Most quantum information optimisation problems involve information encoded in quantum systems, either by construction in an experiment involving quantum systems themselves, or via encoding exogenous or classical information into quantum systems (such as qubits) in order to leverage the benefits of quantum computation. Both approaches involve the input into quantum states in a process known as state preparation. The way in which data is encoded in quantum systems affects the performance and expressiveness many quantum algorithms [Schuld 2021] . Information is usually encoded using one of four standard encoding methods including <cit.>: (a) basis encoding, (b) amplitude encoding, (c) qsample encoding and (d) dynamic encoding. The first of these, basis encoding, is a technique that encodes classical information into quantum basis states. Usually, such procedures involve transformation of data into classical binary bit-strings $(x_1,...,x_d), b_i \in \{0,1\}$ then mapping each bit string to the quantum basis state of a set of qubits of a composite system. For example, for $x \in \R^N$, say a set of decimals, is converted into a $d$-dimensional bit string (e.g. $0.1\to 00001..., -0.6 \to 11001..$) suitably normalised such that $x = \sum_k^d (1/2^k)x_k$. The sequence $x$ is given a representation via $\ketpsi = \ket{000001\,11001}$ (see <cit.>). Amplitude encoding associates normalised classical information e.g. for an $n$-qubit system (with $2^n$ different possible (basis) states $\ket{j}$), a normalised classical sequence $x\in \C^{2^n}, \sum_k |x_k|^2=1$ (possibly with only real parts) with quantum amplitudes $x=(x_1,...,x_{2^n})$ can be encoded as $\ket{\psi_x}=\sum_j^{2^n} x_j \ket{j}$. Other examples of sample-based encoding (e.g. Qsample and dynamic encoding are also relevant but not addressed here). From a classical machine learning perspective, such encoding regimes also enable both features and labels to be encoded into quantum systems.

§.§ QDataset evolution, noise and measruement

§.§.§ Evolution, Hamiltonians and control

The QDataSet is constructed on the basis of a typical quantum control <cit.> problem, where it is assumed there exist a set of controls (such as pulses or voltages) with which the quantum system may be controlled or steered towards a target state (or unitary). In density matrix formalism, the Schrödinger equation is:.

\begin{align}
i\hbar \frac{d \rho}{dt} = [H(t), \rho(t)],
\end{align}

where the commutator $[A,B] = AB-BA$, $\hbar$ is Planck's constant (we can always choose the units such that $\hbar=1$ which will be the convention in this paper), and $H(t)$ is a Hermitian operator called the Hamiltonian. In physical systems, it corresponds to the total energy (sum of kinetic and potential energies) of the system under consideration. For a closed system (i.e. a noiseless isolated system with no interaction with the surrounding environment), it can be expressed in the general form

\begin{align}
H(t) = H_0(t) \doteq H_d(t) + H_{\text{ctrl}}(t).
\end{align}

$H_d(t)$ is called the drifting Hamiltonian and corresponds to the natural evolution of the system in the absence of any control. The second term $H_{\text{ctrl}}(t)$ is called the control Hamiltonian and corresponds to the controlled external forces we apply to the system (such as electromagnetic pulses applied to an atom, or a magnetic field applied to an electron).
The solution of the evolution equation at time $t=T$ is given by:

\begin{align}
\rho(T) = U(T) \rho(0) U^{\dagger}(T),
\end{align}

where $\rho(0)$ is the initial state of the system, the unitary evolution matrix $U(t)$ is given by:

\begin{align}
U(T) &= \mathcal{T}_{+} e^{-i\int_0^T H(t) dt}\\
&\doteq \lim_{\Delta t\to 0} e^{-i H(T) \Delta t} \cdots e^{-i H(3\Delta t) \Delta t} e^{-i H(2\Delta t) \Delta t} e^{-i H(\Delta t) \Delta t}
\end{align}

The symbol $\mathcal{T}_+$ is called a time-ordering operator.  The time-ordering is needed because in general the Hamiltonian is time-dependent and does not commute at different time instants (i.e. $[H(t_i), H(t_j)]\neq 0$). The second line is a time-independent approximation of the time-dependent form based on a Suzuki-Trotter decomposition <cit.>:

\begin{align}
e^{A + B} = \lim_{n\to \infty}\left( e^{A/n}e^{B/n}\right)^n
\end{align}

which under certain conditions allows the time-varying Hamiltonian to be approximated by a piece-wise constant Hamiltonian. In this case where the time interval $[0,T]$ is divided into equal segments of length $\Delta t$. As a special case if the Hamiltonian commute at different time instants then the evolution matrix can be simplified to:

\begin{align}
U(T) &= e^{-i\int_0^T H(t) dt},
\end{align}

which also reduces to:

\begin{align}
U(T) &= e^{-i H(T) T},
\end{align}

in the case of a time-independent Hamiltonian.

On the other hand when the system is open (i.e. uncontrollable interactions with the environment), then the Hamiltonian takes the form:

\begin{align}
H(t) = H_0(t) + H_1(t) \doteq H_d(t) + H_{\text{ctrl}}(t) + H_{SE}(t) + H_E(t).
\end{align}

$H_0(t)$ is defined as before. The new term $H_1(t)$ now consists of two terms: $H_{SE}(t)$ represents an interaction term with the environment, while $H_E(t)$ represents the free evolution of the environment in the absence of the system. In this case, the  Hamiltonian controls the dynamics of both the system and environment combined in a highly non-trivial way. In other words, the state becomes the joint state between the system and environment. The combined system and environment then become closed. Modelling such open quantum systems is complex and challenging and is typically undertaken using a variety of stochastic master equations <cit.> or sophisticated noise spectroscopy. As detailed below, the QDataSet contains a variety of noise realisations for one and two qubit systems together with details of a recent novel operator <cit.> for characterising noise in quantum systems.

§.§.§ Hamiltonians: drift, control, noise

The QDataSet comprises datasets for 1- and 2- qubits systems evolving in the presence and absence of noise. The canonical forms of Hamiltonian in the QDataSet are that given in <cit.>. In that work, a limited set of single-qubit systems subject to external environmental noise (baths) was used as input training data for a novel greybox machine learning alternative method for characterising noise (`beyond' conventional spectroscopic techniques). In this paper, the underlying simulation was modified to generate a greater variety of qubit-noise examples for the single qubit case. The simulation was then extended beyond that in <cit.> to generate examples for the two-qubit case (in the presence or absence of noise). As discussed above, the evolution of closed and open quantum systems is described by Hamiltonian dynamics, which encode time-dependent functions into operators which are the generators of time-translations (operators) acting on quantum states. The general form of the Hamiltonian for the QDataSet is given by:

\begin{align}
H(t)&= H_{\text{drift}}(t) + H_{\text{ctrl}}(t) + H_1(t)
\end{align}

The Hamiltonian comprises three elements: (i) a drift Hamiltonian $H_d(t)$, encoding the endogenous evolution of the quantum system; (ii)  a control Hamiltonian $H_{\text{ctrl}}(t)$, encoding evolution due to the application of classical controls which may be applied to the quantum system to steer its evolution; and (iii) and an interaction (noise) Hamiltonian $H_1(t)$, encoding the effects of coupling of the quantum system to its environment, such as a decohering noise source (a bath). We express the Hamiltonians in the Pauli operator basis which forms a complete basis for our one- and two-qubit systems. Our control functions are represneted as $f_\alpha(t)$ for $\alpha = x,y,z$ where the subscript indicates which generator the control applies to. Concretely, for example, $\sigma_z$ control is denoted $f_z(t) \sigma_z$. In general, continuous time-dependent control formulations are difficult - or infeasible - to solve analytically, where solving here means finding a suitable representation of the control unitary:

\begin{align}
U_{\text{ctrl}} = \mathcal{T}_{+} e^{-i\int_0^T f_\alpha \sigma_\alpha/2 dt}
\end{align}

The control functions act on the The simplest control functional form is fixed amplitude control <cit.> or what is also described as a square pulse, where a constant energy (expressed as amplitudes) is expressed for a discrete time-step $\Delta t$. Most controls are usually [classical] i.e. $f_\alpha(t) \in \R$. Other control waveforms include Gaussian pulses which are characterised by [insert]. Moreover, quantum control in the QDataSet context has two primary imperatives. The first is the use of control in closed noise-free idealised quantum systems where the objective is the use of controls to steer the quantum system to some desired objective state. This is equivalent to the synthesis of quantum circuits (sequences of quantum gates) from the identity $I$ to a target unitary $U_T$. The second is the use of controls in the presence of noise, where the quantum system is coupled to an environment that potentially decoheres the system. In this second case, ideally the controls are tailored to neutralise the effect of noise on the evolution of the quantum system - a process typically described by dynamic decoupling <cit.> (see for example Hahn echo or other examples). Crafting suitable controls to mitigate noise effects is challenging. One must properly time and select appropriate amplitudes for the application of controls to counteract decohering interference. Typically, it requires information about the noise spectrum itself, obtained using techniques from quantum noise spectroscopy <cit.>. It also requires an understanding of how control and noise signals convolve in the context of quantum evolution (see [ref]). Dealing with noise is a central imperative of quantum information processing and the engineering of quantum systems where the aim is to preserve and extend coherence times of quantum information). The simulations in the QDataSet are based upon an alternative technique for quantum control in the presence of a variety of noise <cit.>, where a greybox neural network is used to learn only those characteristics of the noise spectrum relevant to the application of controls (a comparatively simpler problem than seeking to determine the entire spectrum). For machine learning practitioners, the QDataSet thus provides a useful way to seek to apply advanced classical machine learning techniques to the challenging but important problem.

§.§.§ One- and two-qubit Hamiltonians

We begin by describing the single-qubit Hamiltonian and then move to an exposition of the two-qubit case. For the single-qubit system, the drifting Hamiltonian is fixed in the form:

\begin{align}
H_d(t) = H_d =  \frac{1}{2} \Omega \sigma_z.
\end{align}

The $\Omega$ term represents the energy gap of the quantum system (the difference in energy between, for example, the ground and excited state of the qubit, recalling qubits are characterised by having two distinct quantum states). The single-qubit drift Hamiltonian for the QDataSet is time-independent for simplicity, though in realistic cases it will contain a time-dependent component. For the single-qubit control and noise Hamiltonians we have two cases based upon the concept of which axes controls and noise are applied. Recall we can represent a single qubit system on a Bloch sphere, with axes corresponding to the expectations of each Pauli operator and where operations of each Pauli operator constitute rotations about the respective axis. Our controls are control functions, mostly time-dependent, that apply to each Pauli operator (generator). They act to affect the amplitude over time of rotations about the respective Pauli axes. More detailed treatments of noise in quantum systems and quantum control contexts can be found in <cit.>.

As discussed above, the functional form of the control functions $f_\alpha(t)$ varies. We select both square pulses and Gaussian pulses as the form (see below). Each different noise function $B_\alpha(t)$ is parameterised differently depending on various assumptions that are more specifically detailed in <cit.> and summarised below.  Noise and control functions are applied to different qubit axes in the single-qubit and two-qubit cases. For a single qubit, we first have single-axis control along $x$-direction:

\begin{align}
H_{\text{ctrl}}(t) &= \frac{1}{2} f_x(t) \sigma_x
\end{align}

with the noise (interaction) Hamiltonian $H_1(t)$ along $z$-direction (the quantification axis):

\begin{align}
H_1(t) &= \frac{1}{2} \beta_z(t) \sigma_z
\end{align}

Here the function $\beta_z(t)$ (a classical noise function $\beta(t)$ applied along the $z$-axis) may take a variety of forms depending on how the noise was generated (see below for a discussion of noise profiles e.g. N1-N6). It should be noted (for researchers unfamiliar with noise) noise rarely has a functional form and is itself difficult to characterise (so $\beta(t)$ should not be thought of as a simple function). For the second case, we implement multi-axis control along $x$- and $y$- directions and noise along $x$- and $z$-directions in the form:

\begin{align}
H_{\text{ctrl}}(t) &= \frac{1}{2} f_x(t) \sigma_x + \frac{1}{2} f_y(t) \sigma_y \\
H_1(t) &= \frac{1}{2} \beta_x(t) \sigma_x + \frac{1}{2} \beta_z(t) \sigma_z.
\end{align}

Noiseless evolution may be recovered by choosing $H_1(t)=0$.

For the 2-qubit system, we chose the drifting Hamiltonian in the form:

\begin{align}
H_d(t) = \frac{1}{2} \Omega_1 \sigma_z \otimes \sigma_0 + \frac{1}{2} \Omega_2 \sigma_0 \otimes \sigma_z.
\end{align}

For the control Hamiltonians, we also have two cases. The first one is local control along the $x$-axis of each individual qubit, akin to the single-qubit case each. In the notation, $f_{1\alpha}(t)$ indicates that the control function is applied to, in this case, the second qubit, while the first qubit remains unaffected (denoted by the `1' in the subscript and by the identity operator $\sigma_0$). We also introduce an interacting control. This is a control that acts simultaneously on the $x$-axis of each qubit, denoted by $f_{xx}(t)$:

\begin{align}
H_{\text{ctrl}}(t) = \frac{1}{2} f_{x1}(t) \sigma_x \otimes \sigma_0 + \frac{1}{2} f_{1x} \sigma_0 \otimes \sigma_x + f_{xx}(t) \sigma_x \otimes \sigma_x.
\end{align}

The second two-qubit case is for local-control along the $x-$axis of each qubit only and is represented as:

\begin{align}
H_{\text{ctrl}}(t) = \frac{1}{2} f_{x1}(t) \sigma_x \otimes \sigma_0 + \frac{1}{2} f_{1x} \sigma_0 \otimes \sigma_x,
\end{align}

For the noise, we fix the Hamiltonian to be along the $z$-axis of both qubits, in the form:

\begin{align}
H_{1}(t) &= \frac{1}{2} \beta_{z1}(t) \sigma_z \otimes \sigma_0 + \frac{1}{2} \beta_{1z} \sigma_0 \otimes \sigma_z.
\end{align}

Notice, that for the case of local-only control and noiseless evolution, this will correspond to two completely-independent qubits and thus we do not include this case, as it is already covered by the single-qubit datasets.

To summarise, the QDataSet includes four categories for the datasets set-out in Table <ref>. The first two categories are for 1-qubit systems, the first is single axis control and noise, while the second is multi-axis control and noise. The third and fourth categories are 2-qubit systems with local-only control or with an additional interacting control together with noise.

Category      Qubits     Drift     Control     Noise

1     1     $(z)$     $(x)$     $(z)$

2     1     $(z)$     $(x,y)$     $(x,z)$

3     2     $(z1, 1z)$     $(x1, 1x)$     $(z1, 1z)$

4     2     $(z1, 1z)$     $(x1, 1x, xx)$     $(z1, 1z)$

The general categorization of the provided datasets.

§.§.§ Control

The control pulse sequences in the QDataSet consist of two types of waveforms. The first is a train of Gaussian pulses, and the other is a train of square pulses, both of which are very common in actual experiments. Square pulses are the simplest waveforms, consisting of a constant amplitude $A_k$ applied for a specific time interval $\Delta t_k$:

\begin{align}
f(\Delta t_k) = f = A_k \Delta t_k
\end{align}

where $k$ runs over the total number of time-steps in the sequence. The three parameters of such square pulses are the amplitude $A_k$, the position in the sequence $k$ and the time duration over which the pulse is applied $\Delta t$. In the QDataSet, the pulse parameters are stored in a sequence of vectors $\{a_n \}$. Each vector $a_n$ is of dimension $r$ parameters of each pulse (e.g. the Gaussian pulse vectors store the amplitude, mean and variance, the square pulse vectors store pulse position, time interval and amplitude), enabling reconstruction of each pulse from those parameters if desired. For simplicity, we assume constant time intervals such that $\Delta t_k = \Delta t$. The Gaussian waveform can be expressed as:

\begin{align}
f(t) = \sum_{k=1}^{n} A_k e^{-(t-\mu_k)^2 /2 \sigma_k^2}.
\end{align}

A sequence of $n$ Gaussian pulses is defined by $3n$ parameters, being (for each pulse): (i) the amplitude $A_k$, (ii) the mean $\mu_k$ and (iii) the variance $\sigma_k$ of the pulse sequence.
The amplitudes for both Gaussian and square pulses are chosen uniformly at random from the interval $[A_{\text{min}}, A_{\text{max}}]$, the standard deviation for all Gaussian pulses in the train is fixed to $\sigma_k = \sigma$, and the means are chosen randomly such that there is no overlap between the pulses in the train. The square pulses are generated similarly to the Gaussian pulses, with the pulse width chosen to be equal to $6\sigma$. The pulse sequences can be represented in the time or frequency domains <cit.>. The QDataSet pulse sequences are represented using the time-domain as it has been found to be more efficient feature for machine learning algorithms <cit.>.

As discussed in <cit.>, the choice of structure and characteristics of quantum datasets depends upon the particular objectives and uses cases in question, the laboratory quantum control parameters and experimental limitations. Training datasets in machine learning should ideally be structured so as to enhance the generalisability. In the language of statistical learning theory, datasets should be chosen so as to minimise the empirical risk associated with candidate sets of classifiers <cit.>. In a quantum control context, this will include understanding for example the types of controls available to researchers or in experiments, often voltage or (microwave) pulse-based <cit.>. The temporal spacing and amplitude of each pulse in a sequence of controls applied during an experiment may vary by design or to some degree uncontrollably. Pulse waveforms can also differ. For example, the simplest pulse waveform is a constant-amplitude pulse applied for some time $\Delta t$ <cit.>. Such pulses are characterised by for example a single parameter, being the amplitude of the waveform applied to the quantum system (this manifests as we discuss below as an amplitude applied to the algebraic generators of unitary evolution (see <cit.> for an example). Other models of pulses (such as Gaussian) are more complex and require more sophisticated parametrisation and encoding with machine learning architectures in order to simulate. More detail on such considerations and the particular pulse characteristics in the QDataSet are set-out in Table (<ref>).

§.§ QDataSet Noise

§.§.§ Noise characteristics

Noise applicable to quantum systems is generally classified as either classical or quantum <cit.>. Classical noise is represented typically as a stochastic process <cit.> and can include, for example (i) slow noise which is pseudo-static and not varying much over the characteristic time scale of the quantum system and (ii) fast or `white' noise with a high frequency relative to the charateristic frequencies (energy scales) of the system <cit.>. The effect of quantum noise quantum systems of noise is usually characterised into two forms. The first is dephasing ($T_2$) noise, which characteristically causes quantum systems to decohere, thus destroying or degrading quantum information encoded within qubits. Such noise is usually characterised as an operator acting transverse to the quantisation axis of chosen angular momentum.

What this means in practice is usefully construed as follows using a Bloch sphere. Once an orientation ($x,y,z$-axes) is chosen, one is effectively choosing a choice of basis i.e. the basis of a typical qubit $\ketpsi = a\kz + b \ko$ is the basis of eigenstates of the $\sigma_z$ operator. When noise acts along the $z$-axis (i.e. is associated to the $\sigma_z$ operator), then it has the potential to (if the energy of the noise is sufficient) shift the energy state in which the quantum system is in, represented by a `flip' in the basis from $\kz$ to $\ko$ for example. This type of noise is $T_1$ noise. By contrast, noise may act along $x$- and $y$-axes of a qubit, which is represented as being associated with the $\sigma_x$ and $\sigma_y$ operators. These axes are `transverse' to the quantisation axis. Noise along these axis has the effect of dephasing a qubit, thus affecting the coherences encoded in the relative phases of the qubit. Such noise is denoted $T_2$ noise. Open quantum systems' research and understanding noise in quantum systems is a vast and highly specialised topic. As we describe below, the QDataSet adopts the novel approach outlined in <cit.> where, rather than seeking to fully characterise noise spectra, the only the information about noise relevant to the application of controls (to dampen noise) is sought. Such information is encoded in the $V_O$ operator, which is an expectation that encodes the influence of noise on the quantum system. In a quantum control problem using the QDataSet samples containing noise, for example, the objective would then be to select controls that neutralise such effects.

§.§.§ QDataSet noise profiles

In this paper, we chose a set of noise profiles with different statistical properties. The selected noise profiles have been chosen to emulate commonplace types of noise in experimental settings. Doing so improves the utility of algorithms trained using the QDataSet for application in actual experimental and laboratory settings. While engineers and experimentalists across quantum disciplines will usually be familiar with the theoretical and practical aspects of noise in quantum systems, many machine learning and other researchers to whom the QDataSet is directed will not. To assist machine learning practitioners whom may not be familiar with elementary features of noise, it is useful to understand a number of conceptual classifications related to noise used in the QDataSet as follows: (i) power spectral density (which describes the distribution of the noise signal over frequency); (ii) white noise (usually high-frequency noise with a flat frequency); (iii) colored noise, a stochastic process where values are correlated spatially or temporally; (iv) autocorrelated stochasticity, which describes where the noise waveform characteristics are biased by tending to be short (blue) or long (red) as distinct from unautocorrelated noise, where waveforms are relatively uniformly distributed across wavelengths; and (v) stationary noise (a waveform with  a constant time period) and non-stationary noise (a waveform with a varying time period). See literature on noise in signal processing for more detail. The noise realizations are generated in time domain following one of these profile listed as follows (see <cit.> for specific functional forms):

* N0: this is the noiseless case (indicated in the QDataSet parameters as set out in Table (<ref>);

* N1: the noise $\beta(t)$ is described by its power spectral density (PSD) $S_1(f)$, a form of $1/f$ noise with a Gaussian bump;

* N2: here $\beta(t)$ is stationary Gaussian colored noise described by its autocorrelation matrix; chosen such that it is colored, Gaussian and stationary (typically lower frequency) and is produced via convolving Gaussian white noise with a deterministic signal;

* N3: here the noise $\beta(t)$ is non-stationary Gaussian colored noise, again described by its autocorrelation matrix which is chosen such that it is colored , Gaussian and non-stationary. The noise is simulated via multiplication of a deterministic time-domain signal with stationary noise;

* N4: in this case, the noise $\beta(t)$ is described by its autocorrelation matrix chosen such that it is colored, non-Gaussian and non-stationary. The non-Gaussianity of the noise is achieved via squaring the Gaussian noise so as to achieve requisite non-linearities;

* N5: a noise described by its power spectral density (PSD) $S_5(f)$, differing from N1 only via the location of the Gaussian bump; and

* N6: this profile is to model a noise source that is correlated to one of the other five sources (N1 - N5) through a squaring operation. If the $\beta(t)$ is the realization of one of the five profiles, N6 will have realizations of the form $\beta^2(t)$. This profile is used for multi-axis and multi-qubit systems.

The N1 and N5 profiles can be generated following the method described in <cit.>. Regarding the other profiles, any standard numerical package can generate white Gaussian stationary noise. The QDataSet noise realisations were encoded using the Numpy package of Python. We deliberately did so in order to avoid various assumptions used in common quantum programming packages, such as Qutip. To add coloring, we convolve the time-domain samples of the noise with some signal. To generate non-stationarity, we multiply the time-domain samples by a some signal. finally, to generate non-Gaussianity, we start with a Gaussian noise and apply non-linear transformation such as squaring. The last noise profile is used to model the case of two noise sources that are correlated with each other. In this case we generate the first one using any of the profiles N1-N5, and the other source is completely determined.

§.§.§ Distortion

In physical experiments, there control pulses are physical signals (such as microwave pulses), which propagate along cables and get processed by different devices. This introduces distortions which cannot be avoided in any real devices. However, by properly engineering the systems, the effects of these distortions can be minimized and/or compensated for in the design. In this paper, we used a linear-time invariant system to model distortions of the control pulses, and the same filter is used for all datasets. We chose a Chebychev analog filter <cit.>, the undistorted control signal is the input, and thus the filter output is the distorted signal. Table (<ref>) sets out a summary of key parameters.

§.§.§ QDataSet noise operators

The QDataSet comprises datasets for single-qubit and two-qubit systems. In developing the datasets, we have assumed that the environment affecting the qubit is classical and stochastic, namely that $H_1(t)$ will be a stochastic term that acts directly on the the system. The stochasticity of $H_1(t)$ means that the expectation of any observable measured experimentally will be given as:

\begin{align}
\braket{O}_c = \braket{\trace{(\rho(T) O)}}_c,
\end{align}

where $O$ represents the measurement operator corresponding to the observable of interest (e.g. $O \dot{=} M_m$ in notation above) and the $\braket{\cdot}_c$ is a classical expectation over the distribution of the noise realizations. It can then be shown <cit.> that this can be expressed in term of the initial state $\rho(0)$, and the evolution fixed over the time interval $[0,T]$ as

\begin{align}
\braket{O(T)} = \trace{\left(V_O(T) U_0^{\dagger}(T) \rho(0) U_0(T) O\right)}
\label{eqn:voeqn}
\end{align}

where $U_0(T) = \mathcal{T}_{+} e^{-i\int_0^T H_0(t) dt}$ is the evolution matrix in the absence of noise, and

\begin{align}
V_O(T) &= \braket{W_O(T)}_c\\
&\doteq \braket{O^{-1}\tilde{U}_I^{\dagger}(T) O \tilde{U}_I(T)}_c.
\end{align}

is a novel noise operator introduced in <cit.> which characterises the expectation of noise relevant to synthesising counteracting control pulses. We encapsulate the full quantum evolution via the operator $W_O(T)$. Note that $V_O$ is formed via the partial tracing out of the effect of the noise (bath) and its interaction with the control pulses, so encodes only those features of importance or relevance to impact of noise (not the full noise spectrum). Importantly, the use of the $V_O$ operator is designed to allow information about noise and observables to be separated (a hallmark of dynamic decoupling approaches).

The modified interaction unitary $\tilde{U}_I(T)$ is defined such that

\begin{align}
U(T) = \tilde{U}_I(T) U_0(T),
\end{align}

where $U(T)= \mathcal{T}_{+} e^{-i\int_0^T H(t) dt} $ is the full evolution matrix. This contrasts the conventional definition of the interaction unitary which takes the form $U(T) = U_0(T) U_I(T) $. The $V_O$ operator is used in the simulation code for the QDataSet to generate [for each time step and for each noise type] [expectation] values. Ideally, in a noise-free scenario, those expectations should tend to zero (representative of the absence of noise). The idea for including such noise operators is that this data can then be input into machine learning models to assist the algorithms to learn appropriate, for example, pulse sequences or controls that send $V_O \to I$ (neutralising the noise).

A detailed explanation and example of the use of the $V_O$ operator is provided in <cit.>. For machine learning practitioners, the operator $V_O$ may, for example, be used in an algorithm that seeks to negate the effect of $V_O$ i.e. learning the sequence of control pulses necessary such that by applying learnt sequences of pulses, $V_O$ effectively becomes $I$. The utility of this approach is that full noise spectroscopy is not required.

§.§ QDataset Measurement

§.§.§ QDataSet POVMs

As discussed above, quantum measurements are inherently probabilistic. Measurement of quantum systems yields an underlying probability distribution over the possible measurement outcomes (observables) $m_i$ of the system which are in turn determined by the state of the system and the measurement process.  There are several ways to describe quantum measurements mathematically. The most common way (which will be used in this paper), is called protective measurements. In this case, an observable $O$ is described mathematically by a Hermitian operator. The eigendecomposition of the operator can be expressed in the form $O= \sum_{m} m P_m$, where $m$ are the eigenvalues, and $P_m$ are the associated projectors into the corresponding eigenspace. The projectors $P_m$ must satisfy that $P_m^2 = P_m$, and that $\sum_m P_m = I$ (the identity operator), to ensure we get a correct distribution for the outcomes. In more sophisticated treatments, the operators $O$ belong to a POVM described above which partition the Hilbert space $\mathcal{H}$ into distinct projective subspaces $\mathcal{H}_m$ associated with each POVM operator $O$. The probability of measuring an observable is given by:

\begin{align}
\Pr(m) = \trace(\rho P_m),
\end{align}

for a system in the state $\rho$. The expectation value of the observable is given by:

\begin{align}
\braket{O}= \trace{(\rho O)} = \trace{\left(\rho \sum_m m P_m\right)} = \sum_m m \Pr(m).
\end{align}

As detailed below, the QDataSet contains measurement outcomes for a variety of noiseless and noisy systems. The POVM chosen is the set of Pauli operators for one and two-qubit systems. The measurement operators chosen are the Pauli operators described below and the QDataSet contains the expectation values for each Pauli measurement operator. In a classical machine learning context, these measurement statistics form training data labels in optimisation problems, such as designing algorithms that can efficiently sequence control pulses in order to efficiently (time-minimally) synthesise a target state or unitary (and thus undertake a quantum computation) of interest.

§.§.§ Pauli matrices

The POVM set for the QDataSet is the set of Pauli operators. Pauli operators are important operators in quantum information processing involving qubit systems in part because such qubit systems can be usefully decomposed into a Pauli operator basis via the Pauli matrices:

\begin{align}
\sigma_x = \begin{pmatrix}0 & 1 \\ 1 & 0\end{pmatrix}, \sigma_y = \begin{pmatrix}0 & -i \\ i & 0\end{pmatrix}, \sigma_z &= \begin{pmatrix}1 & 0 \\ 0 & -1\end{pmatrix}
\end{align}

together with the identity (denoted $\sigma_0$). Pauli operators are Hermitian (with eigenvalues $+1$ and $-1$), traceless and satisfy that $\sigma_i^2 = I$. Together with the identity matrix (which is sometime denoted by $\sigma_0$), they form an orthonormal basis known as the Pauli basis (with respect to the Hilbert-Schmidt product defined as $\braket{A,B} = \trace{(A^{\dagger}B)}$) for any $2 \times 2$ Hermitian matrix. Qubit states can then be expressed in this basis via the density matrix:

\begin{align}
\rho = \frac{1}{2}\left(I + \mathbf{r}\cdot\bm{\sigma}\right),
\end{align}

where the vector $\mathbf{r}=(r_x, r_y, r_z)$ is a unit vector called the Bloch vector, and the vector $\bm{\sigma}=(\sigma_x, \sigma_y, \sigma_z)$. The dot product of these two vectors is just a shorthand notation for the expression $\mathbf{r}\cdot\bm{\sigma}= r_x\sigma_x + r_y\sigma_y+ r_z \sigma_z$. Any time-dependent Hamiltonian of a qubit can be expressed as

\begin{align}
H(t) = \sum_{i\in\{x,y,z\}} {\alpha_i(t) \sigma_i},
\end{align}

with the time-dependence absorbed in the coefficients $\alpha_i(t)$.

§.§.§ Pauli measurements

As described below, the measurements simulated in the QDataSet are what are known as Pauli measurements. These are formed by taking the expectation value of each Pauli matrix e.g. $\trace{(\rho \sigma_i)}$ for $i \in \{x,y,z\}$ (the identity is omitted). The resultant measurement distributions will typically form labelled data in a machine learning context. As discussed below, measurement distributions are ultimately how various properties of the quantum system are inferred (i.e. via reconstructive inference), such as the characteristics of quantum circuits, evolutionary paths and tomographical quantum state description. As we describe below, measurements in the QDataSet comprise measurements on each eigenstate (six in total) of each Pauli operator by all Pauli operators. [Hermitian] operators have a spectral decomposition in terms of eigenvalues and their corresponding projectors

\begin{align}
P_0 &= \kz \bz = \frac{1}{2}(I + \sigma_z) = \begin{pmatrix}
1 & 0 \\
0 & 0
\end{pmatrix}\\
P_1 &= \kz \bz = \frac{1}{2}(I - \sigma_z) = \begin{pmatrix}
0 & 0 \\
0 & 1
\end{pmatrix}
\end{align}

thus we can write:

\begin{align}
\sigma_z = 1 \times \begin{pmatrix}
1 & 0 \\
0 & 0
\end{pmatrix}  - 1 \times \begin{pmatrix}
0 & 0 \\
0 & 1
\end{pmatrix}
\end{align}

For example, a Pauli meaurement on a qubit in the $-1$ eigenstate with respect to the $\sigma_z$ operator

\begin{align}
\trace( \rho \sigma_z^\dagger) =   \trace \left( \begin{pmatrix}
0 & 0 \\
0 & 1
\end{pmatrix}
\begin{pmatrix}
1 & 0 \\
0 & -1
\end{pmatrix} \right) = \trace \begin{pmatrix}
0 & 0 \\
0 & -1
\end{pmatrix} = -1
\end{align}

which is as expected. The probability of observing $\lambda=-1$ in this state we should expect to be unity (given the state is in the eigenstate):

\begin{align}
Pr(m=-1) = \trace (P_1^2 \rho) = 1
\end{align}

For $n$-qubit systems (such as two-qubit systems in the QDataSet), Pauli measurements are represented by tensor-products of Pauli operators. For example, a $\sigma_z$ measurement on the first qubit and $\sigma_x$ on the second is represented as:

\begin{align}
\sigma_z \otimes \sigma_x
\end{align}

In programming matrix notation, this becomes represented as a $4 \times 4$ matrix (tensor):

\begin{align}
\sigma_z \otimes \sigma_X &=\begin{pmatrix}
1 & 0 \\
0 & -1
\end{pmatrix} \otimes \begin{pmatrix}
0 & 1 \\
1 & 0
\end{pmatrix} \\
1 \times \begin{pmatrix}
0 & 1 \\
1 & 0
\end{pmatrix}  & 0 \\
0 & -1 \times \begin{pmatrix}
0 & 1 \\
1 & 0
\end{pmatrix}
\end{pmatrix} \\
&= \begin{pmatrix}
0 & 1 & 0 & 0\\
1 & 0 & 0 & 0\\
0 & 0 & 0 & -1\\
0 & 0 & -1 & 0
\end{pmatrix}
\end{align}

The Pauli representation of qubits can be usefully visualised via the Bloch sphere as per Figure <ref>. The axes of the Bloch sphere are the expectation values of the Pauli $\sigma_x, \sigma_y$ and $\sigma_z$ operators respectively. As each Pauli operator has eigenvalues $1$ and $-1$, the eigenvalues can be plotted along axes of the 2-sphere. For a pure (non-decohered) quantum state $\rho$, $|\rho|=\sqrt{r_x^2 + r_y^2 + r_z^2}=1$ (as we require $\trace{\rho^2=1}$), thus $\rho$ is represented on the Bloch 2-sphere as a vector originating at the origin and lying on the surface of the Bloch 2-sphere. The evolution of the qubit i.e. a computation according to unitary evolution can then be represented as rotations of $\rho$ across the Bloch sphere. In noisy contexts, decohered $\rho$ are represented whereby $|\rho|<1$ i.e. the norm of $\rho$ shrinks and $\rho$ no longer resides on the surface.

An example of a quantum state rotation on the Bloch sphere. The $\kz,\ko$ indicates the $\sigma_z$-axis, the $X$ and $Y$ the $\sigma_x$ and $\sigma_y$ axes respectively. In (a), the vector is residing in a $+1$ $\sigma_x$ eigenstate. By rotating about the $\sigma_z$ axis by $\pi/4$, the vector is rotated to the right, to the $+1$ $\sigma_y$ eigenstate. A rotation about the $\sigma_Z$ axis by angle $\theta$ is equivalent to the application of the unitary $U(\theta) = \exp(-i \theta_z \sigma_z/2)$.

For machine learning practitioners, it is useful to appreciate the the operation of Pauli operators $\sigma_x, \sigma_y, \sigma_z$ as the generators of rotations about the respective axes of the Bloch sphere (again, this is just a way of representing the evolution of quantum information). Thus the application of $\sigma_z$ to a qubit is akin to rotating the quantum state vector $\rho$ about the $z$-axis. Figure <ref> Conceptually, a qubit is in a $z$-eigenstate if it is lying directly on either the north ($+1$) or south ($-1$) pole. Rotating about the $z$-axis then is akin to rotating the vector on the spot, thus no change in the quantum states (or eigenvalues) for $\sigma_z$ occurs because the system exhibits symmetry under such transformations. This is similarly the case for $\sigma_x, \sigma_y$ generators with respect to their eigenvalues and eigenvectors. However, rotations by $\sigma_\alpha$ will affect the eigenvalues/vectors in the $\sigma_\beta$ basis where $\alpha \neq \beta$ e.g. a $\sigma_x$ rotation will affect the component of the qubit lying along the $\sigma_z$ axis. Similarly, a $\sigma_z$ rotation of a qubit in a $\sigma_x$ eigenstate will alter that state (shown in (a) and (b) of Figure <ref>). An understanding of Pauli operators and conceptualisation of qubit axes is important to the understanding of the simulated QDataSet. An understanding of symmetries of relevance to qubit evolution (and quantum algorithms) is also beneficial. As we describe below, controls or noise are structured to be applied along particular axes of a qubit and thus can be thought of as a way to control or distortions upon the axial rotations of a qubit effected by the corresponding Pauli generator.

There exist higher dimensional generalization to the Pauli matrices that allow forming orthonormal basis to represent operators in these dimensions. In particular if we have a system of $N$ qubits, then one simple generalization is to form the set $\{\sigma_{i_1}^{(1)} \otimes \sigma_{i_2}^{(2)} \otimes \cdots \sigma_{i_N}^{(N)} \}_{i_j \in \{0,x,y,z\}}$. In other words we take tensor products of the Pauli's which gives a set of size $2^N$. For example, for a two-qubit system we can form the $16-$ element set $\{\sigma_0 \otimes \sigma_0, \sigma_0 \otimes \sigma_x, \sigma_0 \otimes \sigma_y, \sigma_0 \otimes \sigma_z, \sigma_x \otimes \sigma_0, \sigma_x \otimes \sigma_x, \sigma_x \otimes \sigma_y, \sigma_x \otimes \sigma_z, \sigma_y \otimes \sigma_0, \sigma_y \otimes \sigma_x, \sigma_y \otimes \sigma_y, \sigma_y \otimes \sigma_z, \sigma_z \otimes \sigma_0, \sigma_z \otimes \sigma_x, \sigma_z \otimes \sigma_y, \sigma_z \otimes \sigma_z\}$. Moreover, for many use cases, we are interested in the minimal number of operators, such as Pauli operators, required to achieve a requisite level of control, such as universal quantum computation. For some machine learning applications, we are interested in minimising the number of controls that must be applied to a quantum system (thus minimising the resources required to control the system). In such cases, we may seek a minimal control algebra or gate set. The minimal number of Pauli operators required to achieve a complete control set of generators for synthesising an arbitrary unitary acting on an $n$ dimensional Hilbert space is given by a bracket-generating set $\Delta \subseteq \mathfrak{su}(2^n)$  <cit.> which can be understood in more complex treatments in the context of Lie algebras and representation theory. Here $\mathfrak{su}(2^n)$ represents the Lie algebra corresponding to the Pauli group SU$(2^n)$, the complete set of generators required to span the $n$ dimensional Hilbert space in the Pauli basis. Given $\Delta$, we can reconstruct the full Pauli basis via the operation of the Lie bracket <cit.>. The QDataSet generators for one- and two-qubit systems are simply one and two (tensor-product) sets of Paulis respectively (i.e. not $\Delta$). For higher-dimensional problems, whether to restrict generators to those within $\Delta$ becomes a consideration for machine learning architectures (see literature on time-optimal quantum control). These generators will typically be used as the tensors or matrices to which classical controls are applied within machine learning architectures.

For the single qubit system, initial states are the two eigenstates of every Pauli operators. As noted above, the quantum state can be decomposed in the Pauli basis as $\rho_j =\frac{1}{2}(I\pm \sigma_j)$, for $j=1,2,3$. This gives a total of 6 states. We perform the three Pauli measurements on each of these states, resulting in a total of 18 possible combinations. These 18 measurements are important to characterize a qubit system. For two-qubits, it will be similar but now we initialize every individual qubit into the 6 possible eigenstates, and we measure all 15 Pauli observables (we exclude identity). This gives a total of 540 possible combinations.

§.§.§ Monte Carlo measurements

Measurements of the one- and two-qubit systems for the QDataSet are undertaken using Monte Carlo techniques. This means that a random Pauli measurement is undertaken multiple times, with the measurement results averaged in order to provide the resultant measurement distribution for each of the POVM operators. The measurement of the quantum systems is contingent on the noise realisations for each system. For the noiseless case, the Pauli measurements are simply the Monte Carlo averages (expectations) of the Pauli operators. Systems with noise will have one or more noise realisations (applications of noise) applied to them. To account for this, we include two separate sets of measurement distribution. The first the expectation value of the three Pauli operators over all possible initial states for each different noise realisation. Thus for each type of noise, there will be a set of measurement statistics. The second is a set of measurement statistics where we average over all noise realisations for the dataset. Doing so enables algorithms trained using the QDataSet to be more fine-grained in their treatment of noise: in some contexts, while noise profiles may be uncertain, it is clear that the noise is of a certain type, so the first set of measurement statistics may be more applicable. These statistics are given by the the set $\{ V_O \}$ in the QDataSet. For other cases, there is almost no information about noise profiles or their sources, in which case the average over all noise realisations may be more appropriate. This second set of measurements is given by the set $\{E_O \}$.

§.§.§ Monte Carlo Simulator

For the benefit of researchers using the QDataSet, we briefly set out a bit more detail of how the datasets were generated. The simulator comprises three main components. The first approximates time-ordered unitary evolution. The second component generates realisations of noise given random parametrisations of the power spectral density (PSD) of the noise. The third component simulates quantum measurement. The simulations are based upon Monte Carlo methods whereby $K$ randomised pulse sequences give rise to noise realisations. The quantum systems are then measured to determine the extent to which the noise realisations affect the expectation values (Pauli measurements). Trial and error indicated a stabilisation of measurement statistics at around $K=500$, thus $K \geq1000$ was chosen for the final simulation run to generate the QDataSet. The final Pauli measurements are then averages over such noise realisations. The parameter $K$ is included for each dataset and example (as described below). For more detail, including useful pseudocode that sets out the relationship between noise realisations, $\beta(t)$ and measurement, see the Supplementary Material in <cit.>.

§ KEY PARAMETERS

Table (<ref>) sets out an example of parameters for various of the datasets.

Parameter     Value

$T$     1

$M$     1024

$K$     2000

$\Omega$     12

$\Omega_1$     12

$\Omega_2$     10

$n$     5

$A_{\text{min}}$     -100

$A_{\text{max}}$     100

$\sigma$     T/(12M)

Dataset Parameters:  $T$: total time, set to unity for standardisation; $M$: the number of time-steps (discretisations); $K$: the number of noise realisations; $\Omega$: the energy gap for the single qubit case (where subscripts 1 and 2 represent the energy gap for each qubit in the single qubit case); $n$: number of control pulses; $A_{\text{max}},A_{\text{min}}$: maximum and minimum amplitude; $\sigma$: standard deviation of pulse spacing (for Gaussian pulses).

§ QDATASET CHARACTERISTICS

2|c|QDataSet File Descriptions

Dataset      Description

G_1q_X      (i) Qubits: one; (ii) Control: $x$-axis, Gaussian; (iii) Noise: none; (iv) No distortion.

G_1q_X_D      (i) Qubits: one; (ii) Control: $x$-axis, Gaussian; (iii) Noise: none; (iv) Distortion.

G_1q_XY      (i) Qubits: one; (ii) Control: $x$-axis and $y$-axis, Gaussian; (iii) Noise: none; (iv) No distortion.

G_1q_XY_D      (i) Qubits: one; (ii) Control: $x$-axis and $y$-axis, Gaussian; (iii) Noise: none; (iv) Distortion.

G_1q_XY_XZ_N1N5      (i) Qubits: one; (ii) Control: $x$-axis and $y$-axis, Gaussian; (iii) Noise: N1 on $x$-axis, N5 on $z$-axis; (iv) No distortion.

G_1q_XY_XZ_N1N5_D      (i) Qubits: one; (ii) Control: $x$-axis and $y$-axis, Gaussian; (iii) Noise: N1 on $x$-axis, N5 on $z$-axis; (iv) No distortion.

G_1q_XY_XZ_N1N6      (i) Qubits: one; (ii) Control: $x$-axis and $y$-axis, Gaussian; (iii) Noise: N1 on $x$-axis, N6 on $z$-axis; (iv) Distortion.

G_1q_XY_XZ_N1N6_D      (i) Qubits: one; (ii) Control: $x$-axis and $y$-axis, Gaussian; (iii) Noise: N1 on $x$-axis, N6 on $z$-axis; (iv) No distortion.

G_1q_XY_XZ_N3N6      (i) Qubits: one; (ii) Control: $x$-axis and $y$-axis, Gaussian; (iii) Noise: N1 on $x$-axis, N6 on $z$-axis; (iv) Distortion.

G_1q_XY_XZ_N3N6_D      (i) Qubits: one; (ii) Control: $x$-axis and $y$-axis, Gaussian; (iii) Noise: N1 on $x$-axis, N6 on $z$-axis; (iv) No distortion.

G_1q_X_Z_N1      (i) Qubits: one; (ii) Control: $x$-axis, Gaussian; (iii) Noise: N1 on $z$-axis; (iv) No distortion.

G_1q_X_Z_N1_D      (i) Qubits: one; (ii) Control: $x$-axis, Gaussian; (iii) Noise: N1 on $z$-axis; (iv) Distortion.

G_1q_X_Z_N2      (i) Qubits: one; (ii) Control: $x$-axis, Gaussian; (iii) Noise: N2 on $z$-axis; (iv) No distortion.

G_1q_X_Z_N2_D      (i) Qubits: one; (ii) Control: $x$-axis, Gaussian; (iii) Noise: N2 on $z$-axis; (iv) Distortion.

G_1q_X_Z_N3      (i) Qubits: one; (ii) Control: $x$-axis, Gaussian; (iii) Noise: N3 on $z$-axis; (iv) No distortion.

G_1q_X_Z_N3_D      (i) Qubits: one; (ii) Control: $x$-axis, Gaussian; (iii) Noise: N3 on $z$-axis; (iv) Distortion.

G_1q_X_Z_N4      (i) Qubits: one; (ii) Control: $x$-axis, Gaussian; (iii) Noise: N4 on $z$-axis; (iv) No distortion.

G_1q_X_Z_N4_D      (i) Qubits: one; (ii) Control: $x$-axis, Gaussian; (iii) Noise: N4 on $z$-axis; (iv) Distortion.

G_2q_IX-XI_IZ-ZI_N1-N6      (i) Qubits: two; (ii) Control: $x$-axis on both qubits, Gaussian; (iii)
Noise: N1 and N6 $z$-axis on each qubit; (iv) No distortion.

G_2q_IX-XI_IZ-ZI_N1-N6_D      (i) Qubits: two; (ii) Control: $x$-axis on both qubits, Gaussian; (iii) Noise: N1 and N6 $z$-axis on each qubit; (iv) Distortion.

G_2q_IX-XI-XX      (i) Qubits: two; (ii) Control: single $x$-axis control on both qubits and $x$-axis interacting control, Gaussian; (iii) Noise: none; (iv) No distortion.

G_2q_IX-XI-XX_D      (i) Qubits: two; (ii) Control: single $x$-axis control on both qubits and $x$-axis interacting control, Gaussian; (iii) Noise: none; (iv) Distortion.

G_2q_IX-XI-XX_IZ-ZI_N1-N5      (i) Qubits: two; (ii) Control: single $x$-axis control on both qubits and $x$-axis interacting control, Gaussian; (iii) Noise: N1 and N5 on $z$-axis noise on each qubit; (iv) No distortion.

G_2q_IX-XI-XX_IZ-ZI_N1-N5      (i) Qubits: two; (ii) Control: single $x$-axis control on both qubits and $x$-axis interacting control, Gaussian; (iii) Noise: N1 and N5 on $z$-axis noise on each qubit; (iv) Distortion.

S_1q_X      (i) Qubits: one; (ii) Control: $x$-axis, square; (iii) Noise: none; (iv) No distortion.

S_1q_X_D      (i) Qubits: one; (ii) Control: $x$-axis, Gaussquaresian; (iii) Noise: none; (iv) Distortion.

S_1q_XY      (i) Qubits: one; (ii) Control: $x$-axis and $y$-axis, square; (iii) Noise: none; (iv) No distortion.

S_1q_XY_D      (i) Qubits: one; (ii) Control: $x$-axis and $y$-axis, Gaussquaresian; (iii) Noise: none; (iv) Distortion.

S_1q_XY_XZ_N1N5      (i) Qubits: one; (ii) Control: $x$-axis and $y$-axis, square; (iii) Noise: N1 on $x$-axis, N5 on $z$-axis; (iv) No distortion.

S_1q_XY_XZ_N1N5_D      (i) Qubits: one; (ii) Control: $x$-axis and $y$-axis, Gaussian; (iii) Noise: N1 on $x$-axis, N5 on $z$-axis; (iv) No distortion.

S_1q_XY_XZ_N1N6      (i) Qubits: one; (ii) Control: $x$-axis and $y$-axis, square; (iii) Noise: N1 on $x$-axis, N6 on $z$-axis; (iv) Distortion.

S_1q_XY_XZ_N1N6_D      (i) Qubits: one; (ii) Control: $x$-axis and $y$-axis, square; (iii) Noise: N1 on $x$-axis, N6 on $z$-axis; (iv) No distortion.

S_1q_XY_XZ_N3N6      (i) Qubits: one; (ii) Control: $x$-axis and $y$-axis, square; (iii) Noise: N1 on $x$-axis, N6 on $z$-axis; (iv) Distortion.

S_1q_XY_XZ_N3N6_D      (i) Qubits: one; (ii) Control: $x$-axis and $y$-axis, square; (iii) Noise: N1 on $x$-axis, N6 on $z$-axis; (iv) No distortion.

S_1q_X_Z_N1      (i) Qubits: one; (ii) Control: $x$-axis, square; (iii) Noise: N1 on $z$-axis; (iv) No distortion.

S_1q_X_Z_N1_D      (i) Qubits: one; (ii) Control: $x$-axis, square; (iii) Noise: N1 on $z$-axis; (iv) Distortion.

S_1q_X_Z_N2      (i) Qubits: one; (ii) Control: $x$-axis, square; (iii) Noise: N2 on $z$-axis; (iv) No distortion.

G_1q_X_Z_N2_D      (i) Qubits: one; (ii) Control: $x$-axis, Gaussian; (iii) Noise: N2 on $z$-axis; (iv) Distortion.

S_1q_X_Z_N3      (i) Qubits: one; (ii) Control: $x$-axis, square; (iii) Noise: N3 on $z$-axis; (iv) No distortion.

S_1q_X_Z_N3_D      (i) Qubits: one; (ii) Control: $x$-axis, square; (iii) Noise: N3 on $z$-axis; (iv) Distortion.

S_1q_X_Z_N4      (i) Qubits: one; (ii) Control: $x$-axis, square; (iii) Noise: N4 on $z$-axis; (iv) No distortion.

S_1q_X_Z_N4_D      (i) Qubits: one; (ii) Control: $x$-axis, square; (iii) Noise: N4 on $z$-axis; (iv) Distortion.

S_2q_IX-XI_IZ-ZI_N1-N6      (i) Qubits: two; (ii) Control: $x$-axis on both qubits, square; (iii) Noise: N1 and N6 $z$-axis on each qubit; (iv) No distortion.

S_2q_IX-XI_IZ-ZI_N1-N6_D      (i) Qubits: two; (ii) Control: $x$-axis on both qubits, square; (iii) Noise: N1 and N6 $z$-axis on each qubit; (iv) Distortion.

S_2q_IX-XI-XX      (i) Qubits: two; (ii) Control: single $x$-axis control on both qubits and $x$-axis interacting control, square; (iii) Noise: none; (iv) No distortion.

S_2q_IX-XI-XX_D      (i) Qubits: two; (ii) Control: single $x$-axis control on both qubits and $x$-axis interacting control, square; (iii) Noise: none; (iv) Distortion.

S_2q_IX-XI-XX_IZ-ZI_N1-N5      (i) Qubits: two; (ii) Control: $x$-axis on both qubits and $x$-axis interacting control, square; (iii) Noise: N1 and N5 $z$-axis on each qubit; (iv) No distortion.

S_2q_IX-XI-XX_IZ-ZI_N1-N5_D      (i) Qubits: two; (ii) Control: $x$-axis on both qubits and $x$-axis interacting control, square; (iii) Noise: N1 and N5 $z$-axis on each qubit; (iv) Distortion.

S_2q_IX-XI-XX_IZ-ZI_N1-N6      (i) Qubits: two; (ii) Control: $x$-axis on both qubits and $x$-axis interacting control, square; (iii) Noise: N1 and N6 $z$-axis on each qubit; (iv) No distortion.

S_2q_IX-XI-XX_IZ-ZI_N1-N6_D      (i) Qubits: two; (ii) Control: $x$-axis on both qubits and $x$-axis interacting control, square; (iii) Noise: N1 and N6 $z$-axis on each qubit; (iv) Distortion.

QDataSet File Description. The left column identifies each dataset in the respective QDataSet examples while the description column describes the profile of the dataset in terms of (i) number of qubits, (ii) axis of control and pulse wave-form (iii) axis and type of noise and (iv) whether distortion is present or absent.


# Preservation of log-Hölder coefficients of the vorticity in the transport
equation

Dongho Chae Department of Mathematics, Chung-Ang University. E-mail:
<EMAIL_ADDRESS>In-Jee Jeong Department of Mathematical Sciences and RIM,
Seoul National University. E-mail<EMAIL_ADDRESS>
###### Abstract

We show that the log-Hölder coefficients of a solution to the transport
equation is preserved in time.

AMS Subject Classification Number: 35Q31, 76B03, 76W05
Keywords: Transport equation, incompressible Euler equations, modulus of
continuity, vorticity

##  1 Introduction

We are concerned on the transport equations in $\mathbb{R}^{n}$:

(1.1)
$\theta_{t}+u\cdot\nabla\theta=0\quad\text{on}\quad\mathbb{R}^{n}\times(0,T),$

where $u=u(x,t)=(u_{1}(x,t),\cdots,u_{n}(x,t))$ for
$(x,t)\in\mathbb{R}^{n}\times[0,+\infty)$, and $\theta=\theta(x,t)$ is a
scalar with $\theta(x,0)=\theta_{0}(x)$.

We say $f$ is Hölder continuous with exponent $\gamma\in(0,1)$ at
$x_{0}\in\mathbb{R}^{n}$ if there exists $B_{r}(x_{0})\subset\mathbb{R}^{n}$
such that

$\sup_{x\in B_{r}(x_{0})}\frac{|f(x)-f(x_{0})|}{|x-x_{0}|^{\gamma}}<+\infty.$

In this case we define the Hölder coefficient $[f]_{\gamma;x_{0}}$ by

(1.2) $[f]_{\gamma;x_{0}}=\lim_{r\to 0}\sup_{x\in
B_{r}(x_{0})}\frac{|f(x)-f(x_{0})|}{|x-x_{0}|^{\gamma}},$

and denote

$C^{\gamma}(x_{0})=\\{f\in
L^{\infty}(\mathbb{R}^{n})\,|\,[f]_{\gamma;x_{0}}<+\infty\\}.$

Note that we are following the notation of [1, pp.52]. One can generalize the
above notion to a general case of functions defining the modulus of continuity
$\delta=\delta(s)$, and denote

(1.3) $[f]_{\delta(s);x_{0}}:=\lim_{r\to 0}\sup_{x\in
B_{r}(x_{0})}\frac{|f(x)-f(x_{0})|}{\delta(|x-x_{0}|)},$

for $x_{0}\in\mathbb{R}^{n}$, and write

$C^{\delta(s)}(x_{0})=\\{f\in
L^{\infty}(\mathbb{R}^{n})\,|\,[f]_{\delta(s);x_{0}}<+\infty\\}.$

Given a vector field $u=u(x,t)\in L^{1}(0,T;Lip(\mathbb{R}^{n}))$, we
introduce the particle trajectory mapping $x_{0}\mapsto\varphi(x_{0},t)$ from
$\mathbb{R}^{n}$ into $\mathbb{R}^{n}$ generated by $u=u(x,t)$, which is
defined as the solution of the ordinary differential equation,

(1.4) $\left\\{\begin{aligned} &\frac{\partial\varphi(x_{0},t)}{\partial
t}=u(\varphi(x_{0},t),t)\quad\text{on}\quad(0,T),\\\
&\varphi(x_{0},0)=x_{0}\in\mathbb{R}^{n}.\end{aligned}\right.$

Our main result states that for the transport equation, the log-Hölder
coefficient is constant in time.

######  Theorem 1.1.

Let $u\in L^{1}(0,T;Lip(\mathbb{R}^{n}))$ and $\theta\in
L^{\infty}(0,T;L^{\infty}(\mathbb{R}^{n}))$ satisfies (2.15) in
$\mathbb{R}^{n}\times(0,T)$. Suppose there exists $\gamma\in(0,\infty)$ such
that $\theta_{0}\in C^{(\log\frac{1}{s})^{-\gamma}}(x_{0})$ for some
$x_{0}\in\mathbb{R}^{n}$. Then,

(1.5)
$[\theta(\cdot,t)]_{{\log(\frac{1}{s})}^{-\gamma};\varphi(x_{0},t)}=[\theta_{0}]_{{\log(\frac{1}{s})}^{-\gamma};x_{0}}\quad\forall
t\in[0,T),$

where $\varphi(x_{0},t)$ denotes the particle trajectory defined in (1.4). In
particular, we have

(1.6)
$\sup_{x\in\mathbb{R}^{n}}[\theta(t)]_{{\log(\frac{1}{s})}^{-\gamma},x}=\sup_{x\in\mathbb{R}^{n}}[\theta_{0}]_{{\log(\frac{1}{s})}^{-\gamma},x}.$

As an application we consider the 2D Euler equations on
$\mathbb{R}^{2}\times[0,+\infty)$.

$(E)\left\\{\begin{aligned} &\omega_{t}+u\cdot\nabla\omega=0,\\\ &\nabla\cdot
u=0,\quad\partial_{1}u_{2}-\partial_{2}u_{1}=\omega,\\\
&\omega(x,0)=\omega_{0}(x).\end{aligned}\right.$

For bounded initial data $\omega_{0}\in L^{1}\cap L^{\infty}(\mathbb{R}^{2})$,
well known Yudovich theory gives a unique solution to $(E)$ in $\omega(t,x)\in
L^{\infty}_{t}(L^{1}\cap L^{\infty})_{x}.$ If $\omega_{0}$ has a little bit of
additional regularity, then it is guaranteed that $u$ is Lipschitz continuous
in space ([6]).

######  Corollary 1.1.

Let $u\in L^{1}(0,T;Lip(\mathbb{R}^{2}))$ be a solution to (E), Suppose
$\omega_{0}\in C^{(\log\frac{1}{s})^{-\gamma}}(x_{0})$ for some
$\gamma\in(0,\infty)$ and $x_{0}\in\mathbb{R}^{2}$. Then,

(1.7)
$[\omega(\cdot,t)]_{{\log(\frac{1}{s})}^{-\gamma};\varphi(x_{0},t)}=[\omega_{0}]_{{\log(\frac{1}{s})}^{-\gamma};x_{0}}\quad\forall
t\in[0,T).$

Remark 1.1  In the proof, we observe the following inequalities.

(1.8)
$\displaystyle[\omega_{0}]_{\beta;x_{0}}\exp\left(-\beta\int_{0}^{t}|\nabla
u(\varphi(x_{0},s),s)|ds\right)$ (1.9)
$\displaystyle\qquad\leq[\omega(\cdot,t)]_{\beta;\varphi(x_{0},t)}\leq[\omega_{0}]_{\beta;x_{0}}\exp\left(\beta\int_{0}^{t}|\nabla
u(\varphi(x_{0},s),s)|ds\right)\quad\forall\beta\in(0,1]$

for all $t\in[0,T)$. We could understand that (1.7) is a limiting relation of
(1.8) as $\beta\to 0$.

Remark 1.2  It would be interesting to compare (1.7) with the result of the
double exponential growth in time of the vorticity gradient by Kieselev and
Šverák[5] for the 2D Euler equations. As far as the authors know we do not
know if the growth of $\sup_{x\in\mathbb{R}^{2}}[\omega(t)]_{\gamma,x}$ (or
$\|\omega(t)\|_{C^{0,\gamma}}$) is slower than the double exponential one. The
above result says that for the log-Hölder coefficient it is constant in time.

Remark 1.3  In the 2D Euler equations, initial vorticity with asymptotics

$\begin{split}|\omega_{0}(x)|\simeq\log(\frac{1}{|x|})^{-\gamma},\qquad
0<|x|\ll 1\end{split}$

for some $\gamma>0$ together with odd-odd symmetry with respect to both axes
and satisfying $\omega_{0}\geq 0$ on $(\mathbb{R}_{+})^{2}$ was considered in
[2].111This type of data could be used to prove ill-posedness of the 2D Euler
equations in the critical Sobolev space $\omega\in H^{1}$. Our main result
shows that whenever $\gamma\leq 1$, if the corresponding velocity field
belongs to $L^{1}(0,T;Lip)$ for any $T>0$, it is forced that
$|\omega(x,t)|\simeq\log(\frac{1}{|x|})^{-\gamma}$. Here, the origin is fixed
by the flow due to the odd symmetry. But then, using this asymptotics with the
explicit formula

$\begin{split}(\partial_{x_{1}}u_{1})(0,t)=c_{0}\int_{(\mathbb{R}_{+})^{2}}\frac{y_{1}y_{2}}{|y|^{4}}\omega(y,t)\,dy\gtrsim\int_{r(t)}^{\infty}\frac{1}{r(\log\frac{1}{r})^{\gamma}}\,dr,\end{split}$

one can prove that $|\nabla u(\cdot,t)|=+\infty$ for $t\in(0,T)$, which is a
contradiction to the $L^{1}(0,T;Lip)$–assumption. Therefore, as a corollary of
our main result, we deduce that for such an initial vorticity,
$\int_{0}^{T}\|\nabla u(\cdot,t)\|_{L^{\infty}}\,dt=+\infty$ for any $T>0$. In
turn, this forces the Lagrangian deformation to be divergent;
$|\partial_{x_{1}}\Phi_{1}(0,t)|=+\infty$ for $t>0$. This is very interesting,
since a formal analysis given in [4] suggests that $\|\nabla
u(\cdot,t)\|_{L^{\infty}}\simeq ct^{-1}$ for some absolute constant $c>0$,
which is barely non-integrable in time. To clarify, in general (without the
odd-odd assumption) it is possible that
$|\omega(x,t)|\simeq\log(\frac{1}{|x|})^{-\gamma}$ and $u(x,t)\in
L^{1}_{t}Lip$ hold at the same time for any $\gamma>0$. This happens for
instance the vorticity satisfies a rotational symmetry; see [3].

For the three-dimensional incompressible Euler equations, we have the
following

######  Theorem 1.2.

Let $(u,\omega)$ be a solution to the 3D Euler equations on
$\mathbb{R}^{3}\times[0,T)$, namely

(1.10) $\left\\{\begin{aligned}
&\partial_{t}\omega+u\cdot\nabla\omega=\omega\cdot\nabla u,\\\ &\nabla\cdot
u=0,\quad\nabla\times u=\omega.\end{aligned}\right.$

Let $\gamma>0$, and suppose there exists $\gamma_{1}>\gamma$ such that

$\int_{0}^{t}[u(\tau)]_{1;\varphi(x_{0},\tau)}[\nabla
u(\tau)]_{{\log(\frac{1}{s})}^{-\gamma_{1}};\varphi(x_{0},\tau)}d\tau<+\infty$

holds for some $0<t<T$. Then, we have

(1.11)
$[\omega(\cdot,t)]_{{\log(\frac{1}{s})}^{-\gamma};\varphi(x_{0},t)}=[\omega_{0}]_{{\log(\frac{1}{s})}^{-\gamma};x_{0}}\quad\forall
x_{0}\in\mathbb{R}^{n}.$

In particular,

(1.12)
$\sup_{x\in\mathbb{R}^{n}}[\omega(\cdot,t)]_{{\log(\frac{1}{s})}^{-\gamma};x}=\sup_{x\in\mathbb{R}^{n}}[\omega_{0}]_{{\log(\frac{1}{s})}^{-\gamma};x}.$

##  2  The Proof of the Main Theorem

Proof of Theorem 1.1  In the proof, we assume that $u(x,t)$ is $C^{1}$–smooth
in $x$, so that the flow map $\varphi(\cdot,t)$ defines a
$C^{1}$–diffeomorphism on $\mathbb{R}^{2}$ for $0\leq t<T$. The case of
$u(x,t)\in L^{1}(0,T;Lip(\mathbb{R}^{n}))$ then follows from a simple
approximation argument.

From $\varphi(\alpha,t)=\alpha+\int_{0}^{t}u(\varphi(\alpha,t),t)ds$ and the
triangle inequality,

(2.1)
$\displaystyle|\alpha-\beta|-\left|\int_{0}^{t}\left(u(\varphi(\alpha,t),t)-u(\varphi(\beta,t),t)\right)ds\right|$
(2.2)
$\displaystyle\qquad\leq|\varphi(\alpha,t)-\varphi(\beta,t)|\leq|\alpha-\beta|+\left|\int_{0}^{t}\left(u(\varphi(\alpha,t),t)-u(\varphi(\beta,t),t)\right)ds\right|$

from which, diving it by $|\alpha-\beta|\not=0$, we have

(2.3) $\displaystyle
1-\int_{0}^{t}\frac{|u(\varphi(\alpha,s),s)-u(\varphi(\beta,s),s)|}{|\varphi(\alpha,s)-\varphi(\beta,s)|}\frac{|\varphi(\alpha,s)-\varphi(\beta,s)|}{|\alpha-\beta|}ds$
(2.4)
$\displaystyle\qquad\leq\frac{|\varphi(\alpha,t)-\varphi(\beta,t)|}{|\alpha-\beta|}$
(2.5) $\displaystyle\qquad\leq
1+\int_{0}^{t}\frac{|u(\varphi(\alpha,s),s)-u(\varphi(\beta,s),s)|}{|\varphi(\alpha,s)-\varphi(\beta,s)|}\frac{|\varphi(\alpha,s)-\varphi(\beta,s)|}{|\alpha-\beta|}ds,$

and

(2.6) $\displaystyle 1-\int_{0}^{t}\|\nabla
u(s)\|_{L^{\infty}}\frac{|\varphi(\alpha,s)-\varphi(\beta,s)|}{|\alpha-\beta|}ds$
(2.7)
$\displaystyle\qquad\qquad\leq\frac{|\varphi(\alpha,t)-\varphi(\beta,t)|}{|\alpha-\beta|}\leq
1+\int_{0}^{t}\|\nabla
u(s)\|_{L^{\infty}}\frac{|\varphi(\alpha,s)-\varphi(\beta,s)|}{|\alpha-\beta|}ds.$

By Gronwall’s inequality we obtain

(2.8)
$\frac{1}{\mu(t)}\leq\frac{|\varphi(\alpha,t)-\varphi(\beta,t)|}{|\alpha-\beta|}\leq\mu(t),\qquad\mu(t)=e^{\int_{0}^{t}\|\nabla
u(s)\|_{L^{\infty}}ds}$

for all $\alpha,\beta\in\mathbb{R}^{n}$. This shows that

$\beta\in B_{\frac{r}{\mu(t)}}(\alpha)\Rightarrow\varphi(\beta,t)\in
B_{r}(\varphi(\alpha,t))\Rightarrow\beta\in B_{\mu(t)r}(\alpha)\quad\forall
r>0,$

which implies that

(2.9) $\varphi(B_{r}(\alpha),t)\subset
B_{\mu(t)r}(\varphi(\alpha,t))\quad\text{and}\quad
B_{r}(\varphi(\alpha,t))\subset\varphi(B_{\mu(t)r}(\alpha),t).$

From this we infer

(2.10) $\sup_{\beta\in
B_{r}(\alpha)}f(\varphi(\beta,t))=\sup_{\varphi(\beta,t)\in\varphi(B_{r}(\alpha),t)}f(\varphi(\beta,t))\leq\sup_{y\in
B_{\mu(t)r}(\varphi(\alpha,t))}f(y),$

and

(2.11) $\sup_{y\in
B_{r}(\varphi(\alpha,t))}f(y)\leq\sup_{y\in\varphi(B_{\mu(t)r}(\alpha),t)}f(y)=\sup_{\beta\in
B_{\mu(t)r}(\alpha)}f(\varphi(\beta,t)).$

Combining (2.10) with (2.11), we have

$\displaystyle\lim_{r\to 0}\sup_{\beta\in B_{r}(\alpha)}f(\varphi(\beta,t))$
$\displaystyle\leq\lim_{r\to 0}\sup_{y\in
B_{\mu(t)r}(\varphi(\alpha,t))}f(y)=\lim_{r\to 0}\sup_{y\in
B_{r}(\varphi(\alpha,t))}f(y)$ $\displaystyle\leq\lim_{r\to 0}\sup_{\beta\in
B_{\mu(t)r}(\alpha)}f(\varphi(\beta,t))=\lim_{r\to 0}\sup_{\beta\in
B_{r}(\alpha)}f(\varphi(\beta,t)).$

Hence,

(2.12) $\lim_{r\to 0}\sup_{\beta\in
B_{r}(\alpha)}f(\varphi(\beta,t))=\lim_{r\to 0}\sup_{y\in
B_{r}(\varphi(\alpha,t))}f(y).$

From (2.8) one has

(2.13)
$\displaystyle\left(1+\frac{\log\mu(t)}{\log\frac{1}{|\alpha-\beta|}}\right)^{-\gamma}\leq\frac{\left(\log\frac{1}{|\varphi(\alpha,t)-\varphi(\beta,t)|}\right)^{-\gamma}}{\left(\log\frac{1}{|\alpha-\beta|}\right)^{-\gamma}}\leq\left(1-\frac{\log\mu(t)}{\log\frac{1}{|\alpha-\beta|}}\right)^{-\gamma},$

and thus

(2.14) $\lim_{r\to 0}\sup_{\beta\in
B_{r}(\alpha)}\frac{\left(\log\frac{1}{|\varphi(\alpha,t)-\varphi(\beta,t)|}\right)^{-\gamma}}{\left(\log\frac{1}{|\alpha-\beta|}\right)^{-\gamma}}=1.$

From $\theta(\varphi(\alpha,t),t)=\theta_{0}(\alpha)$ we deduce

(2.15) $\displaystyle[\theta_{0}]_{(\log\frac{1}{s})^{-\gamma},\alpha}$
$\displaystyle=\lim_{r\to 0}\sup_{\beta\in
B_{r}(\alpha)}\frac{|\theta_{0}(\alpha)-\theta_{0}(\beta)|}{\left(\log\frac{1}{|\alpha-\beta|}\right)^{-\gamma}}$
(2.16) $\displaystyle=\lim_{r\to 0}\sup_{\beta\in
B_{r}(\alpha)}\frac{|\theta(\varphi(\alpha,t),t)-\theta(\varphi(\beta,t),t)|}{\left(\log\frac{1}{|\varphi(\alpha,t)-\varphi(\beta,t)|}\right)^{-\gamma}}\frac{\left(\log\frac{1}{|\varphi(\alpha,t)-\varphi(\beta,t)|}\right)^{-\gamma}}{\left(\log\frac{1}{|\alpha-\beta|}\right)^{-\gamma}}$
(2.17) $\displaystyle=\lim_{r\to 0}\sup_{\beta\in
B_{r}(\alpha)}\frac{|\theta(\varphi(\alpha,t),t)-\theta(\varphi(\beta,t),t)|}{\left(\log\frac{1}{|\varphi(\alpha,t)-\varphi(\beta,t)|}\right)^{-\gamma}}$
(2.18) $\displaystyle=\lim_{r\to 0}\sup_{y\in
B_{r}(\varphi(\alpha,t))}\frac{|\theta(\varphi(\alpha,t),t)-\theta(y,t)|}{\left(\log\frac{1}{|\varphi(\alpha,t)-y|}\right)^{-\gamma}}$
(2.19)
$\displaystyle=[\theta(t)]_{(\log\frac{1}{s})^{-\gamma},\varphi(\alpha,t)},$

where we used (2.14) in the third equality and (2.12) in the fourth equality
respectively. $\square$

Remark 2.1  The result is sharp in the case of the 2D Euler equations in the
sense that for any $\beta>0$, there exists initial data $\omega_{0}\in
C^{\beta}(x_{0})$ with $[\omega(\cdot,t)]_{\beta;\varphi(x_{0},t)}$ growing at
least exponentially in time. To see this, we take $\omega_{0}(x)\in
C^{\beta}_{c}(\mathbb{R}^{2})$ which is odd with respect to both axes, non-
negative on the first quadrant and satisfies

$\omega_{0}(x)=\frac{2x_{1}x_{2}}{4x_{1}^{2}+x_{2}^{2}}|x|^{\beta}$

for $|x|\leq 1$. There is a unique global in time solution (with Lipschitz
continuous velocity) for any $\beta>0$. From the symmetry, $\varphi(0,t)=0$
for all $t$. For fixed $|x|\leq 1$, the maximum of $\omega_{0}(x)$ is achieved
for $2x_{1}=x_{2}$. Using the version of Kiselev-Sverak “Key Lemma” appeared
in Zlatos [7], one can prove at least for a short time interval that

$\begin{split}\varphi_{1}(x,t)\simeq
r\exp\left((\frac{C_{0}}{\beta}+O(1))t\right),\qquad\varphi_{2}(x,t)\simeq
2r\exp\left(-(\frac{C_{0}}{\beta}+O(1))t\right),\end{split}$

for all $x=(r,2r)$ with $0<r<1/10$. In particular, $|\varphi(x,t)|$ is
decreasing in time. Then, using

$\omega(\varphi(x,t),t)=\omega_{0}(x)=\frac{2x_{1}x_{2}}{4x_{1}^{2}+x_{2}^{2}}|x|^{\beta}=\frac{1}{2}(\sqrt{5}r)^{\beta}$

for $x=(r,2r)$ and

$\begin{split}[\omega(\cdot,t)]_{\beta;0}\geq\lim_{r\to
0}\frac{(\sqrt{5}r)^{\beta}}{2|\varphi(x,t)|^{\beta}},\qquad
x=(r,2r),\end{split}$

we can show local in time growth of $[\omega(\cdot,t)]_{\beta;0}$. This
argument could work to prove growth of moduli of continuity which lie
“between” Hölder and log-Hölder.

Proof of Theorem 1.3  From the vorticity form of the Euler equations

$\omega_{t}+u\cdot\nabla\omega=\omega\cdot\nabla u,\quad\nabla\times
u=\omega,\quad\nabla\cdot u=0,$

we have

$\omega(\varphi(\alpha,t),t)=\omega_{0}(\alpha)+\int_{0}^{t}(\omega\cdot\nabla)u(\varphi(\alpha,\tau),\tau)d\tau.$

By the triangular inequality

(2.20)
$\displaystyle\frac{|\omega_{0}(\alpha)-\omega_{0}(\beta)|}{\left(\log\frac{1}{|\alpha-\beta|}\right)^{-\gamma}}-\int_{0}^{t}\frac{|\omega\cdot\nabla
u(\varphi(\alpha,\tau),\tau)-\omega\cdot\nabla
u(\varphi(\beta,\tau),\tau)|}{\left(\log\frac{1}{|\alpha-\beta|}\right)^{-\gamma}}d\tau$
(2.21)
$\displaystyle\qquad\leq\frac{|\omega(\varphi(\alpha,t),t)-\omega(\varphi(\beta,t),t)|}{\left(\log\frac{1}{|\alpha-\beta|}\right)^{-\gamma}}$
(2.22)
$\displaystyle\qquad\leq\frac{|\omega_{0}(\alpha)-\omega_{0}(\beta)|}{\left(\log\frac{1}{|\alpha-\beta|}\right)^{-\gamma}}+\int_{0}^{t}\frac{|\omega\cdot\nabla
u(\varphi(\alpha,\tau),\tau)-\omega\cdot\nabla
u(\varphi(\beta,\tau),\tau)|}{\left(\log\frac{1}{|\alpha-\beta|}\right)^{-\gamma}}d\tau$

We note that

(2.23) $\displaystyle|\omega\cdot\nabla
u(\varphi(\alpha,\tau),\tau)-\omega\cdot\nabla u(\varphi(\beta,\tau),\tau)|$
(2.24) $\displaystyle\leq|\omega(\varphi(\alpha,\tau),\tau)||\nabla
u(\varphi(\alpha,\tau),\tau)-\nabla u(\varphi(\beta,\tau),\tau)|$ (2.25)
$\displaystyle\qquad+|\nabla
u(\varphi(\beta,\tau),\tau)||\omega(\varphi(\alpha,\tau),\tau)-\omega(\varphi(\beta,\tau),\tau)|$
(2.26) $\displaystyle\leq 2|\nabla u(\varphi(\alpha,\tau),\tau)||\nabla
u(\varphi(\alpha,\tau),\tau)-\nabla u(\varphi(\beta,\tau),\tau)|$ (2.27)
$\displaystyle\qquad+2|\nabla u(\varphi(\beta,\tau),\tau)||\nabla
u(\varphi(\alpha,\tau),\tau)-\nabla u(\varphi(\beta,\tau),\tau)|$ (2.28)
$\displaystyle\leq 2(|\nabla u(\varphi(\alpha,\tau),\tau)|+|\nabla
u(\varphi(\beta,\tau),\tau)|)(|\nabla u(\varphi(\alpha,\tau),\tau)-\nabla
u(\varphi(\beta,\tau),\tau)|).$

Hence, we have

(2.29) $\displaystyle\lim_{r\to 0}\sup_{\beta\in
B_{r}(\alpha)}\int_{0}^{t}\frac{|\omega\cdot\nabla
u(\varphi(\alpha,\tau),\tau)-\omega\cdot\nabla
u(\varphi(\beta,\tau),\tau)|}{\left(\log\frac{1}{|\alpha-\beta|}\right)^{-\gamma}}d\tau$
(2.30) $\displaystyle\leq 4\lim_{r\to 0}\sup_{\beta\in
B_{r}(\alpha)}\int_{0}^{t}|\nabla u(\varphi(\alpha,\tau),\tau)|\frac{(|\nabla
u(\varphi(\alpha,\tau),\tau)-\nabla
u(\varphi(\beta,\tau),\tau)|)}{\left(\log\frac{1}{|\alpha-\beta|}\right)^{-\gamma}}d\tau$
(2.31) $\displaystyle\leq 4\lim_{r\to 0}\sup_{\beta\in
B_{r}(\alpha)}\int_{0}^{t}|\nabla u(\varphi(\alpha,\tau),\tau)|\times$ (2.32)
$\displaystyle\qquad\qquad\times\frac{(|\nabla
u(\varphi(\alpha,\tau),\tau)-\nabla
u(\varphi(\beta,\tau),\tau)|)}{\left(\log\frac{1}{|\varphi(\alpha,s)-\varphi(\beta,\tau)|}\right)^{-\gamma_{1}}}\frac{\left(\log\frac{1}{|\varphi(\alpha,\tau)-\varphi(\beta,\tau)|}\right)^{-\gamma_{1}}}{\left(\log\frac{1}{|\alpha-\beta|}\right)^{-\gamma}}d\tau$
(2.33) $\displaystyle\leq 4\int_{0}^{t}[u(t)]_{1;\varphi(\alpha,\tau)}[\nabla
u]_{(\log\frac{1}{s})^{-\gamma_{1}};\varphi(\alpha,\tau)}\lim_{r\to
0}\sup_{\beta\in
B_{r}(\alpha)}\frac{\left(\log\frac{1}{|\varphi(\alpha,\tau)-\varphi(\beta,\tau)|}\right)^{-\gamma_{1}}}{\left(\log\frac{1}{|\alpha-\beta|}\right)^{-\gamma}}d\tau$
(2.34) $\displaystyle=0.$

Taking $\lim_{r\to 0}\sup_{\beta\in B_{r}(\alpha)}$ on (2.20), and
substituting (2.29),

$\lim_{r\to 0}\sup_{\beta\in
B_{r}(\alpha)}\frac{|\omega(\varphi(\alpha,t),t)-\omega(\varphi(\beta,t),t)|}{\left(\log\frac{1}{|\alpha-\beta|}\right)^{-\gamma}}=[\omega(t)]_{(\log\frac{1}{s})^{-\gamma_{1}};\varphi(\alpha,\tau)},$

we obtain (1.11). $\square$

### Acknowledgments

D. Chae was supported partially by NRF grant 2021R1A2C1003234. I.-J. Jeong was
supported by the New Faculty Startup Fund from Seoul National University, the
Science Fellowship of POSCO TJ Park Foundation, and the National Research
Foundation of Korea grant (No. 2019R1F1A1058486).

## References

  * [1] D. Gilbarg and N. S. Truginger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, (2001).
  * [2] T. Elgindi and I.-J. Jeong, Ill-posedness for the incompressible Euler equations in critical Sobolev spaces, Ann. PDE, 3 (2017), 7.
  * [3] T. Elgindi and I.-J. Jeong, Symmetries and Critical Phenomena in Fluids, Comm. Pure Appl. Math. 73 (2020), no. 2, 257–316.
  * [4] T. Elgindi and I.-J. Jeong, On Singular Vortex Patches, I: Well-posedness Issues, Memoirs of the AMS, to appear, arXiv:1903.00833.
  * [5] A. Kieselev and V. Šverák, Small scale creation for solutions of the incompressible two-dimensional Euler equation, Ann. Math. 180, 1205-1220 (2014).
  * [6] A. Majda and A. Bertozzi, Vorticity and Incompressible Flow, Cambridge University Press, Cambridge Univ. Press. (2002).
  * [7] A. Zlatos, Exponential growth of the vorticity gradient for the Euler equation on the torus, Adv. Math., 268 (2015), 396-403.