloch sphere (or equivalently, the
rotations between $\left|S\right\rangle$ and $\left|T_{0}\right\rangle$
states) present a greater challenge. In Ref. [33] it was realized with the
help of an interdot gradient of electron spin splitting $\Delta B_{z}$ caused
by difference of nuclear polarizations in the two dots. If the local fields
(either nuclear Overhauser fields, or magnetic fields from external magnets)
in both dots were identical, $|S\rangle$ and $|T_{0}\rangle$ states would not
experience any dynamics, since the phase acquired by spin-down state
$\left|\downarrow_{L}\right\rangle$ (spin-up state
$\left|\uparrow_{L}\right\rangle$) of the electron in the $L$ dot would be
cancelled by the spin-up state $\left|\uparrow_{R}\right\rangle$ (spin down
state $\left|\downarrow_{R}\right\rangle$) of the electron in the second dot.
The field gradient breaks this symmetry, which can be seen in the following
example, where we allow $|S\rangle$ to evolve for time $t$

$\displaystyle|S\rangle$
$\displaystyle=\frac{1}{\sqrt{2}}\left(\left|\uparrow_{L}\downarrow_{R}\right\rangle-\left|\downarrow_{L}\uparrow_{R}\right\rangle\right)$
$\displaystyle\xrightarrow{t}\frac{1}{\sqrt{2}}\left(\left|\uparrow_{L}\downarrow_{R}\right\rangle-\mathrm{e}^{it{\Delta
B_{z}}}\left|\downarrow_{L}\uparrow_{R}\right\rangle\right)\,.$ (2)

We can see that the initial state transforms back-and-forth between singlet
and the unpolarized triplet, as the phase factor oscillates between $1$ and
$-1$ with frequency set by $\Delta B_{z}$. The field gradient $\Delta B_{z}$
contributes the following term to the qubit’s Hamiltonian:

$\hat{H}_{\Delta B_{z}}=\Delta B_{z}\frac{\hat{\sigma}_{x}}{2}\,\,,$ (3)

where $\hat{\sigma}_{x}=\left|S\right\rangle\left\langle
T_{0}\right|+\left|T_{0}\right\rangle\left\langle S\right|$.

The difficulty in the realization of precise singlet-to-triplet transitions
lies in the fact that the necessary field gradient, in most cases, is
generated by the slowly fluctuating Overhauser field established by the
nuclear spins of the atoms comprising the sample [55]. Therefore, $\Delta
B_{z}$ must be treated as given (in fact, it must be measured beforehand, and
the value of $\Delta B_{z}$ varies in different repetitions of the
experiment), hence, it would induce an ongoing transition between
$\left|S\right\rangle$ and $\left|T_{0}\right\rangle$, which is undesirable in
the context of the entangling procedure. Instead, such a procedure requires
precise state transformations on demand. For example, if one desires to
execute a rotation from $\left|S\right\rangle$ to $\left|T_{0}\right\rangle$
at a given moment $t^{\prime}$ it would be ideal if $\Delta B_{z}$ could be
turned on only for an interval $[t^{\prime},t^{\prime}+\pi/\Delta B_{z}]$, so
that the phase in Eq. (2) is $\mathrm{e}^{i\Delta
B_{z}\cdot(t^{\prime}+\frac{\pi}{\Delta
B_{z}}-t^{\prime})}=\mathrm{e}^{i\pi}=-1$. Since there is no practical way to
change the value of $\Delta B_{z}$ during a single realization of the
procedure, one cannot simply turn off the gradient and stop the transition.
Nevertheless, the transition can be effectively blocked by overshadowing
$\Delta B_{z}$ with strong enough splitting $J$, which can be controlled with
relative ease, and it can be switched on and off at will. In order to realize
the idealized scenario such as the one described above, the time scale on
which $J$ is manipulated must be the shortest time scale in the problem. In
particular, this time scale must be much shorter than the period of rotations
around $x$-axis $1/\Delta B_{z}$ as well as the period of $z$-rotations set by
the maximal value of singlet-triplet splitting $1/J_{\mathrm{max}}$. Note that
pulse-shaping techniques for gate error mitigation were derived for
$S$-$T_{0}$ qubits [56, 57, 58], but their implementation has proven to be
challenging so far, and we focus here on the simplest control scheme used in
experiment on creation of entanglement in Ref. [39]. Let us also note that
entanglement of two $S$-$T_{0}$ qubits in the situation in which $\Delta
B_{z}$ is always larger than $J$ was generated using a scheme involving ac
control of $J$ [59], which is distinct from the one used in Ref. [39] and
analyzed here.

###  II.2 Decoherence of a single $S$-$T_{0}$ qubit

####  II.2.1 The nature of noisy terms in the Hamiltonian

A qubit evolving with nominally constant $\Delta B_{z}$ and $J$ is undergoing
decoherence due to uncontrolled fluctuations of these parameters. When the
magnetic field gradient $\Delta B_{z}$ is due to the difference of the $z$
components of nuclear Overhauser fields in the two dots (which is the case on
which we focus here), the main mechanism responsible for its fluctuations is
the spin diffusion process [55, 60] caused by dipolar interactions between the
nuclear spins. Note that we focus here on $J\\!\gg\Delta B_{z}$ regime, in
which only the large-amplitude classical fluctuations of the Overhauser fields
can affect the qubit. This is in contrast to $J\\!\ll\\!\Delta B_{z}$ case, in
which a quantum treatment of nuclear fluctuations is necessary [61, 62, 63].
The large-amplitude fluctuations have a characteristic decorrelation time
scale of about one second [55, 60], which is much longer than the time scale
of a single run of the experiment, i.e. a single repetition of qubit’s
initialization – evolution – measurement cycle. Therefore, $\Delta B_{z}(t)$
can be treated as quasistatic [64, 19, 51], i.e. it is considered as a
constant (i.e. time independent) random variable with certain probability
distribution $p(\Delta B_{z})$. For a large number of nuclei, this
distribution is normal [65], with the average value $\overline{\Delta B_{z}}$
and dispersion (standard deviation) $\sigma_{\Delta
B_{z}}=\sqrt{\overline{\Delta B_{z}^{2}}-(\overline{\Delta B_{z}})^{2}}$.
Within this approximation, the result of the experiments are interpreted as
follows. Given the initial density operator of the qubit,
$\hat{\rho}(0)=\hat{\rho}_{\mathrm{ini}}$, its evolution during the experiment
run number $n$, is described by the qubit Hamiltonian
$\hat{H}_{\mathrm{q}}(t)=\hat{H}_{J}(t)+\hat{H}_{\Delta B_{z}}$ with an
unknown, but constant, value of $\Delta B_{z}=\Delta B_{z}^{n}$ drawn from the
distribution $p(\Delta B_{z})$,

$\displaystyle\hat{\rho}(t|\Delta B_{z}^{n})$
$\displaystyle=\mathrm{e}^{-it(J\hat{\sigma}_{z}+\Delta
B_{z}\hat{\sigma}_{x})/2}\hat{\rho}_{\mathrm{ini}}$
$\displaystyle\times\mathrm{e}^{it(J\hat{\sigma}_{z}+\Delta
B_{z}\hat{\sigma}_{x})/2}\Big{|}_{\Delta B_{z}=\Delta B_{z}^{n}}\,.$ (4)

The final results are derived from the whole series of $N$ experiment
repetitions with the same initial state. Therefore, the expectation values of
the measured quantities are calculated with the help of the averaged density
operator according to

$\displaystyle\hat{\rho}_{\mathrm{fin}}(t)$
$\displaystyle=\frac{1}{N}\sum_{n=1}^{N}\hat{\rho}(t|\Delta B_{z}^{n})$
$\displaystyle\xrightarrow{N\to\infty}\int_{-\infty}^{\infty}d(\Delta
B_{z})p(\Delta B_{z})\hat{\rho}(t|\Delta B_{z})$
$\displaystyle=\overline{\hat{\rho}(t|\Delta B_{z})}\,.$ (5)

On the other hand, the exchange splitting $J(t)$ changes due to fluctuations
of local electric fields, i.e. due to charge noise, which is ubiquitous in
semiconductor nanostructures. The charge noise typically has its spectral
weight concentrated at low frequencies. Contributions of noise at frequencies
corresponding to the inverse of typical qubit coherence are typically
negligible when considering free evolution of the qubit (hence the noise can
be treated then as quasistatic), but they have to be taken into account when
modeling the spin echo experiment [49], in which the influence of the lowest-
frequency noise is removed, and coherence times are longer. In order to model
the experimental data, one has to average the qubit’s evolution over many
realizations of the stochastic process $J(t)$. If the noise statistics is
assumed to be Gaussian (which is natural for noise consisting of many
independent contributions), and if the evolution can be treated in the pure
dephasing approximation (i.e. neglecting $\Delta B_{z}\hat{\sigma}_{x}$ term
when $\Delta B_{z}\\!\ll\\!J(t)$), the averaging can be done analytically. In
the case of non-Gaussian noise and when keeping the general form of the
Hamiltonian, one has to resort to numerical simulations [66, 67].

It was shown [49] that the GaAs/AlGaAs $S$-$T_{0}$ qubit is affected by noise
having power spectral density of $1/f^{\beta}$ form with
$\beta\\!\approx\\!0.7$ in the range of frequencies relevant for correct
description of spin echo signal. It is unclear if this value of $\beta$ is
specific to this material or only to the device used in that experiment.

####  II.2.2 Decoherence during free evolution of the qubit

We now assume that the initial state of a qubit is
$|{-y}\rangle=\frac{1}{\sqrt{2}}(|S\rangle-i|T_{0}\rangle)$. This choice is
connected with the entangling procedure used in Ref. [39], in which this
single-qubit state is used as the initial one for both qubits. In a free
induction decay (FID) experiment, the qubit undergoes evolution without any
manipulations between its initialization and the coherence readout at time
$\tau$. With fixed values of $J$ and $\Delta B_{z}$, the expectation values of
qubit observables
$\langle\hat{\sigma}_{i}^{\mathrm{FID}}(\tau)\rangle=\langle{-y}|\hat{\sigma}_{i}(\tau)|{-y}\rangle$
are given by

$\displaystyle\langle\hat{\sigma}_{x}^{\mathrm{FID}}(\tau)\rangle$
$\displaystyle=\frac{J}{\sqrt{\Delta B_{z}^{2}+J^{2}}}\sin\Big{(}\sqrt{\Delta
B_{z}^{2}+J^{2}}\tau\Big{)}\,\,,$ (6)
$\displaystyle\langle\hat{\sigma}_{y}^{\mathrm{FID}}(\tau)\rangle$
$\displaystyle=-\cos\Big{(}\sqrt{\Delta B_{z}^{2}+J^{2}}\tau\Big{)}\,\,,$ (7)
$\displaystyle\langle\hat{\sigma}_{z}^{\mathrm{FID}}(\tau)\rangle$
$\displaystyle=-\frac{\Delta B_{z}}{\sqrt{\Delta
B_{z}^{2}+J^{2}}}\sin\Big{(}\sqrt{\Delta B_{z}^{2}+J^{2}}\tau\Big{)}\,\,.$ (8)

The qubit evolves under the influence of $J\\!\gg\\!\Delta B_{z}$, so to the
lowest order in $\Delta B_{z}/J$ the initial amplitude of $x$ and $y$ signals
is $\approx 1$, while the $z$ signal has amplitude $\approx\\!\Delta B_{z}/J$.
In the presence of fluctuations of both $J$ and $\Delta B_{z}$ all these
signals will average to zero at long times, at which the arguments of the
oscillatory functions taken from an appropriate distribution will have
relative phases randomly distributed between $0$ and $2\pi$.

We can average the above expressions over $\Delta B_{z}$ and $J$ assuming
Gaussian and quasistatic fluctuations of either of these parameters. We focus
on the largest observable,
$\langle\hat{\sigma}_{y}^{\mathrm{FID}}(\tau)\rangle$, and we approximate
$\sqrt{\Delta B_{z}^{2}+J^{2}}\tau\approx J\tau+\Delta B_{z}^{2}\tau/2J$,
which is valid for $J\gg\Delta B_{z}$ and for short durations $\tau\ll
8J^{3}/\Delta B_{z}^{4}$. For $J$ noise we get then a simple Gaussian decay:

$\left\langle\langle\hat{\sigma}_{y}^{\mathrm{FID}}(\tau)\rangle\right\rangle_{J}\approx-\frac{1}{2}\exp\Big{[}-(\tau/T^{*}_{2,J})^{2}\Big{]}\cos(\bar{J}\tau)\,\,,$
(9)

where $\bar{J}$ is the average value of $J$ and the decay time scale is

$T^{*}_{2,J}=\frac{\sqrt{2}}{\sigma_{J}}\,\,,$ (10)

where $\sigma_{J}$ is the standard deviation of $J$.

Averaging over $\Delta B_{z}$ gives

$\displaystyle\left\langle\langle\hat{\sigma}_{y}^{\mathrm{FID}}(\tau)\rangle\right\rangle_{\Delta
B_{z}}$ $\displaystyle\approx-\exp\left(-\frac{\tau^{2}\overline{\Delta
B_{z}}^{2}\sigma_{\Delta B_{z}}^{2}}{2\left(J^{2}+(\sigma_{\Delta
B_{z}}^{2}\tau)^{2}\right)}\right)$
$\displaystyle\times\frac{1}{\big{(}1+(\sigma_{\Delta
B_{z}}^{2}\tau/J)^{2}\big{)}^{1/4}}$
$\displaystyle\times\cos\Big{(}r(\tau)+s(\tau)\Big{)},$ (11)

where $r(\tau)=\frac{1}{2}\arctan\left(\sigma_{\Delta
B_{z}}^{2}\tau/J\right)$, and $s(\tau)=J\tau\Big{(}\overline{\Delta
B_{z}}^{2}+2J^{2}+2(\sigma_{\Delta
B_{z}}^{2}\tau)^{2}\Big{)}/\Big{(}2\big{(}J^{2}+(\sigma_{\Delta
B_{z}}^{2}\tau)^{2}\big{)}\Big{)}$. The decay envelope is then a product of
two factors. The first one dominates the decay when $\overline{\Delta
B_{z}}\\!\gg\\!\sigma_{\Delta B_{z}}$, i.e. in the situation in which a finite
$\Delta B_{z}$ is used for coherent control of the qubit. Then at long $\tau$
this factor saturates at $\exp\big{(}-\overline{\Delta
B_{z}}^{2}/2\sigma^{2}_{\Delta B_{z}}\big{)}\\!\ll\\!1$, and the qubit loses
most of its coherence at time scale $\tau\\!\ll\\!J/\sigma^{2}_{\Delta
B_{z}}$, at which the factor can be approximated as
$\exp\Big{(}-(\tau/T_{2,\Delta B_{z}}^{*})^{2}\Big{)}$, where

$T^{*}_{2,\Delta B_{z}}\\!=\\!\frac{\sqrt{2}}{\sigma_{\Delta
B_{z}}}\cdot\frac{J}{\overline{\Delta B_{z}}}\,\,.$ (12)

On the other hand, when $\overline{\Delta B_{z}}\\!\ll\\!\sigma_{\Delta
B_{z}}$, the second factor dominates, and the signal envelope decays in power
law fashion $\propto\\!\sqrt{\tau_{\Delta B_{z}}/\tau}$, where the
characteristic time $\tau_{\Delta B_{z}}\\!=\\!J/\sigma^{2}_{\Delta B_{z}}$.

It is important to note now that in the regime of
$\bar{J}\\!\gg\\!\overline{\Delta B_{z}}$ and $\overline{\Delta
B_{z}}\\!\gg\\!\sigma_{\Delta B_{z}}$ (that is relevant for experiments of
interest in this paper), we typically have $\sigma_{J}\\!\gg\\!\sigma_{\Delta
B_{z}}$. This is due to the observed relation between $J$ and detuning
$\epsilon$: $J\\!\approx\\!J_{0}\exp(\epsilon/\epsilon_{0})$, and the fact
that the noise in $J$ comes mostly from fluctuations of $\epsilon$. We have
thus $\delta J/J\sim\delta\epsilon/\epsilon_{0}$, so for constant level of
detuning noise the standard deviation $\sigma_{J}$ increases with $J$ [49]. It
is then a reasonable assumption to neglect the effect of fluctuations of
$\Delta B_{z}$ in the calculation of decoherence. Furthermore, for
$\bar{J}\\!\gg\\!\overline{\Delta B_{z}}$ the main effect of $\overline{\Delta
B_{z}}\hat{\sigma}_{x}$ term is a slight tilt in $xOz$ plane of the axis about
which the qubit’s Bloch vector is precessing. Neglecting this effect, we
arrive at the pure dephasing approximation to the qubit’s Hamiltonian:

$\hat{H}(t)\approx\left(\bar{J}+\delta
J(t)\right)\frac{\hat{\sigma}_{z}}{2}\,\,,$ (13)

where we have now included the time dependence of $J$ noise. The off-diagonal
element of the qubit’s density operator is given by

$\displaystyle\rho_{ST_{0}}(\tau)$
$\displaystyle=\rho_{ST_{0}}(0)\,\mathrm{e}^{-i\bar{J}\tau}$
$\displaystyle\times\left\langle\exp\left(-i\int_{0}^{\infty}f_{\mathrm{FID}}(t;\tau)\delta
J(t)\mathrm{d}t\right)\right\rangle_{\delta J}\,\,,$ (14)

where $\langle...\rangle_{\delta J}$ denotes averaging over different
realizations of $\delta J(t)$ noise, and
$f_{\mathrm{FID}}(t;\tau)\\!=\\!\theta(t)\theta(\tau-t)$ is the FID time-
domain filter function, where $\theta(t)$ is Heaviside step function. The
transverse components of the qubit state are given by
$\left\langle\langle\hat{\sigma}_{x}^{\mathrm{FID}}(\tau)\rangle\right\rangle\\!=\\!2\mathrm{Re}\rho_{ST_{0}}(\tau)$
and
$\left\langle\langle\hat{\sigma}_{y}^{\mathrm{FID}}(\tau)\rangle\right\rangle\\!=\\!2\mathrm{Im}\rho_{ST_{0}}(\tau)$.
For Gaussian $\delta J(t)$ noise only the second cumulant of the random phase
is nonzero [68, 69, 70], and we have a closed formula for coherence:

$\rho_{ST_{0}}(\tau)=\rho_{ST_{0}}(0)\,\mathrm{e}^{-i\bar{J}\tau}\,\mathrm{e}^{-\chi_{\mathrm{FID}}(\tau)}\,\,,$
(15)

in which the attenuation factor $\chi_{\mathrm{FID}}$ is defined in the
following way

$\displaystyle\chi_{\mathrm{FID}}$
$\displaystyle=\int_{0}^{\infty}\mathrm{d}t_{1}\int_{0}^{\infty}\mathrm{d}t_{2}f_{\mathrm{FID}}(t_{1})f_{\mathrm{FID}}(t_{2})\left\langle\delta
J(t_{1})\delta J(t_{2})\right\rangle_{\delta J}$
$\displaystyle=\int_{0}^{\infty}S(\omega)|\tilde{f}_{\mathrm{FID}}(\omega)|^{2}\frac{\mathrm{d}\omega}{\pi}\,\,,$
(16)

where in the second line we have assumed that the noise is stationary, so that
its autocorrelation function is $C(t_{1}-t_{2})\\!=\\!\left\langle\delta
J(t_{1})\delta J(t_{2})\right\rangle_{\delta J}$. The spectral density of the
noise is $S(\omega)\\!=\\!\int_{-\infty}^{\infty}C(t)\mathrm{e}^{i\omega
t}\mathrm{d}t$ and the frequency domain filter function [68, 69] is

$\displaystyle\tilde{f}_{\mathrm{FID}}(\omega)$
$\displaystyle=\int_{-\infty}^{\infty}f_{\mathrm{FID}}(t;\tau)\mathrm{e}^{i\omega
t}\mathrm{d}t$
$\displaystyle=2\sin^{2}\left(\frac{\omega\tau}{2}\right)\frac{1}{\omega^{2}}\,\,.$
(17)

In the case of $S(\omega)\\!\sim\\!\frac{1}{f^{\beta}}$ with $\beta=0.7$, the
attenuation factor defined in Eq. (16) depends on $\tau$ as a power function:
$\chi_{\mathrm{FID}}\propto\tau^{1.7}$ (see Appendix B for a detailed
calculation).

Note that for $\beta\\!\geqslant\\!1$ the integral in Eq. (16) diverges.
However, in a real experimental setting, the total time of data acquisition,
$T_{M}$, involving many repetitions of cycles of qubit initialization,
evolution for time $\tau$, and measurement, sets the low-frequency cutoff,
$\omega_{\mathrm{min}}\\!\sim\\!1/T_{M}$, for frequencies of noise that
actually affect the qubit [70, 49, 25]. Consequently, the lower limit of the
integral in Eq. (16) should be $\omega_{\mathrm{min}}>0$ instead of zero,
making the attenuation factor finite. In this paper we are setting
$\omega_{\mathrm{min}}/2\pi\\!=\\!1$ mHz, corresponding to $T_{M}$ in a
perfectly realistic range of tens of minutes.

Figure 1:  (Color online) FID and SE signals of a single $S$-$T_{0}$ qubit for
the case of $1/f^{\beta}$ noise in exchange splitting $J$, with fluctuations
in $\Delta B_{z}$ neglected. The power of the noise was chosen to be such that
ensures decay of SE signal as observed in the experiment [49] (such a power
that leads to half-decay of SE signal in 1.5 $\mu s$ for $J\\!\approx\\!1$
$\mu$eV). Note that the improvement in coherence time due to echo relative to
FID is larger when $\beta$ is larger, i.e. when there is more noise power at
the lowest frequencies.

####  II.2.3 Decoherence in spin echo protocol

The influence of very low frequency noise on qubit dephasing can be removed by
employing a spin echo protocol [68], in which a qubit is subjected to a $\pi$
rotation about an axis perpendicular to the axis along which the noise is
coupled (here we focus on $\pi$ rotations about the $x$ axis) at time
$t=\frac{\tau}{2}$, and the coherence is read out at the final time $t=\tau$.

Within the pure dephasing approximation introduced above, and for perfect
short ($\delta$ function like) pulses, the calculation of echoed coherence
signal as a function of duration $\tau$ of the echo procedure amounts to
replacing the $f_{\mathrm{FID}}(t)$ function in Eq. (14) by the
$f_{\mathrm{SE}}(t)$ function which is nonzero for $t\in[0,\tau]$ and changes
its value from $1$ to $-1$ at $t=\tau/2$, see e.g. Refs. [68, 70]. The
coherence at the final readout time is given by a formula analogous to Eq.
(15), only with $\chi_{\mathrm{FID}}(\tau)$ replaced by
$\chi_{\mathrm{SE}}(\tau)$ in which
$\tilde{f}_{\mathrm{SE}}(\omega)=4\sin^{4}\frac{\omega\tau}{4}/\omega^{2}$
appears. The latter filter function strongly suppresses very low $\omega$
contribution (precisely from $\omega\\!\ll\\!4\tau$ range) to the attenuation
factor $\chi_{\mathrm{SE}}(\tau)$. For quasistatic charge noise we have
$S(\omega)\\!\approx\\!\sigma_{J}\delta(\omega)$, and the echo protocol leads
to a complete recovery of the initial coherence.

For $J$ noise with nontrivial spectrum, but with a lot of noise power at low
frequencies, the echo decay time is expected to be much longer than the FID
decay time, see Fig. 1 for illustration. Recall that our justification for the
neglect of quasistatic $\Delta B_{z}$ fluctuations was that in the
experimentally relevant parameter regime they lead to much slower FID decay
than the $J$ fluctuations. Echo-induced suppression of $J$ noise effects could
be suspected of leading to breakdown of that assumption. This is not the case:
the echo protocol is also strongly suppressing the effects of quasistatic
transverse noise, provided that the effective transverse field $\Delta B_{z}$
is much smaller than the typical longitudinal field. For given values of $J$
and $\Delta B_{z}$ we have

$\displaystyle\langle\hat{\sigma}_{y}^{\mathrm{SE}}(\tau)\rangle$
$\displaystyle=\langle{-y}|\mathrm{e}^{i\hat{H}\frac{\tau}{2}}(i\hat{\sigma}_{x})\mathrm{e}^{i\hat{H}\frac{\tau}{2}}\hat{\sigma}_{y}$
$\displaystyle\times\mathrm{e}^{-i\hat{H}\frac{\tau}{2}}(-i\hat{\sigma}_{x})\mathrm{e}^{-i\hat{H}\frac{\tau}{2}}|{-y}\rangle$
$\displaystyle=\frac{1}{\Delta B_{z}^{2}+J^{2}}\Big{(}J^{2}+\Delta
B_{z}^{2}\cos\Big{(}\sqrt{\Delta B_{z}^{2}+J^{2}}\tau\Big{)}\Big{)}\,\,.$ (18)

As before, this expression can be analytically averaged over $\Delta B_{z}$
for short durations $\tau\ll 8J^{3}/\Delta B_{z}^{4}$. The full result is more
complicated than Eq. (11) (see Appendix A for full expression), but for
$\sigma^{2}_{\Delta B_{z}}\tau/J\\!\gg\\!(\Delta B_{z}/\sigma_{\Delta
B_{z}})^{2}$ one can arrive at

$\displaystyle\left\langle\langle\hat{\sigma}_{y}^{\mathrm{SE}}(\tau)\rangle\right\rangle_{\Delta
B_{z}}$ $\displaystyle\approx\frac{J^{2}}{J^{2}+\overline{\Delta B_{z}}^{2}}$
$\displaystyle+\frac{\sigma^{2}_{\Delta B_{z}}}{2\left(J^{2}+\overline{\Delta
B_{z}}^{2}\right)}\left(\frac{J}{\sigma^{2}_{\Delta B_{z}}\tau}\right)^{3/2}$
$\displaystyle\times\cos\left(J\tau\right)\,\,,$ (19)

where we see that at long times the only effect of quasistatic transverse
noise is decrease of the coherence signal by an amount
$\approx(\overline{\Delta B_{z}}/J)^{2}\\!\ll\\!1$ compared to its initial
value.

The echo signal averaged over quasistatic fluctuations of $J$ looks similarly

$\displaystyle\langle\langle\hat{\sigma}_{y}^{\mathrm{SE}}(\tau)\rangle\rangle_{J}$
$\displaystyle\approx\frac{\bar{J}^{2}+\sigma_{J}^{2}}{\bar{J}^{2}+\Delta
B_{z}^{2}}$ $\displaystyle+\frac{\Delta B_{z}^{2}}{\bar{J}^{2}+\Delta
B_{z}^{2}}\exp\left(-\frac{\sigma_{J}^{2}\tau^{2}}{2}\right)$
$\displaystyle\times\cos\left(\bar{J}\tau\right).$ (20)

For long times, the signal
$\langle\langle\sigma_{y}^{\mathrm{SE}}(\tau)\rangle\rangle_{J}$ remains close
to its initial value ${(\bar{J}^{2}+\sigma_{J}^{2})}/{(\bar{J}^{2}+\Delta
B_{z}^{2})}\approx 1$.

##  III The procedure for entangling two $S$-$T_{0}$ qubits

Now we proceed with the description of the procedure designed in Ref. [39]
aimed to create maximally entangled two-qubit states out of an initial product
state. Here, let us focus on an idealized setting in which both $J$ and
$\Delta B_{z}$ are piecewise-constant and fully controlled, so that no
averaging over their values is performed. Of course, in reality only $J$ is
controlled, and it furthermore fluctuates – and the consequences of this will
be the main subject of subsequent sections.

Entanglement generation requires some kind of qubit-qubit interaction, and in
the case of $S$-$T_{0}$ qubits one utilizes the coupling between electric
dipoles induced by state-dependent charge distributions in each DQD [37]. Only
if both qubits are in state $\left|S\right\rangle$, and their exchange
splittings are finite, the charge distributions are asymmetric (due to mixing
of the singlet state relevant here with a singlet state of two electrons
localized in a single dot), and hence, each DQD possesses a nonzero electric
dipole moment. Therefore, the effective Hamiltonian of this interaction is
given by (in units of $\hbar$)

$\hat{H}_{\mathrm{int}}=\frac{1}{4}J_{12}\left|SS\right\rangle\left\langle
SS\right|=\frac{1}{4}J_{12}(\hat{\sigma}_{z}+\hat{\mathds{1}})\otimes(\hat{\sigma}_{z}+\hat{\mathds{1}})\,.$
(21)

It was established empirically in Ref. [39] that for values of splittings
$J_{i}$ used there the strength of the interaction is given by

$J_{12}=\frac{J_{1}J_{2}}{K}\,\,,$ (22)

where parameter $K$ is a constant. Thus, the control over the exchange
splittings (described in the previous section) simultaneously allows to modify
the value of the coupling $J_{12}$. Note that the fact that $J_{12}\propto
J_{1}J_{2}$ exposes single-qubit gates to crosstalk when nearby $S$-$T_{0}$
qubits have finite $J_{i}$ splittings, and methods for dealing with this issue
have been discussed [71]. Furthermore, configuration interaction calculations
have suggested the existence of parameter regions for two capacitively coupled
$S$-$T_{0}$ qubits in which the relation between $J_{12}$ and $J_{i}$ is more
complicated, leading e.g. to predictions of “sweet spots” at which charge-
noise induced fluctuations of $J_{i}$ and/or $J_{12}$ are suppressed [72, 73].
Here we focus on noisy behavior of $J_{i}$ measured in Ref. [49] and the
resulting noisy behavior of $J_{12}$ following from Eq. (22).

The procedure consists of the following steps. Before each run the value of
the gradients of magnetic field $\Delta B_{z,i}$ in each DQD is established
(e.g. by a measurement or even setting the value with a dedicated procedure).
Then each $S$-$T_{0}$ qubit is independently initialized in
$\left|S\right\rangle$ state, yielding a separable two-qubit state

$\left|\psi(t=0)\right\rangle=\left|S\right\rangle\otimes\left|S\right\rangle\equiv\left|SS\right\rangle\,.$
(23)

For each qubit, the exchange splitting is turned off for a time interval
corresponding to $\frac{\pi}{2}$ rotation around $x$ axis due to magnetic
gradients. Generally $\Delta B_{z,1}\neq\Delta B_{z,2}$ (for concreteness,
suppose that $\Delta B_{z,1}\geqslant\Delta B_{z,2}$) so that each splitting
has to be kept turned off for different time interval
$t_{i}=\frac{\pi}{2\Delta B_{z,i}}$. After time $t_{1}$ one obtains

$\displaystyle\left|\psi(t_{1})\right\rangle=$
$\displaystyle{}\,\mathrm{e}^{-\frac{i}{2}\frac{\pi}{2}\hat{\sigma}_{x}}\otimes\mathrm{e}^{-\frac{i}{2}\Delta
B_{z,2}t_{1}\hat{\sigma}_{x}}\left|SS\right\rangle$ $\displaystyle=$
$\displaystyle{}\,\frac{1}{\sqrt{2}}\Big{(}\left|S\right\rangle-i\left|T_{0}\right\rangle\Big{)}\otimes\mathrm{e}^{\frac{i}{2}\frac{\pi}{2}\frac{\Delta
B_{z,2}}{\Delta B_{z,1}}\hat{\sigma}_{x}}\left|S\right\rangle\,.$ (24)

Then, the splitting $J_{1}$ is raised to suppress the rotation of qubit $1$
about $x$ axis, while the rotation of qubit $2$ is being completed in time
$\delta t=t_{2}-t_{1}$:

$\displaystyle\left|\psi(t_{1}+\delta t=t_{2})\right\rangle$
$\displaystyle=\mathrm{e}^{-\frac{i}{2}J_{1}\delta
t\hat{\sigma}_{z}}\otimes\mathrm{e}^{-\frac{i}{2}\Delta B_{z,2}\delta
t\hat{\sigma}_{x}}\left|\psi(t_{1})\right\rangle$
$\displaystyle=\frac{1}{\sqrt{2}}\Big{(}\mathrm{e}^{-i\frac{J_{1}\delta
t}{4}}\left|S\right\rangle-i\mathrm{e}^{+i\frac{J_{1}\delta
t}{4}}\left|T_{0}\right\rangle\Big{)}$
$\displaystyle\otimes\frac{1}{\sqrt{2}}\Big{(}\left|S\right\rangle-i\left|T_{0}\right\rangle\Big{)}\,.$
(25)

In order to remove the phases imprinted on qubits due to finite $\Delta
B_{z,i}$, the spin echo sequences are carried out on each qubit. (Of course,
in the more realistic setting in which $J$ fluctuates, the need to remove the
influence of the slowest of these fluctuations on the final two-qubit state is
a much stronger motivation to employ spin echo.) The SE sequence consists of
three steps: the evolution with splitting $J_{i}$ over a chosen time interval,
$\pi$ rotation of a state about $x$ axis (i.e. $2t_{i}$ interval when the
splitting $J_{i}$ is turned off), which is followed by the evolution for the
same time interval. In this case, the durations of SE on each qubit are chosen
so that both sequences terminate at the same instant $\tau$. Since $t_{1}\neq
t_{2}$ this requirement implies that each SE starts at different time, and
they last for unequal durations. Simultaneously during the evolution intervals
excluding the periods of evolution corresponding to the $\pi$ rotations both
splittings are on and hence the two-qubit coupling $\hat{H}_{\mathrm{int}}$ is
on as well, thus allowing for qubits to entangle. Figure 2 showcases the time
dependence of splittings $J_{1}$ and $J_{2}$ for the entirety of the
entangling procedure. At the end of the procedure, the following state is
produced:

$\displaystyle\left|\tau\right\rangle\equiv|\psi(\tau)\rangle$
$\displaystyle=\frac{1}{2}\mathrm{e}^{-\frac{1}{2}iJ_{12}(\tau-3t_{2})}\left|SS\right\rangle$
$\displaystyle+\frac{i}{2}\left|ST_{0}\right\rangle+\frac{i}{2}\left|T_{0}S\right\rangle$
$\displaystyle-\frac{1}{2}\mathrm{e}^{-\frac{1}{2}iJ_{12}(\tau-3t_{2})}\left|T_{0}T_{0}\right\rangle\,.$
(26)

Recall that $t_{2}>t_{1}$ because it was assumed that $\Delta
B_{z,1}\geqslant\Delta B_{z,2}$ without loss of generality (if $\Delta
B_{z,2}>\Delta B_{z,1}$ simply relabel the qubits to obtain the same result).
Note that in the currently considered case of fixed $J$ and $\Delta B_{z}$ the
latter drops out from the final state $|\tau\rangle$.

It should be noted that for the discrete set of durations
$\tau=\tau_{\mathrm{ent}}^{(a)}=(2a-1)\frac{\pi}{J_{12}}+3t_{2}$, where $a$ is
a natural number, the state $|\tau\rangle$ is maximally entangled,
specifically, for odd or even $a$ the entangling procedure generates the state

$|\psi_{\mathrm{o}}\rangle=\frac{i}{2}\left(-|SS\rangle+|ST_{0}\rangle+|T_{0}S\rangle+|T_{0}T_{0}\rangle\right)$
(27)

or

$|\psi_{\mathrm{e}}\rangle=\frac{i}{2}\left(|SS\rangle+|ST_{0}\rangle+|T_{0}S\rangle-|T_{0}T_{0}\rangle\right),$
(28)

respectively. One can easily notice that the entangling procedure realizes a
CPHASE gate: $-\mathrm{e}^{-iJ_{12}\frac{\tau-3t_{2}}{2}}|SS\rangle\langle
T_{0}T_{0}|-|ST_{0}\rangle\langle T_{0}S|-|T_{0}S\rangle\langle
ST_{0}|-\mathrm{e}^{-iJ_{12}\frac{\tau-3t_{2}}{2}}|T_{0}T_{0}\rangle\langle
SS|$.

Figure 2:  (Color online) The temporal control of the exchange splittings
$J_{1}(t),~{}J_{2}(t)$ during execution of the entangling procedure when the
magnetic field gradients $\Delta B_{z,1}\neq\Delta B_{z,2}$. Thin vertical
blue dashed and red dash-dotted lines are positioned at the middle of the
durations of $\pi$ pulses for qubit 1 and 2, respectively.

##  IV The quantification of two-qubit entanglement

The next step is to quantify the level of entanglement possessed by the two-
qubit state $\left|\tau\right\rangle$ produced at the end of the entangling
procedure. For two-qubit states described by the density operator $\hat{\rho}$
the most commonly used measure of entanglement is the concurrence [74]. It is
defined as

$C(\hat{\rho})=\mathrm{max}\left\\{0,\lambda_{1}-\lambda_{2}-\lambda_{3}-\lambda_{4}\right\\}\,,$
(29)

where
$\lambda_{1}\geqslant\lambda_{2}\geqslant\lambda_{3}\geqslant\lambda_{4}$ are
the square roots of the eigenvalues of the matrix
$\hat{\rho}\hat{\tilde{\rho}}$, where
$\hat{\tilde{\rho}}=(\hat{\sigma}_{y}\otimes\hat{\sigma}_{y})\hat{\rho}^{*}(\hat{\sigma}_{y}\otimes\hat{\sigma}_{y})$.
Here ${\hat{\rho}}^{*}$ denotes the operation of complex conjugation of each
element of $\hat{\rho}$. The concurrence ranges from $C=0$ for separable
states to $C=1$ for maximally entangled states. The concurrence of a pure
state is given by a simple formula:
$C=|\langle\psi|\hat{\sigma}_{y}\otimes\hat{\sigma}_{y}|\psi^{*}\rangle|$,
where $|\psi^{*}\rangle$ is a complex conjugation of $|\psi\rangle$.

For the family of output states $\hat{\rho}(\tau)=|\tau\rangle\langle\tau|$
the concurrence is given by

$C\left(\hat{\rho}(\tau)\right)=\left|\sin\left[\frac{1}{2}J_{12}(\tau-3t_{2})\right]\right|\,.$
(30)

One can make use of an alternative strategy to check to what degree the
resulting state is entangled: high level (greater than 1/2) of fidelity
calculated between the actual state $|\tau\rangle$ and the expected entangled
state $|\psi_{\mathrm{o}}\rangle$ or $|\psi_{\mathrm{e}}\rangle$, defined as
$F(|\tau\rangle,|\psi_{\mathrm{o}}\rangle)=|\langle\tau|\psi_{\mathrm{o}}\rangle|^{2}$,
confirms the entanglement [39]. In general, for mixed states, fidelity is
defined as
$F(\hat{\rho}(\tau),\hat{\rho}_{\mathrm{o}})=\mathrm{Tr}\\{\hat{\rho}(\tau)\hat{\rho}_{\mathrm{o}}\\}$.
If one of the states is pure, as we will be having below, fidelity is given by
$F(\hat{\rho}(\tau),|\psi_{\mathrm{o}}\rangle)=\langle\psi_{\mathrm{o}}|\hat{\rho}(\tau)|\psi_{\mathrm{o}}\rangle$.

In Fig. 3 we show the results of numerical calculation of concurrence of the
final two-qubit state as a function of procedure duration $\tau$ in a model
less idealized than in the previous section. While we are still assuming that
$J_{i}$ and $\Delta B_{z,i}$ do not experience any fluctuations, we now keep
$\Delta B_{z,i}$ fixed during the evolution (as it is the case in experiment),
so that the evolution with finite $J_{i}$ does not amount to phase evolution
in the computational basis of $S/T_{0}$ states of two qubits, thus
necessitating numerical evaluation of the entanglement measure.

Although the impact of always-on $\Delta B_{z,i}$ terms amounts to a small
drop of level of entanglement of the resulting state compared to that from the
idealized scenario described in the previous section (see Fig. 3), it reveals
also another delicate effect: level of entanglement of the resulting state
$|\tau\rangle$ starts to oscillate with frequency of precession of single
qubit states $\omega_{i}=\sqrt{\Delta B_{z,i}^{2}+J_{i}^{2}}$ due to the fact
that in such conditions the qubits’ states rotate around the axis which does
not coincide with $z$ axis exactly, but it is tilted in $xz$ plane because of
presence $\Delta B_{z}$. Characteristically, the oscillations of entanglement
of the state $\hat{\rho}(\tau)$ gradually increase their amplitude with
increasing durations for $0<\tau\lesssim 2\frac{\pi}{J_{12}}$ and reach their
maximum amplitude at $\tau\approx 2\frac{\pi}{J_{12}}$. Then the amplitude of
oscillations decreases for $2\frac{\pi}{J_{12}}\lesssim\tau\lesssim
4\frac{\pi}{J_{12}}$. The observed pattern of fast oscillation of two-qubit
entanglement is periodic in duration $\tau$ of the entangling procedure with a
period of about $4\frac{\pi}{J_{12}}$. Such a pattern of oscillation amplitude
$\tau$-dependence is a consequence of the entangling procedure design: at
$\tau=2\frac{\pi}{J_{12}}$ the ideal resulting state
$|\tau\rangle=-\frac{1}{2}\left(|S\rangle-i|T_{0}\rangle\right)^{\otimes 2}$
is unentangled due to a very specific combination of phases generated before
and after the $\pi$ rotations of qubits’ states. The two-qubit state that is
produced in the middle of the idealized realization of the entangling
procedure (just before the step of $\pi$ rotations) is fully entangled.
However, when the axes around which qubit states precess are tilted from $z$
direction 111 For typical values of $\Delta B_{z,i}\approx 0.12$ $\mu$eV and
$J_{i}\approx 1.2$ $\mu$eV, the angle $\alpha$ between the $z$ axis and the
actual axis of rotations is $\alpha=\arcsin\left(\Delta B_{z,i}/\sqrt{\Delta
B_{z,i}^{2}+J_{i}^{2}}\right)\approx 6^{\circ}$. , the initial superposition
states of qubits do not rotate in the equatorial plane (as intended) but in a
slightly tilted plane. After the $\pi$ rotation around $x$ axes those states
land on the plane which is tilted off from the equatorial plane in the
opposite direction 222 Using a geometrical interpretation of qubit’s state
space, one can notice that $\pi$ rotation about $x$ axis,
$\hat{R}_{x}(\pi)=\mathrm{diag}(1,-1,-1)$, changes the sign of $z$ component
of the qubit’s Bloch vector. Hence, for example, the maximally distant from
the equatorial plane state
$|\tilde{x}_{-}\rangle=\cos\left(\theta/2\right)|S\rangle+\sin\left(\theta/2\right)|T_{0}\rangle$,
where $\theta=2\arccos\left(\sqrt{\Delta B_{z,i}^{2}+J_{i}^{2}-\Delta
B_{z,i}\sqrt{\Delta B_{z,i}^{2}+J_{i}^{2}}}\right.$
$\left./\sqrt{2\left(\Delta
B_{z,i}^{2}+J_{i}^{2}\right)}\right)>\frac{\pi}{2}$, with corresponding Bloch
vector
$\mathbf{r}_{|\tilde{x}_{-}\rangle}=\cos\alpha\,\mathbf{i}-\sin\alpha\,\mathbf{k}=\begin{pmatrix}\cos\alpha\\\
0\\\ -\sin\alpha\end{pmatrix}$, which is obtained from the initial state
$|{-y}\rangle$ after a period $t=\pi/\left(2\sqrt{\Delta
B_{z,i}^{2}+J_{i}^{2}}\right)$ of the free evolution (the rotation by $\pi/2$
about an axis tilted from the $z$ direction with both present $\Delta B_{z,i}$
and $J_{i}$), transforms into
$\hat{R}_{x}(\pi)|\tilde{x}_{-}\rangle=|{\tilde{x}}_{+}\rangle$ with
corresponding Bloch vector
$\mathbf{r}_{|{\tilde{x}}_{+}\rangle}=\cos\alpha\,\mathbf{i}+\sin\alpha\,\mathbf{k}=\begin{pmatrix}\cos\alpha\\\
0\\\ \sin\alpha\end{pmatrix}$, where $\mathbf{i}$, $\mathbf{k}$ are the unit
vectors of $x$, $z$ directions, respectively. The only qubit’s states
generated in the free evolution starting from $|{-y}\rangle$ which stay in the
equatorial plane after application of $\hat{R}_{x}(\pi)$ are states $|{\pm
y}\rangle=\frac{1}{\sqrt{2}}(|S\rangle\pm i|T_{0}\rangle)$ with corresponding
Bloch vectors $\mathbf{r}_{|{\pm y}\rangle}=\pm\mathbf{j}=\begin{pmatrix}0\\\
\pm 1\\\ 0\end{pmatrix}$, where $\mathbf{j}$ is the unit vector of $y$
direction. , and subsequently the phases which qubits’ states acquire during
the second half of the procedure are no longer in a perfect correspondence to
the previously obtained phases and now they do not counterbalance each other,
so the final two-qubit state manifests unexpected entanglement. One can also
notice that in the presence of constant $\Delta B_{z,i}$ the period of slow
entanglement oscillations is slightly longer compared to the idealized case
(see Fig. 3). This fact cannot be illustrated with the help of Bloch sphere as
it originates from a two-qubit interaction, but it is evident from the
numerical diagonalization of the full two-qubit Hamiltonian: when $\Delta
B_{z,i}$ are present, the rate of acquisition of the desired two-qubit phase
is slower than that of the ideal case (CPHASE gate is rotated from the basis
of
$\\{|SS\rangle,~{}|ST_{0}\rangle,~{}|T_{0}S\rangle,~{}|T_{0}T_{0}\rangle\\}$
to the basis of eigenvectors of the full Hamiltonian, and as a result the rate
of two-qubit phase acquisition becomes lower).

Figure 3:  (Color online) Top panel: Concurrence of the two-qubit state
generated in the idealized realization of the entangling procedure described
in Sec. III ($\Delta B_{z,i}$ are on during the qubit rotations only, the
rotations assumed to be perfect) plotted as red dashed line, and the same for
a more realistic procedure considered in Sec. IV ($\Delta B_{z,i}$ are always
on, albeit none of the parameters of the Hamiltonian are fluctuating) plotted
as a solid blue line. The parameters are close to those from the experiment
[39]: $J_{1}\\!=\\!J_{2}\\!=\\!1.2$ $\mu$eV (with corresponding frequency
$J_{i}/h\\!=\\!300$ MHz), $J_{12}\\!=\\!1.29\cdot 10^{-2}$ $\mu$eV, $\Delta
B_{z,1}\\!=\\!\Delta B_{z,2}\\!=0.12$ $\mu$eV. Bottom panel: zoomed in region
of the top panel in the vicinity of $\tau\approx 2\frac{\pi}{J_{12}}$, red
dotted line is the concurrence obtained in the idealized case, blue solid line
is the concurrence obtained in a more realistic case with constant parameters
$J_{i},~{}J_{12}$, $\Delta B_{z,i}$, and green dash-dotted line is the
concurrence obtained in the idealized case but with value of $J_{12}$ from the
latter case.

##  V Entangling procedure in the presence of decoherence

In the experiment [39] it turned out that the two-qubit states obtained as
outcome of the procedure were indeed entangled but only for short durations
$\tau$ of the procedure. Furthermore, the maximal level of entanglement of the
generated states was decreased compared to that of the expected ones. We
consider below possible factors that preclude one from obtaining the maximally
entangled two-qubit states.

###  V.1 Influence of fluctuations of magnetic field gradients on efficiency
of the entangling procedure

The magnetic field gradients between the QDs, which are the sources of finite
$\Delta B_{z,i}$ (with $i=1$, $2$), are often produced by the polarized in an
appropriate way nuclear spins of the atoms from which the sample is built [33,
39]. The spin bath is dynamically polarized before each iteration of the
experiment of entangling of qubits. Due to the slowness of the intrinsic
dynamics of the nuclear spin bath, one does not expect any fluctuation on the
time scale of a single run of the experiment ($\sim 1~{}\mu\mathrm{s}$).
However, possible variations of the values of $\Delta B_{z,i}$ from one run of
the experiment to another is the factor which can preclude from obtaining the
maximally entangled states. This effect is mainly caused by imprecise
rotations of qubits’ states around $x$ axis, which are performed just after
the initializations of the qubits in $|SS\rangle$ state ($\frac{\pi}{2}$
rotations) and in the middle of the entangling procedure ($\pi$ rotations).
Such systematical errors lead to forming unequal superposition states of
$|S\rangle$ and $|T_{0}\rangle$. The influence of quasistatic fluctuations of
$\Delta B_{z,i}$ on the efficiency of the entangling procedure results in a
decrease of the overall efficiency independently of $\tau$, i.e. the
fluctuating quasistatically $\Delta B_{z,i}$ influences in a similar way the
outcomes for all duration $\tau$ of the entangling procedure.

This effect can be easily seen when one considers the idealized realization of
the procedure: for simplicity, let us assume that only the first rotation was
not exactly $\frac{\pi}{2}$ around $x$ axis (errors of rotations
$\hat{R}_{x}(\frac{\pi}{2})$ and $\hat{R}_{x}(\pi)$ will accumulate – there is
no possibility that the next rotation cancels the error of the previous one).
In such a case, the procedure will produce the state
$|\tau(\theta_{1},\theta_{2})\rangle=|SS\rangle(-i)\sin\frac{\theta_{1}}{2}\sin\frac{\theta_{2}}{2}\exp(-i\frac{\tau}{2}J_{12})+|ST_{0}\rangle(-i)\sin\frac{\theta_{1}}{2}\cos\frac{\theta_{2}}{2}+|T_{0}S\rangle(-i)\cos\frac{\theta_{1}}{2}\sin\frac{\theta_{2}}{2}+|T_{0}T_{0}\rangle\cos\frac{\theta_{1}}{2}\cos\frac{\theta_{2}}{2}\exp(-i\frac{\tau}{2})$,
where $\theta_{i}$ is the actual angle of rotation of the state of $i$th qubit
around $x$ axis. This state $|\tau(\theta_{1},\theta_{2})\rangle$ is maximally
entangled when superpositions of $|S\rangle$ and $|T_{0}\rangle$, created in
each qubit from the separable two-qubit state $|SS\rangle$ after
$\hat{R}_{x}(\frac{\pi}{2})$ rotation, are equal (i.e. all components have the
same amplitudes): at $\tau=\pi/J_{12}$ the state
$|\tau(\theta_{1},\theta_{2})\rangle$ should become
$|\psi_{\mathrm{o}}\rangle$, which is maximally entangled, so one can estimate
entanglement of $|\tau(\theta_{1},\theta_{2})\rangle$ by calculating fidelity
$F(|\tau(\theta_{1},\theta_{2})\rangle,|\psi_{\mathrm{o}}\rangle)=|\langle\tau(\theta_{1},\theta_{2})|\psi_{\mathrm{o}}\rangle|^{2}=\frac{1}{4}|\sin\frac{\theta_{1}}{2}\sin\frac{\theta_{2}}{2}+\sin\frac{\theta_{1}}{2}\cos\frac{\theta_{2}}{2}+\cos\frac{\theta_{1}}{2}\sin\frac{\theta_{2}}{2}+\cos\frac{\theta_{1}}{2}\cos\frac{\theta_{2}}{2}|^{2}$,
which has its maximum $F=1$ when $\theta_{1}=\theta_{2}=\frac{\pi}{2}$. Any
deviation of $\theta_{1},\theta_{2}\in[0,\pi]$ will reduce the degree of
entanglement of the state $|\tau(\theta_{1},\theta_{2})\rangle$. On the other
hand, having a pure state $|\tau(\theta_{1},\theta_{2})\rangle$ it is possible
to calculate analytically its concurrence
$C(|\tau(\theta_{1},\theta_{2})\rangle)=|\langle\tau(\theta_{1},\theta_{2})|\hat{\sigma}_{y}\otimes\hat{\sigma}_{y}|\tau^{*}(\theta_{1},\theta_{2})\rangle|=4\sin\frac{\theta_{1}}{2}\cos\frac{\theta_{1}}{2}\sin\frac{\theta_{2}}{2}\cos\frac{\theta_{2}}{2}\sin(\frac{\tau}{2}J_{12})$,
which, when $\tau=\pi/J_{12}$, varies from $C=1$ for
$\theta_{1}=\theta_{2}=\frac{\pi}{2}$ to $C=0$ when
$\theta_{1},\theta_{2}\rightarrow 0\mathrm{~{}or~{}}\pi$. Hence, the impact of
quasistatical fluctuations of magnetic field gradients $\Delta B_{z,i}$
amounts to a loss of the maximal level of produced entanglement (see Fig. 4).
This effect is independent of the duration $\tau$ of the procedure and does
not lead to a complete inability to yield some entaglement when fluctuations
of $\Delta B_{z,i}$ are moderate or small. Another important observation which
comes from Fig. 4 is that the presence of quasistatically fluctuating magnetic
gradients $\Delta B_{z,i}$ during the entire entangling procedure, does not
degrade the efficiency of the procedure by a significant amount while
rotations of qubit states around $x$ axis are accurate (cf. line c, which is
very close to line b, with line d in Fig. 4).

Consequently, in the rest of the paper, where we will focus on influence of
fluctuations of $J_{i}$ that will lead to complete decay of entanglement, we
will neglect all the above-discussed effects of quasistatic fluctuations of
$\Delta B_{z,i}$ and rotation errors caused by $\Delta B_{z,i}$ being finite,
albeit small in comparison to $J_{i}$.

Figure 4:  (Color online) Concurrence of the two-qubit state
$\hat{\rho}(\tau)$ obtained in (a) idealized realization of entangling
procedure (the same as red dashed line in Fig. 3), (b) entangling procedure
with always-on constant $\Delta B_{z,i}$ (the same as blue solid line in Fig.
3), (c) entangling procedure in which magnetic field gradients $\Delta
B_{z,i}$ are constant during a single run of entangling procedure, but
fluctuate quasistatically from one run to another (average over 10,000
realizations, $\Delta B_{z,i}\sim\mathcal{N}(\overline{\Delta
B_{z,i}},\,\sigma_{\Delta B_{z,i}})$, $\sigma_{\Delta
B_{z,i}}=15\%\,\overline{\Delta B_{z,i}}$), here rotations around $x$ axis
artificially kept perfect (i.e. precisely $\frac{\pi}{2}$ at the beginning and
$\pi$ in the middle of the procedure), (d) the same as (c) but with imprecise
rotations around $x$ axis which arise as a result of mismatch of the rotation
time and the actual value of $\Delta B_{z,i}$. All results are obtained under
the assumption that $S$-$T_{0}$ splittings $J_{i}$ do not fluctuate and are
switched on when needed (as shown in Fig. 2). Values of parameters are the
same as in Fig. 3.

###  V.2 Influence of fluctuations of exchange splittings on efficiency of the
entangling procedure

####  V.2.1 Quasistatic fluctuations of exchange splittings

To begin with, we consider the influence of quasistatic fluctuations of
exchange splittings $J_{i}$ on the entanglement generation. The influence of
quasistatically fluctuating exchange splittings $J_{1},~{}J_{2}$ can be
estimated by disregarding off-diagonal terms of the Hamiltonian (but the qubit
rotations involved in the entangling procedure are assumed to be perfect) and
performing the averaging of the density operator Eq. (31) over the
distribution of $J_{1},~{}J_{2}$. We assume that $J_{1},~{}J_{2}$ fluctuate
according to normal distribution with mean values $\bar{J}_{1},~{}\bar{J}_{2}$
and standard deviations $\sigma_{J_{1}},~{}\sigma_{J_{2}}$, respectively.

The idealized entangling procedure generates states which are described by the
following density operator:

$\displaystyle\hat{\rho}(\tau)$
$\displaystyle=|\psi(\tau)\rangle\langle\psi(\tau)|$
$\displaystyle=\frac{1}{4}\begin{pmatrix}1&-i\phi(\tau)&-i\phi(\tau)&-1\\\
i\phi^{*}(\tau)&1&1&-i\phi^{*}(\tau)\\\
i\phi^{*}(\tau)&1&1&-i\phi^{*}(\tau)\\\
-1&i\phi(\tau)&i\phi(\tau)&1\end{pmatrix},$ (31)

where $\phi(\tau)=\exp\left(-i\frac{\tau}{2}J_{12}\right)$. After averaging
over quasistatic fluctuations of $J_{1},~{}J_{2}$ we obtain the density
operator

$\displaystyle\langle\hat{\rho}(\tau)\rangle=\frac{1}{4}\begin{pmatrix}1&-i\langle\phi(\tau)\rangle&-i\langle\phi(\tau)\rangle&-1\\\
i\langle\phi^{*}(\tau)\rangle&1&1&-i\langle\phi^{*}(\tau)\rangle\\\
i\langle\phi^{*}(\tau)\rangle&1&1&-i\langle\phi^{*}(\tau)\rangle\\\
-1&i\langle\phi(\tau)\rangle&i\langle\phi(\tau)\rangle&1\end{pmatrix},$ (32)

where

$\displaystyle\langle\phi(\tau)\rangle$
$\displaystyle=\frac{2K}{\sqrt{4K^{2}+\sigma_{J_{1}}^{2}\sigma_{J_{2}}^{2}\tau^{2}}}\exp\left(-i\frac{4\bar{J}_{1}\bar{J}_{2}K\tau}{8K^{2}+2\sigma_{J_{1}}^{2}\sigma_{J_{2}}^{2}\tau^{2}}\right)$
$\displaystyle\times\exp\left(-\frac{(\bar{J}_{1}^{2}\sigma_{J_{2}}^{2}+\bar{J}_{2}^{2}\sigma_{J_{1}}^{2})\tau^{2}}{8K^{2}+2\sigma_{J_{1}}^{2}\sigma_{J_{2}}^{2}\tau^{2}}\right),$
(33)

where constant $K~{}=~{}{\bar{J}_{1}\bar{J}_{2}}/{\bar{J}_{12}}$, and
$\bar{J}_{12}=\frac{\pi}{\tau_{\mathrm{ent}}}$. In the experiment [39] the
values of parameters were as follows: $\bar{J}_{1}\\!=\\!1.16$ $\mu$eV,
$\bar{J}_{2}\\!=\\!1.32$ $\mu$eV, $\bar{J}_{12}\\!=\\!1.29\cdot 10^{-2}$
$\mu$eV, $\overline{\Delta B_{z,1}}\\!=\\!\overline{\Delta B_{z,2}}\\!=0.12$
$\mu$eV (so $t_{2}=0$ in the Eq. (30)). Note that we are using $J_{12}$ twice
larger than the value reported in Ref. [39]. However, with this value we
obtain the period of oscillations of concurrence in agreement with
experimental data, i.e. the first maximum of entanglement occurs at
$\tau\\!=\\!\pi/J_{12}\\!\approx 160$ ns.

Entanglement of Eq. (32) as a function of duration $\tau$ is shown in the top
panel of Fig. 5 for $J_{i}$ drawn from normal distribution with standard
deviations $\sigma_{i}=15\%\bar{J}_{i}$. Due to quasistatic fluctuations of
$J_{i}$ the overall efficiency of the entangling procedure decreases with
increase of its duration $\tau$. Although the direct impact of the quasistatic
fluctuations of $J_{1}$, $J_{2}$ on the resulting two-qubit state is
completely removed by utilizing simultaneous Hahn echo sequence on each qubit,
the entangling interaction between qubits, which is determined by two-qubit
interaction energy $J_{12}\propto J_{1}J_{2}$, remains sensitive to the
fluctuations, and this causes decay of the entangling procedure efficiency
with increasing duration $\tau$.

####  V.2.2 Dynamical fluctuations of exchange splittings

In the experiment [39] the procedure of entangling two $S$-$T_{0}$ qubits was
based on the SE procedure. While the SE perfectly cancels the impact of
quasistatic single-qubit noises on the end state, in the case of dynamical
fluctuations it helps to refocus the state of the qubits only partially.
Moreover, the two-qubit interaction part of the evolution operator that
describes the entangling procedure (which is responsible for the entanglement
generation) is not affected by the SE procedure, and consequently it is
susceptible to noisy electric fields (leading to noise in $J_{i}$ and
$J_{12}$) in the same way as in FID experiment. As a result, the overall
efficiency of the entangling procedure decays with increasing its duration
$\tau$.

In order to estimate analytically the influence of dynamical fluctuations of
exchange splittings, we approximate the Hamiltonian of the system by its
diagonal neglecting the off-diagonal terms associated with magnetic field
gradients $\Delta B_{z,i}$, which were an order of magnitude smaller than
exchange splittings $J_{i}$ in the experiment [39]:

$\displaystyle\hat{H}_{\mathrm{2q}}\approx\hat{H}_{\mathrm{2q}}^{\mathrm{diag}}$
$\displaystyle=\frac{1}{2}\Big{(}J_{1}(t)\hat{\sigma}_{z}\otimes\mathds{1}+J_{2}(t)\mathds{1}\otimes\hat{\sigma}_{z}$
$\displaystyle+\frac{1}{2}J_{12}(t)(\hat{\sigma}_{z}+\mathds{1})\otimes(\hat{\sigma}_{z}+\mathds{1})\Big{)}.$
(34)

Assuming perfect rotations of qubits’ states, averaged density operator
elements of the resulting two-qubit state after performing the entangling
procedure are

$\displaystyle\left\langle\rho_{ab,cd}(\tau)\right\rangle=\left\langle\langle
ab|\hat{\rho}(\tau)|cd\rangle\right\rangle$ $\displaystyle=\langle\langle
ab|\hat{U}_{\mathrm{SE}}(\tau)\hat{\rho}^{\mathrm{initial}}\hat{U}_{\mathrm{SE}}^{\dagger}(\tau)|cd\rangle\rangle$
$\displaystyle=\langle\langle
ab|\hat{U}_{\mathrm{FID}}\Big{(}\frac{\tau}{2},\tau\Big{)}\Big{(}(-i\sigma_{x})\otimes(-i\sigma_{x})\Big{)}\hat{U}_{\mathrm{FID}}\Big{(}0,\frac{\tau}{2}\Big{)}$
$\displaystyle\times\sum_{kl,mn}\rho_{kl,mn}^{\mathrm{initial}}|kl\rangle\langle
mn|$
$\displaystyle\times\hat{U}_{\mathrm{FID}}^{\dagger}\Big{(}0,\frac{\tau}{2}\Big{)}\Big{(}(i\sigma_{x})\otimes(i\sigma_{x})\Big{)}\hat{U}_{\mathrm{FID}}^{\dagger}\Big{(}\frac{\tau}{2},\tau\Big{)}|cd\rangle\rangle,$
(35)

where evolution operator
$\hat{U}_{\mathrm{FID}}(t_{1},t_{2})=\exp\left({-i\int_{t_{1}}^{t_{2}}\hat{H}_{\mathrm{2q}}(t)\mathrm{d}t}\right)\approx\exp\left({-i\int_{t_{1}}^{t_{2}}\hat{H}_{\mathrm{2q}}^{\mathrm{diag}}(t)\mathrm{d}t}\right)$.

Analyzing the two-qubit system, we consider two distinct possibilities of
dynamical fluctuations: splitting energies $J_{1}$, $J_{2}$ could fluctuate
independently, i.e. $J_{i}(t)=\bar{J}_{i}+\delta J_{i}(t)$, or their
fluctuations may have a common source $J_{i}(t)=\bar{J}_{i}+s_{i}\delta J(t)$,
where $s_{i}\in[0,1]$ is a coupling of $i$th qubit to the noise. Note that
correlations of low-frequency charge noises affecting two quantum dots
separated by $\sim\\!100$ nm distance have been observed in experiments [77,
78]. Correspondingly, the two-qubit coupling in the former case reads

$\displaystyle J_{12}(t)$
$\displaystyle=\frac{J_{1}(t)J_{2}(t)}{K}=\frac{1}{K}\left[\bar{J}_{1}+\delta
J_{1}(t)\right]\left[\bar{J}_{2}+\delta J_{2}(t)\right]$
$\displaystyle\approx\frac{1}{K}\left[\bar{J}_{1}\bar{J}_{2}+\bar{J}_{2}\delta
J_{1}(t)+\bar{J}_{1}\delta J_{2}(t)\right]\,\,,$ (36)

and in the latter case

$\displaystyle J_{12}(t)$
$\displaystyle=\frac{J_{1}(t)J_{2}(t)}{K}=\frac{1}{K}\left[\bar{J}_{1}+s_{1}\delta
J(t)\right]\left[\bar{J}_{2}+s_{2}\delta J(t)\right]$
$\displaystyle\approx\frac{1}{K}\left[\bar{J}_{1}\bar{J}_{2}+\left(s_{1}\bar{J}_{1}+s_{2}\bar{J}_{2}\right)\delta
J(t)\right]\,\,.$ (37)

Note that we neglect here the quadratic in noises terms $\propto\delta
J_{1}(t)\delta J_{2}(t)$ and $\propto(\delta J(t))^{2}$ .

For the case of independent (completely uncorrelated) charge noises that
affect $J_{1}(t)$ and $J_{2}(t)$, the average density operator elements of the
generated state are

$\displaystyle\langle\rho_{ab,cd}(\tau)\rangle=\langle\langle
ab|\hat{\rho}(\tau)|cd\rangle\rangle$
$\displaystyle=\rho_{-a-b,-c-d}^{\mathrm{initial}}\mathrm{e}^{-i\frac{\bar{J}_{1}\bar{J}_{2}}{4K}(ab-
cd)\tau}$
$\displaystyle\times\left\langle\exp\Big{[}-\frac{i}{2}\Big{(}\frac{\bar{J}_{2}}{2K}(ab-
cd)\int_{0}^{\tau}\mathrm{d}t\,\delta J_{1}(t)f_{\mathrm{FID}}(t)\right.$
$\displaystyle\left.+\big{(}c-a+\frac{\bar{J}_{2}}{2K}(c+d-a-b)\big{)}\int_{0}^{\tau}\mathrm{d}t\,\delta
J_{1}(t)f_{\mathrm{SE}}(t)\Big{)}\Big{]}\right\rangle$
$\displaystyle\times\left\langle\exp\Big{[}-\frac{i}{2}\Big{(}\frac{\bar{J}_{1}}{2K}(ab-
cd)\int_{0}^{\tau}\mathrm{d}t\,\delta J_{2}(t)f_{\mathrm{FID}}(t)\right.$
$\displaystyle\left.+\big{(}d-b+\frac{\bar{J}_{1}}{2K}(c+d-a-b)\big{)}\int_{0}^{\tau}\mathrm{d}t\,\delta
J_{2}(t)f_{\mathrm{SE}}(t)\Big{)}\Big{]}\right\rangle$ (38)
$\displaystyle=\rho_{-a-b,-c-d}^{\mathrm{initial}}\mathrm{e}^{-i\frac{\bar{J}_{1}\bar{J}_{2}}{4K}(ab-
cd)\tau}$
$\displaystyle\times\exp\Big{[}-\frac{1}{2}\Big{\\{}\Big{(}\frac{\bar{J}_{2}}{4K}(ab-
cd)\Big{)}^{2}\chi_{\mathrm{FID},J_{1}}(\tau)$
$\displaystyle+\Big{(}\frac{1}{2}\big{(}c-a+\frac{\bar{J}_{2}}{2K}(c+d-a-b)\big{)}\Big{)}^{2}\chi_{\mathrm{SE},J_{1}}(\tau)\Big{\\}}\Big{]}$
$\displaystyle\times\exp\Big{[}-\frac{1}{2}\Big{\\{}\Big{(}\frac{\bar{J}_{1}}{4K}(ab-
cd)\Big{)}^{2}\chi_{\mathrm{FID},J_{2}}(\tau)$
$\displaystyle+\Big{(}\frac{1}{2}\big{(}d-b+\frac{\bar{J}_{1}}{2K}(c+d-a-b)\big{)}\Big{)}^{2}\chi_{\mathrm{SE},J_{2}}(\tau)\Big{\\}}\Big{]},$
(39)

where $a,\,b,\,c,\,d\,\in\,\\{1,-1\\}$ (these parameters code the basis
$\\{|SS\rangle,|ST_{0}\rangle,|T_{0}S\rangle,|T_{0}T_{0}\rangle\\}$ as
$\\{{11},{1-\\!1},{-11},{-1-\\!1}\\}$ in indices of density operator
elements), and
$\chi_{\mathrm{FID},J_{i}}(\tau),~{}\chi_{\mathrm{SE},J_{i}}{(\tau)}$ are the
attenuation factors that describe the influence of dynamical noise of
$J_{i}(t)$ in the case of FID (with constant time domain filter function
$f_{\mathrm{FID}}(t)$) or SE (with time domain filter function with a single
sign inversion $f_{\mathrm{SE}}(t)$). For noise spectrum of the form
$S(\omega)\\!=\\!A/\omega^{\beta}$ they are given by

$\displaystyle\chi_{\mathrm{FID}}(\tau)$
$\displaystyle=\frac{4A}{\pi}\int_{0}^{\infty}\frac{\mathrm{d}\omega}{\omega^{2+\beta}}\sin^{2}\frac{\omega\tau}{2}\,\,,$
(40) $\displaystyle\chi_{\mathrm{SE}}(\tau)$
$\displaystyle=\frac{16A}{\pi}\int_{0}^{\infty}\frac{\mathrm{d}\omega}{\omega^{2+\beta}}\sin^{4}\frac{\omega\tau}{4}\,\,,$
(41)

see Appendix B for the details of derivation and simple analytical
approximations in considered here cases of $\beta\\!=\\!0.7$ and $2$.

Figure 5:  (Color online) Concurrence of the two-qubit state
$\langle\hat{\rho}(\tau)\rangle$ as a function of the duration $\tau$ of the
entanglement generation procedure. Lines are results for the case in which
fluctuations of $J_{i}$ and $J_{12}$ are taken into account, while results in
which the fluctuations of $J_{12}$ were artificially turned off are shown with
open symbols. Top panel: The case of quasistatically fluctuating $J_{1}$,
$J_{2}$ with standard deviations $\sigma_{i}=0.15\bar{J}_{i}$ (Eq. (32)).
Middle and bottom panels: The cases of dynamically fluctuating $J_{1}$,
$J_{2}$, and hence $J_{12}$, due to $1/f^{\beta}$ noise that is uncorrelated
(blue solid line) or perfectly correlated (red dashed lines) for the two
qubits. The power of the noise affecting each qubit was chosen to be such that
ensures the time scale of decay of single-qubit SE signal like in the
experiment [49]: $S$-$T_{0}$ qubit having $J\\!=\\!1.16$ $\mu$eV shows
$T_{\mathrm{SE}}\approx 1.6~{}\mu s$, $S$-$T_{0}$ qubit having $J\\!=\\!1.32$
$\mu$eV shows $T_{\mathrm{SE}}\approx 1.4~{}\mu s$.

Owing to the fact that the approximated Hamiltonian is diagonal, the density
operator undergoes the decoherence of pure dephasing type. There are two
essentially distinct types of off-diagonal elements of two-qubit density
operator. The density operator elements with a single spin flip:
$\langle\rho_{11,1-1}(\tau)\rangle$, $\langle\rho_{11,-11}(\tau)\rangle$,
$\langle\rho_{1-1,-1-1}(\tau)\rangle$, $\langle\rho_{-11,-1-1}(\tau)\rangle$
and their Hermitian conjugated partners diminish mainly due to decrease of
single-qubits’ signals
$\propto\exp\left(-\frac{1}{2}\chi_{\mathrm{SE}}(\tau)\right)$, whereas
elements with two spin flips: $\langle\rho_{11,-1-1}(\tau)\rangle$,
$\langle\rho_{1-1,-11}(\tau)\rangle$ and their Hermitian conjugated partners
decay two times faster as both qubits make their contribution to the decay
$\propto\exp\left(-\chi_{\mathrm{SE}}(\tau)\right)$. Hence, the scale on which
one can expect the generation of entangled state is limited from above by the
single-qubit SE decay time.

In the case of correlated noises, $J_{i}(t)=\bar{J}_{i}+s_{i}\delta J(t)$, the
averaged density operator elements are of the following form:

$\displaystyle\langle\rho_{ab,cd}(\tau)\rangle=\langle\langle
ab|\hat{\rho}(\tau)|cd\rangle\rangle$
$\displaystyle=\rho_{-a-b,-c-d}^{\mathrm{initial}}\mathrm{e}^{-i\frac{\bar{J}_{1}\bar{J}_{2}}{4K}(ab-
cd)\tau}$
$\displaystyle\times\left\langle\exp\Big{[}-\frac{i}{2}\Big{\\{}\frac{s_{1}\bar{J}_{2}+s_{2}\bar{J}_{1}}{2K}(ab-
cd)\right.$ $\displaystyle\times\int_{0}^{\tau}\mathrm{d}t\,\delta
J(t)f_{\mathrm{FID}}(t)$
$\displaystyle+\Big{(}s_{1}\big{(}c-a+\frac{\bar{J}_{2}}{2K}(c+d-a-b)\big{)}$
$\displaystyle+s_{2}\big{(}d-b+\frac{\bar{J}_{1}}{2K}(c+d-a-b)\big{)}\Big{)}$
$\displaystyle\left.\times\int_{0}^{\tau}\mathrm{d}t\,\delta
J(t)f_{\mathrm{SE}}(t)\Big{\\}}\Big{]}\right\rangle$ (42)
$\displaystyle=\rho_{-a-b,-c-d}^{\mathrm{initial}}\mathrm{e}^{-i\frac{\bar{J}_{1}\bar{J}_{2}}{4K}(ab-
cd)\tau}$
$\displaystyle\times\exp\Big{[}-\frac{1}{2}\Big{\\{}\Big{(}\frac{s_{1}\bar{J}_{2}+s_{2}\bar{J}_{1}}{4K}(ab-
cd)\Big{)}^{2}\chi_{\mathrm{FID}}(\tau)$
$\displaystyle+\Bigg{(}\frac{1}{2}\Big{(}s_{1}\big{(}c-a+\frac{\bar{J}_{2}}{2K}(c+d-a-b)\big{)}$
$\displaystyle+s_{2}\big{(}d-b+\frac{\bar{J}_{1}}{2K}(c+d-a-b)\big{)}\Big{)}\Bigg{)}^{2}\chi_{\mathrm{SE}}(\tau)\Big{\\}}\Big{]}.$
(43)

The key qualitative feature of Eqs. (39) and (43) is the presence of terms
proportional to $\chi_{\mathrm{SE}}$, in which the low-frequency noise is
suppressed by the echo procedure, and of terms proportional to
$\chi_{\mathrm{FID}}$, related to fluctuations of interqubit interaction, in
which the low-frequency noise spectrum fully contributes to dephasing.

In Fig. 5 the amount of entanglement is presented in the case of uncorrelated
noises for two exponents characterizing $1/f^{\beta}$ noise, $\beta=2.0$ and
$0.7$ (middle and bottom panels, respectively). As can be seen in the bottom
panel of Fig. 5, for $\beta\\!<\\!1$ the decay of the overall efficiency of
the entangling procedure is mainly caused by influence of fluctuations of
splittings of individual qubits (which also fully determines the decay of the
fidelity of single-qubit coherence). On the other hand, for $\beta>1$, (e.g.
$\beta=2.0$, the middle panel of Fig. 5), the decay of the overall
entanglement generation efficiency is mostly due to the infidelity of the
entangling gate, which is realized by dynamically fluctuating two-qubit term.
This is due to the fact that for noise that is very strongly concentrated at
lowest frequencies, single-qubit noise is very efficiently suppressed by the
echo procedure, and the non-echoed fluctuations of two-qubit interaction,
which are the sources of terms $\propto\chi_{\mathrm{FID}}/K$ in the above
expressions for two-qubit coherences, are dominating the dephasing of the
final state. Thus, the effect of dynamical noise of $J_{i}$ with high value of
$\beta$ on the final state is qualitatively the same as that of quasistatic
fluctuations of splittings $J_{i}$ (see the top panel of Fig. 5)), where
single-qubit terms cancel out perfectly (thanks to applying Hahn echo on each
qubit) and the fidelity of two-qubit entangling gate, which is susceptible to
fluctuations as in FID experiment, is diminishing when the duration $\tau$ of
the procedure becomes longer.

Figure 6:  (Color online) Pauli set for the case of $1/f^{0.7}$ noise. Blue
lines show calculated Pauli set for two independent noises of
$J_{1}(t),J_{2}(t)$, red lines show calculated Pauli set for correlated noises
$J_{1}(t)=J_{2}(t)$. In the top panel for the case of independent noises the
following measures are shown: blue solid line is concurrence of
$\langle\hat{\rho}(\tau)\rangle$, brown dashed line is fidelity
$\langle\psi_{\mathrm{o}}|\langle\hat{\rho}(\tau)\rangle|\psi_{\mathrm{o}}\rangle$
(see Eq. (27)), purple dash-dotted line is fidelity
$\langle\psi_{\mathrm{e}}|\langle\hat{\rho}(\tau)\rangle|\psi_{\mathrm{e}}\rangle$
(see Eq. (28)), turquoise dotted line is fidelity
$\langle\psi(\tau)|\langle\hat{\rho}(\tau)\rangle|\psi(\tau)\rangle$ (see Eq.
(26); and red dotted line is concurrence calculated for correlated noises
$J_{1}(t)=J_{2}(t)$. The power of the noise affecting each qubit was chosen to
be such that ensures the time scale of decay of single-qubit SE signal like in
the experiment [49]: $S$-$T_{0}$ qubit having $J\\!=\\!1.16$ $\mu$eV shows
$T_{\mathrm{SE}}\approx 1.6~{}\mu s$, $S$-$T_{0}$ qubit having $J\\!=\\!1.32$
$\mu$eV shows $T_{\mathrm{SE}}\approx 1.4~{}\mu s$. In the case of fully
correlated noises $J_{1}(t)=J_{2}(t)$, the higher power of the noise has been
chosen for both qubits.

In Fig. 6 the two Pauli sets for two-qubit states created in the entangling
procedure are presented for the case of two uncorrelated $1/f^{0.7}$ noises
(blue lines) and for the case of two fully correlated $1/f^{0.7}$ noises (red
lines). In noise-free experiment, one expects that the only nonzero two-qubit
correlations
$\langle\hat{\sigma}_{i}\otimes\hat{\sigma}_{j}\rangle=\mathrm{Tr}\\{\left(\hat{\sigma}_{i}\otimes\hat{\sigma}_{j}\right)\hat{\rho}(\tau)\\}$
are

$\displaystyle\langle\hat{\sigma}_{x}\otimes\hat{\sigma}_{z}\rangle$
$\displaystyle=\langle\hat{\sigma}_{z}\otimes\hat{\sigma}_{x}\rangle=-\sin
J_{12}\frac{\tau}{2},$ (44)
$\displaystyle\langle\hat{\sigma}_{z}\otimes\mathds{1}\rangle$
$\displaystyle=\langle\mathds{1}\otimes\hat{\sigma}_{y}\rangle=\cos
J_{12}\frac{\tau}{2},$ (45)
$\displaystyle\langle\hat{\sigma}_{y}\otimes\hat{\sigma}_{y}\rangle$
$\displaystyle=1.$ (46)

Dynamical fluctuations of splittings $J_{i}$ destroy these correlations and
diminish their amplitude with increasing duration $\tau$. It is important to
notice that fully correlated noises always lead to decreased but nonzero value
of
$\langle\hat{\sigma}_{y}\otimes\hat{\sigma}_{y}\rangle\stackrel{{\scriptstyle\tau\rightarrow\infty}}{{=}}2\operatorname{\mathrm{Re}}\rho_{1-1,-11}^{\mathrm{initial}}=\frac{1}{2}$,
and at the same time new two-qubit correlation
$\langle\hat{\sigma}_{x}\otimes\hat{\sigma}_{x}\rangle\stackrel{{\scriptstyle\tau\rightarrow\infty}}{{=}}2\operatorname{\mathrm{Re}}\rho_{1-1,-11}^{\mathrm{initial}}=\frac{1}{2}$
is generated. Therefore, the spatial correlations of noises can have a visible
impact on the evolution of resulting state $\hat{\rho}(\tau)$ and components
of Pauli set. Hence, one can make use of this fact to estimate to what degree
the noises were correlated in the experiment. By comparison of experimental
data (Fig. 3 in Ref. [39]) with simulated results (Fig. 6) one may deduce that
in the experiment [39] noises of splittings $J_{1}$ and $J_{2}$ were
uncorrelated.

##  VI Conclusions

We have theoretically analyzed the creation and evolution of entanglement of
two double quantum dot-based $S$-$T_{0}$ qubits measured in Ref. [39] while
taking into account realistic charge and nuclear noise affecting the qubits.
We have confirmed that it is possible to have nearly maximal coherence signal
of a single $S$-$T_{0}$ qubit in the presence of quasistatic fluctuations of
either exchange splitting $J$ or magnetic-field gradient $\Delta B_{z}$ by
performing spin echo procedure on the qubit. Then, we have shown that in the
system of two $S$-$T_{0}$ qubits quasistatic fluctuations of $\Delta B_{z,i}$
lead only to partial decrease of overall efficiency of the entangling
procedure due to imprecise rotations of the qubits’ states.

Both quasistatic or dynamical fluctuations of exchange splittings
$J_{1},~{}J_{2}$, and two-qubit coupling $J_{12}\propto J_{1}J_{2}$ lead to
decay of overall efficiency of the entangling procedure with increasing its
duration $\tau$. The level of correlation of charge noises, as well as their
exact functional form (i.e. value of parameter $\beta$ characterizing the
$1/f^{\beta}$ noise affecting $J$) translates in a distinctive manner into the
shape of decay of two-qubit entanglement as a function of procedure duration
$\tau$. Decay of the overall efficiency of entangling procedure may arise as a
result of the infidelity of single qubit operations (due to dynamical
fluctuations of splittings $J_{1},~{}J_{2}$) or may be caused by infidelity of
entangling gate (due to fluctuations of two-qubit coupling $J_{12}$).

Comparison of experimental data from Ref. [39] with our calculations shows
that the charge noises in the system of two $S$-$T_{0}$ qubit investigated
there were uncorrelated. The main reason of the decreased level of
entanglement of the resulting two-qubit state is infidelity of single-qubit
operations (as we have obtained for $1/f^{\beta}$ noise with $\beta\\!=\\!0.7$
consistent with noise observed in other experiments [49] on samples similar to
those used in Ref. [39]), whereas contribution of the non-ideal two-qubit gate
is negligible in the considered entangling procedure in the regime $\Delta
B_{z,i}\ll J_{i}$. We predict that for $J$ noises of more prominently low-
frequency character (i.e. $1/f^{\beta}$ with $\beta$ closer to $2$ than $1$),
the fluctuations of the two-qubit interactions, which are not echoed by $\pi$
pulses applied to the two qubits separately, will become the main factor
suppressing the maximal entanglement achievable in the considered procedure.
We have also identified qualitative features of long-time behavior of two-
qubit observables from the Pauli set that should be visible when the exchange
splitting noises for the two qubits are correlated.

## Acknowledgements

Authors thank Piotr Szańkowski for helpful discussions. This work was
supported by Polish National Science Centre (NCN), grant no.
DEC-2012/07/B/ST3/03616.

##  Appendix A Components of a Single $S$-$T_{0}$ Qubit as Functions of
Duration $\tau$

In the ideal case, FID signals (i.e. average
$\langle\hat{\sigma}_{i}^{\mathrm{FID}}(\tau)\rangle$ components of
$S$-$T_{0}$ qubit) evolve as follows.

$\displaystyle\langle\hat{\sigma}_{x}^{\mathrm{FID}}(\tau)\rangle=\langle{-y}|\mathrm{e}^{i\hat{H}\tau}\hat{\sigma}_{x}\mathrm{e}^{-i\hat{H}\tau}|{-y}\rangle$
$\displaystyle=\langle{-y}|\Bigg{(}\cos\Big{(}\frac{\tau}{2}\sqrt{\Delta
B_{z}^{2}+J^{2}}\Big{)}\mathds{1}$
$\displaystyle+i\sin\Big{(}\frac{\tau}{2}\sqrt{\Delta
B_{z}^{2}+J^{2}}\Big{)}\frac{\Delta B_{z}}{\sqrt{\Delta
B_{z}^{2}+J^{2}}}\hat{\sigma}_{x}$
$\displaystyle+i\sin\Big{(}\frac{\tau}{2}\sqrt{\Delta
B_{z}^{2}+J^{2}}\Big{)}\frac{J}{\sqrt{\Delta
B_{z}^{2}+J^{2}}}\hat{\sigma}_{z}\Bigg{)}$
$\displaystyle\times\hat{\sigma}_{x}\Bigg{(}\cos\Big{(}\frac{\tau}{2}\sqrt{\Delta
B_{z}^{2}+J^{2}}\Big{)}\mathds{1}$
$\displaystyle-i\sin\Big{(}\frac{\tau}{2}\sqrt{\Delta
B_{z}^{2}+J^{2}}\Big{)}\frac{\Delta B_{z}}{\sqrt{\Delta
B_{z}^{2}+J^{2}}}\hat{\sigma}_{x}$
$\displaystyle-i\sin\Big{(}\frac{\tau}{2}\sqrt{\Delta
B_{z}^{2}+J^{2}}\Big{)}\frac{J}{\sqrt{\Delta
B_{z}^{2}+J^{2}}}\hat{\sigma}_{z}\Bigg{)}|{-y}\rangle$
$\displaystyle=\frac{J}{\sqrt{\Delta B_{z}^{2}+J^{2}}}\sin\Big{(}\sqrt{\Delta
B_{z}^{2}+J^{2}}\tau\Big{)}.$ (47)

$\displaystyle\langle\hat{\sigma}_{y}^{\mathrm{FID}}(\tau)\rangle$
$\displaystyle=-\cos\Big{[}\sqrt{\Delta B_{z}^{2}+J^{2}}\tau\Big{]}$ (48)
$\displaystyle\approx-\cos\Big{[}\left(J+\frac{\Delta
B_{z}^{2}}{2J}\right)\tau\Big{]},$ (49)

the approximation is good for $\tau\ll\frac{8J^{3}}{\Delta B_{z}^{4}}$.

$\langle\hat{\sigma}_{z}^{\mathrm{FID}}(\tau)\rangle=-\frac{\Delta
B_{z}}{\sqrt{\Delta B_{z}^{2}+J^{2}}}\sin\Big{(}\sqrt{\Delta
B_{z}^{2}+J^{2}}\tau\Big{)}.$ (50)

In the ideal case, SE signals (i.e. average
$\langle\hat{\sigma}_{i}^{\mathrm{SE}}(\tau)\rangle$ components of $S$-$T_{0}$
qubit) evolve as follows.

$\displaystyle\langle\hat{\sigma}_{x}^{\mathrm{SE}}(\tau)\rangle$
$\displaystyle=\langle{-y}|\mathrm{e}^{i\hat{H}\frac{\tau}{2}}(i\hat{\sigma}_{x})\mathrm{e}^{i\hat{H}\frac{\tau}{2}}\hat{\sigma}_{x}$
$\displaystyle\times\mathrm{e}^{-i\hat{H}\frac{\tau}{2}}(-i\hat{\sigma}_{x})\mathrm{e}^{-i\hat{H}\frac{\tau}{2}}|{-y}\rangle$
$\displaystyle=\frac{8\Delta B_{z}^{2}J}{(\Delta
B_{z}^{2}+J^{2})^{3/2}}\cos\Big{(}\frac{1}{4}\sqrt{\Delta
B_{z}^{2}+J^{2}}\tau\Big{)}$
$\displaystyle\times\sin^{2}\Big{(}\frac{1}{4}\sqrt{\Delta
B_{z}^{2}+J^{2}}\tau\Big{)}.$ (51)

$\displaystyle\langle\hat{\sigma}_{y}^{\mathrm{SE}}(\tau)\rangle$
$\displaystyle=\frac{1}{\Delta B_{z}^{2}+J^{2}}$
$\displaystyle\times\Big{(}J^{2}+\Delta B_{z}^{2}\cos\Big{(}\sqrt{\Delta
B_{z}^{2}+J^{2}}\tau\Big{)}\Big{)}.$ (52)

$\displaystyle\langle\hat{\sigma}_{z}^{\mathrm{SE}}(\tau)\rangle$
$\displaystyle=\frac{\Delta B_{z}}{(\Delta B_{z}^{2}+J^{2})^{3/2}}$
$\displaystyle\times\left[2J^{2}\sin^{2}\left(\frac{1}{2}\sqrt{\Delta
B_{z}^{2}+J^{2}}\tau\right)\right.$ $\displaystyle+\left.\Delta
B_{z}^{2}\sin\left(\sqrt{\Delta B_{z}^{2}+J^{2}}\tau\right)\right].$ (53)

Assuming that $\bar{J}\gg\overline{\Delta B_{z}},\,\sigma_{J}$, after
averaging over quasistatic fluctuations of the parameter $\Delta B_{z}$ or $J$
one obtains following approximate expressions for $y$ qubit component during
SE.

$\displaystyle\langle\langle\hat{\sigma}_{y}^{\mathrm{SE}}(\tau)\rangle\rangle_{\Delta
B_{z}}$ $\displaystyle\approx\frac{J^{2}}{J^{2}+\overline{\Delta B_{z}}^{2}}$
$\displaystyle+\frac{1}{J^{2}+\overline{\Delta
B_{z}}^{2}}\exp\left(-\frac{\overline{\Delta B_{z}}^{2}\sigma_{\Delta
B_{z}}^{2}\tau^{2}}{2(J^{2}+(\sigma_{\Delta B_{z}}^{2}\tau)^{2})}\right)$
$\displaystyle\times\Bigg{[}\frac{J^{3/2}(J(\overline{\Delta
B_{z}}^{2}+\sigma_{\Delta B_{z}}^{2})+i\sigma_{\Delta
B_{z}}^{4}\tau)}{2(J+i\sigma_{\Delta B_{z}}^{2}\tau)^{5/2}}\Bigg{.}$
$\displaystyle\times\exp\left(-iJ\tau\frac{\overline{\Delta
B_{z}}^{2}+2J^{2}+2(\sigma_{\Delta
B_{z}}^{2}\tau)^{2}}{2(J^{2}+(\sigma_{\Delta B_{z}}^{2}\tau)^{2})}\right)$
$\displaystyle\Bigg{.}+\,\mathrm{c.c.}\Bigg{]}.$ (54)

$\displaystyle\langle\langle\hat{\sigma}_{y}^{\mathrm{SE}}(\tau)\rangle\rangle_{J}$
$\displaystyle\approx\frac{\bar{J}^{2}+\sigma_{J}^{2}}{\bar{J}^{2}+\Delta
B_{z}^{2}}$ $\displaystyle+\frac{\Delta B_{z}^{2}}{\bar{J}^{2}+\Delta
B_{z}^{2}}\exp\left(-\frac{\sigma_{J}^{2}\tau^{2}}{2}\right)\cos\left(\bar{J}\tau\right).$
(55)

##  Appendix B $S$-$T_{0}$ Qubit Attenuation Factors Derived for Dynamically
Fluctuating Exchange Splitting

We present the calculation of the attenuation factors
$\chi_{\mathrm{FID}}(\tau)$ and $\chi_{\mathrm{SE}}(\tau)$ that account for
the effect of dynamically fluctuating exchange splitting $J$. The attenuation
factor $\chi_{\mathrm{FID}}(\tau)$ is calculated as follows.

$\displaystyle\chi_{\mathrm{FID}}(\tau)$
$\displaystyle=\int_{0}^{\tau}\mathrm{d}t_{1}\int_{0}^{\tau}\mathrm{d}t_{2}\langle\delta
J(t_{1})\delta J(t_{2})\rangle f_{\mathrm{FID}}(t_{1})f_{\mathrm{FID}}(t_{2})$
$\displaystyle=\int_{-\infty}^{\infty}\mathrm{d}t_{1}\int_{-\infty}^{\infty}\mathrm{d}t_{2}\int_{-\infty}^{\infty}\frac{\mathrm{d}\omega}{2\pi}\mathrm{e}^{-i\omega(t_{1}-t_{2})}S(\omega)$
$\displaystyle\times\int_{-\infty}^{\infty}\frac{\mathrm{d}\omega_{1}}{2\pi}\tilde{f}_{\mathrm{FID}}(\omega_{1})\mathrm{e}^{-i\omega_{1}t_{1}}\int_{-\infty}^{\infty}\frac{\mathrm{d}\omega_{2}}{2\pi}\tilde{f}_{\mathrm{FID}}(\omega_{2})\mathrm{e}^{-i\omega_{2}t_{2}}$
$\displaystyle=\int_{-\infty}^{\infty}\mathrm{d}\omega_{1}\int_{-\infty}^{\infty}\mathrm{d}\omega_{2}\int_{-\infty}^{\infty}\frac{\mathrm{d}\omega}{2\pi}\delta(\omega+\omega_{1})\delta(\omega-\omega_{2})$
$\displaystyle\times\tilde{f}_{\mathrm{FID}}(\omega_{1})\tilde{f}_{\mathrm{FID}}(\omega_{2})S(\omega)$
$\displaystyle=\int_{-\infty}^{\infty}\frac{\mathrm{d}\omega}{2\pi}\tilde{f}_{\mathrm{FID}}(-\omega)\tilde{f}_{\mathrm{FID}}(\omega)S(\omega)=\int_{-\infty}^{\infty}\frac{\mathrm{d}\omega}{2\pi}\left|\tilde{f}_{\mathrm{FID}}(\omega)\right|^{2}S(\omega)$
$\displaystyle=\int_{-\infty}^{\infty}\frac{\mathrm{d}\omega}{2\pi}\frac{2F_{\mathrm{FID}}(\omega\tau)}{\omega^{2}}S(\omega)=2\int_{0}^{\infty}\frac{\mathrm{d}\omega}{\pi}\frac{F_{\mathrm{FID}}(\omega\tau)}{\omega^{2}}S(\omega).$
(56)

Using the explicit analytical expressions for the filter function
$F_{\mathrm{FID}}(\omega\tau)=2\sin^{2}\frac{\omega\tau}{2}$ and the spectral
density of the noise $S(\omega)=\frac{A}{\omega^{\beta}}$, where $A$ is a
constant corresponding to the power of the noise, we obtain the following
expression for the attenuation factor

$\chi_{\mathrm{FID}}(\tau)=\frac{4A}{\pi}\int_{0}^{\infty}\mathrm{d}\omega\sin^{2}\frac{\omega\tau}{2}\frac{1}{\omega^{2+\beta}},$
(57)

which for $\beta=0.7$ can be evaluated giving finally:

$\chi_{\mathrm{FID}}(\tau)=\frac{2A}{\pi}\cos\left(\frac{3\pi}{20}\right)\Gamma\left(-1.7\right)\tau^{1.7}\approx
2.24\,\frac{2A}{\pi}\tau^{1.7},$ (58)

as the definite integral can be calculated analytically

$\int_{0}^{\infty}\mathrm{d}\omega\sin^{2}\frac{\omega\tau}{2}\frac{1}{\omega^{2+0.7}}=\frac{1}{2}\cos\left(\frac{3\pi}{20}\right)\Gamma\left(-1.7\right)\tau^{1.7}.$
(59)

Similarly, the attenuation factor $\chi_{\mathrm{SE}}(\tau)$ reads as follows.

$\displaystyle\chi_{\mathrm{SE}}(\tau)$
$\displaystyle=\int_{0}^{\tau}\mathrm{d}t_{1}\int_{0}^{\tau}\mathrm{d}t_{2}\langle\delta
J(t_{1})\delta J(t_{2})\rangle f_{\mathrm{SE}}(t_{1})f_{\mathrm{SE}}(t_{2})$
$\displaystyle=2\int_{0}^{\infty}\frac{\mathrm{d}\omega}{\pi}\frac{F_{\mathrm{SE}}(\omega\tau)}{\omega^{2}}S(\omega).$
(60)

Using the explicit analytical expressions for the filter function
$F_{\mathrm{SE}}(\omega\tau)=8\sin^{4}\frac{\omega\tau}{4}$ and the spectral
density of the noise $S(\omega)=\frac{A}{\omega^{\beta}}$, we obtain the
following expression for the attenuation factor

$\chi_{\mathrm{SE}}(\tau)=\frac{16A}{\pi}\int_{0}^{\infty}\mathrm{d}\omega\sin^{4}\frac{\omega\tau}{4}\frac{1}{\omega^{2+\beta}},$
(61)

which for $\beta=0.7$ can be evaluated giving finally:

$\chi_{\mathrm{SE}}(\tau)=\frac{2A}{\pi}\left(2^{0.3}-1\right)\cos\left(\frac{3\pi}{20}\right)\Gamma\left(-1.7\right)\tau^{1.7}\approx
0.52\,\frac{2A}{\pi}\tau^{1.7},$ (62)

as the definite integral can be calculated analytically

$\int_{0}^{\infty}\mathrm{d}\omega\sin^{4}\frac{\omega\tau}{4}\frac{1}{\omega^{2+0.7}}=\frac{1}{8}\left(2^{0.3}-1\right)\cos\left(\frac{3\pi}{20}\right)\Gamma\left(-1.7\right)\tau^{1.7}.$
(63)

The attenuation factor $\chi_{\mathrm{SE}}(\tau)$ for noise with $\beta=2.0$
is $\chi_{\mathrm{SE}}(\tau)=\frac{A}{12}\tau^{3}$ as the following definite
integral can be calculated analytically
$\int_{0}^{\infty}\mathrm{d}\omega\sin^{4}\frac{\omega\tau}{4}\frac{1}{\omega^{2+2.0}}=\frac{\pi}{192}\tau^{3}$.

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  * Note [1] For typical values of $\Delta B_{z,i}\approx 0.12$ $\mu$eV and $J_{i}\approx 1.2$ $\mu$eV, the angle $\alpha$ between the $z$ axis and the actual axis of rotations is $\alpha=\arcsin\left(\Delta B_{z,i}/\sqrt{\Delta B_{z,i}^{2}+J_{i}^{2}}\right)\approx 6^{\circ}$.
  * Note [2] Using a geometrical interpretation of qubit’s state space, one can notice that $\pi$ rotation about $x$ axis, $\hat{R}_{x}(\pi)=\mathrm{diag}(1,-1,-1)$, changes the sign of $z$ component of the qubit’s Bloch vector. Hence, for example, the maximally distant from the equatorial plane state $|\tilde{x}_{-}\rangle=\cos\left(\theta/2\right)|S\rangle+\sin\left(\theta/2\right)|T_{0}\rangle$, where $\theta=2\arccos\left(\sqrt{\Delta B_{z,i}^{2}+J_{i}^{2}-\Delta B_{z,i}\sqrt{\Delta B_{z,i}^{2}+J_{i}^{2}}}\right.$ $\left./\sqrt{2\left(\Delta B_{z,i}^{2}+J_{i}^{2}\right)}\right)>\frac{\pi}{2}$, with corresponding Bloch vector $\mathbf{r}_{|\tilde{x}_{-}\rangle}=\cos\alpha\,\mathbf{i}-\sin\alpha\,\mathbf{k}=\begin{pmatrix}\cos\alpha\\\ 0\\\ -\sin\alpha\end{pmatrix}$, which is obtained from the initial state $|{-y}\rangle$ after a period $t=\pi/\left(2\sqrt{\Delta B_{z,i}^{2}+J_{i}^{2}}\right)$ of the free evolution (the rotation by $\pi/2$ about an axis tilted from the $z$ direction with both present $\Delta B_{z,i}$ and $J_{i}$), transforms into $\hat{R}_{x}(\pi)|\tilde{x}_{-}\rangle=|{\tilde{x}}_{+}\rangle$ with corresponding Bloch vector $\mathbf{r}_{|{\tilde{x}}_{+}\rangle}=\cos\alpha\,\mathbf{i}+\sin\alpha\,\mathbf{k}=\begin{pmatrix}\cos\alpha\\\ 0\\\ \sin\alpha\end{pmatrix}$, where $\mathbf{i}$, $\mathbf{k}$ are the unit vectors of $x$, $z$ directions, respectively. The only qubit’s states generated in the free evolution starting from $|{-y}\rangle$ which stay in the equatorial plane after application of $\hat{R}_{x}(\pi)$ are states $|{\pm y}\rangle=\frac{1}{\sqrt{2}}(|S\rangle\pm i|T_{0}\rangle)$ with corresponding Bloch vectors $\mathbf{r}_{|{\pm y}\rangle}=\pm\mathbf{j}=\begin{pmatrix}0\\\ \pm 1\\\ 0\end{pmatrix}$, where $\mathbf{j}$ is the unit vector of $y$ direction.
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