\section*{Introduction} 
The world around us is constantly changing, which makes dynamics of the artificial and natural systems at different spatial scales is of great interest to the scientific community. Traditionally, to describe a dynamical system one explores a time-dependent behaviour of a relevant physical quantity. In certain situations the choice of such a quantity is obvious, for instance description of a planetary motion is fulfilled by analyzing the trajectories, time-varying coordinates of the planets. On the other hand, in the case of the systems out of equilibrium the search for a suitable quantity could be a non-trivial task. Here, a bright primer known from the school textbooks is a microparticle in a liquid environment characterized by Brownian motion \cite{Brown}. It was found that observations over the trajectory of an individual particle does not allow to characterize the random motion, the only time-dependent mean-square displacement \cite{Einstein, Langevin} or velocity autocorrelation function averaged over many particles trajectories matter. Importantly, this fundamental finding has determined a general strategy for the analysis of non-equilibrium motion in physical systems of one or a few bodies within the standard statistical approaches or advanced machine learning techniques\cite{Granik, ASCnano}. 

\begin{figure}[!hb]
		\centering
		\includegraphics[width=0.45\textwidth]{fig1.png}
		\caption{\label{fig1} Pictorial representation of the protocol realizing the concept one process - one number. A physical process of our interest is recorded (A) and then is converted into a set of frames with one-dimensional digital representations (B) of the system at each time moment. (C) These frames are merged in a space-time grid. (D) By using the renormalization procedure with spatio-temporal filters discussed in the text we attribute the initial physical process to a single number (spatio-temporal dissimilarity) characterzing the dissimilarity of the patterns in the digital system's representation on the space-time grid.}
	\end{figure}

A wide diversity of non-equilibrium processes in classical systems stimulates permanent search for distinct measures \cite{compression1, compression2, Falk} that allow  theoretical characterization of the observed dynamics. Of particular difficulty is the description and prediction of a collective behavior of classical systems consisting of many interacting and simultaneously moving elements\cite{protein, colloid} or participants\cite{flocks, sheeps, pedestrian1,pedestrian2}. The standard choice here is the calculations of correlation functions that are non-local in both time and space for some physical quantities characterized the system in question. Depending on the situation these quantities can be taken directly from the experiment as they are (for instance, time-dependent positions of the particles obtained with a tracking system \cite{colloid,Scalliet}), or one could define some useful features \cite{Gnesotto} on the basis of available experimental data.  The problem is that even when defining the spatio-temporal correlators there is no a universal method for extracting a meaningful information from them, for instance, a signature of a hidden dynamical order. One of the possible solutions to simplify the consideration is to decouple the intertwined spatial and temporal correlations and explore them separately \cite{flocks, sheeps}. It means that potentially important information on dynamics of the system in question may be lost. In certain situations the choice of measures taken from information theory and related to entropy or complexity gives very promising results \cite{fallpaper1, fallpaper2, Mutual_info, Palmer} in solving the characterization task.  

Non-equilibrium physics is not limited to the case of the classical systems. At the moment, significant theoretical and experimental efforts are concentrated around quantum matter and phenomena out of equilibrium for which none classical analogues exist. Exploration of thermalization dynamics \cite{thermalization1}, hydrodynamic transport \cite{transport},  quantum information scrambling \cite{scrabmling_Google}, non-equilibrium phases that are formed in periodically-driven quantum systems \cite{CNN_MBL, DTC_IBM, DTC_Google}, topological order emerge in out-of-equilibrium states of programmable Rydberg atom arrays \cite{Lukin1}, quantum chaos \cite{quantum_chaos} and other non-trivial findings play a crucial role in further developing quantum technologies, information theory and condensed matter physics. In this regard, quantum simulators  and quantum computers facilitate the modeling of the non-equilibrium quantum dynamics \cite{Google_Majorana} and represent a promising alternative to classical supercomputers whose using is fundamentally limited due to the exponential growth of the Hilbert space. Similar to the classical case, theoretical analysis of non-equilibrium quantum systems is mainly based on calculations of temporal autocorrelators that are local in space and the spatial correlation functions that are local in time. Such a separate consideration of the spatial and temporal correlations inevitably leads to loss of information on interplay between space and time degrees of freedom.

In this paper we propose an alternative approach for exploring the non-equilibrium dynamics of classical and quantum systems by considering space and time degrees of freedom simultaneously and, thus, without uncoupling spatio-temporal correlations that could contain important information on the state of the dynamical system. To this aim, we follow the idea one process - one number which is schematically visualized in Fig.\ref{fig1}. It includes recording a process at given external conditions (Fig.\ref{fig1} A), converting the recorded movie into digital representation as a spatio-temporal image (Figs.\ref{fig1} B and C) and estimating patterns of the image by means of the structural complexity (dissimilarity) measure recently introduced in Refs.\cite{PNASComplexity, Dissimilarity} (Fig.\ref{fig1} D). As we will show by classical and quantum examples such a consideration of the dynamical process eases the search for critical values of the external parameters at which there are global changes in the dynamical properties of the physical system in question. Generality and flexibility in exploring completely different non-equilibrium quantum or classical systems within the same framework are important features of the proposed scheme. Despite of the fact that there is no correlation function in our work, we perform a characterization of discrete time crystal - a complex out-of-equilibrium phase of the quantum matter. For that we use the spatio-temporal dissimilarity introduced in this investigation and some measures taken from the information theory. It shows that the theoretical description of non-equilibrium dynamical processes without calculating correlation functions is possible.    

\section*{Results}
\subsection*{Spatio-temporal dissimilarity}

Our approach to the analysis of the quantum or classical dynamics out of equilibrium is formulated on the basis of the recently proposed renormalization group (RG) algorithm \cite{PNASComplexity} for estimating the structural complexity of an object. The essence of this method is to quantify dissimilarities between the patterns that occur at different spatial scales.  Such an approach was found extremely useful in context of defining equilibrium  phase transitions of various nature~\cite{PNASComplexity, Khatami} as well as verification of quantum wave functions \cite{Dissimilarity}. Since in quantum case this approach is hardly related to the complexity of a quantum state itself and operate with the array of single measurements (bitstrings) the term dissimilarity used to name the procedure\cite{Dissimilarity}. 
    
In this paper, to characterize a dynamical process we propose the spatio-temporal dissimilarity measure. The idea is based on dealing with two-dimensional patterns that occur when one puts the digital representations of the system states at different steps of its time evolution together. Having a one-dimensional discrete system of size $L$ and its evolution over $T$ time steps we can construct a matrix $B^0_{L\times T}$. At every step $k>0$ of a coarse-graining procedure we create a matrix $B^k_{L\times T}$ of the same size, whose elements are calculated as follows
    \begin{equation}\label{Coarse-graining procedure}
        b^k_{ij} = \frac{1}{\Lambda^k_x\Lambda^k_t} \sum_{l} \sum_{m} b^{k-1}_{\Lambda^k_x i+m, \Lambda^k_t j+l}.
    \end{equation} 
This means that the whole matrix is divided into blocks of $\Lambda^k_x\times\Lambda^k_t$ size (in this particular case, $k$ is a degree, not an index) and the elements within each block are substituted with its average value. In this work $\Lambda_{\rm x} = \Lambda_{\rm t} = 2$ space-time filters were used, since, as we will show below, they provide a complete description of the one-dimensional gas (classical case) and discrete time crystal (quantum case). However, for arbitrary non-equilibrium system one needs to test different spatio-temporal filters to be sure on extracting maximal useful information with dissimilarity measure. Since we are dealing with quantum or quantum-like systems, the initial matrix is filled with either $-1$ or $1$. 

After that we compute the partial spatio-temporal dissimilarities for the patterns separated by one step of RG. In case of simple averaging scheme used in our calculations, this can be done using the following equation
    
   \begin{equation}\label{D_k}
		\mathcal{D}_{\rm xt}^k = \frac{1}{2LT}\left|\left(\sum_{i,j=1}^{L,T} \left[(b^{k+1}_{ij})^2 - (b^{k}_{ij})^2 \right]\right) \right|.
   \end{equation}
	
	Finally, given features emerging at every new scale, we obtain
	\begin{equation}\label{Complexity}
		\mathcal{D}_{\rm xt} = \sum_{k=1}^{N}\mathcal{D}_{\rm xt}^k ,
	\end{equation} where $N$ is the total number of RG steps.  Thus, within this protocol each dynamical process can be associated to a single number characterizing the dissimilarity of the system's patterns in the spatio-temporal grid.
 The described procedure is then technically identical to the calculation of the structural complexity of the two-dimensional grayscale image \cite{PNASComplexity}.  

An independent choice of the initial filter sizes ($\Lambda_{\rm x}$ and $\Lambda_{\rm t}$) allows a selective quantifying either spatial or temporal patterns. For instance, if one chooses $\Lambda_{\rm x} = 1$ and $\Lambda_{\rm t} > 1$, then $\mathcal{D}_{\rm xt}$ boils down to $\mathcal{D}_{\rm t}$ that estimates temporal patterns individually for each spatial degree of freedom and then averages over them. On the other hand, for $\Lambda_{\rm x} > 1$ and $\Lambda_{\rm t} = 1$, one gets $\mathcal{D}_{\rm x}$ that probes spatial patterns in the non-equilibrium system. 

Thus, the complete set of dissimilarity functions, $\mathcal{D}_{\rm xt}$, $\mathcal{D}_{\rm t}$ and $\mathcal{D}_{\rm x}$ including their partial contributions can be used for a low-level characterization of non-equilibrium dynamics in a quantum 
 or classical system without searching for quantities that contain basic information on the system and calculating correlation functions for them.


\subsection*{1D lattice gas model}

\begin{figure}[t]
\centering
\includegraphics[width=0.8\columnwidth]{fig2.png}
\caption{\label{fig:clg_model} (Top) Schematic representation of the CLG model on a chain with periodic boundaries. At each time step one randomly chosen active particle (depicted in orange) moves to the unoccupied position. The dynamic stops when there are no active particles (t = 5). (Bottom) Examples of the time evolution of the L = 16 chain with N = 7 (adsorbing state) and N = 9 (active state) occupied sites.}
\end{figure}

\begin{figure}[t]
\centering
\includegraphics[width=1.0\columnwidth]{fig3.png}
\caption{\label{fig:1d_gas} Comparison of the $\mathcal{D}_{\rm xt}$ (red squares) and CID (blue circles) behaviour in case of CLG problem on 1D chains of different size.}
\end{figure}

To assess the viability of the spatio-temporal dissimilarity approach we first consider a simple conserved lattice gas (CLG) model in 1D~\cite{clg1, clg2} which is one of the sandpile models to study self-organised criticality. The initial state of the system in question is a random distribution of $N$ occupied sites on the $L\ge N$ chain. Particles, for which one of its neighbouring site is occupied are considered as active. Dynamics in the CLG model can be described in the following way: at each time step one should randomly choose an active particle and move it to the unoccupied neighbouring position. For $\rho = N/L \le 0.5$ the system ends up in a steady adsorbing state without active sites meanwhile for higher densities the dynamical fluctuations will take place at any time. Thus, we have a nonequilibrium phase transition with the order parameter $f_a$ denoting a fraction of active states on the final lattice configuration. Schematic representation of the CLG model as well as the examples of real time evolution of the system belonging to different phases are given in Fig.~\ref{fig:clg_model}. 

The authors of Ref.~\onlinecite{clg_cid} have shown, that the phase transition between the adsorbing and active states in this model can be easily defined in the thermodynamic limit with using a computable information density (CID). This quantity is simply equal to the ratio between the sizes of the losslessly compressed and the initial data files, containing a final state of the chain. The idea is based on the intuitive assumption that the more structured the file, the less information is needed to reproduce its content, which is similar to the concept of Kolmogorov complexity~\cite{Kolmogorov}. It was demonstrated that the method facilitates accurate description of the thermodynamic phase transitions in equilibrium magnetic systems\cite{compression2}, reveals hidden dynamical ordering in classical systems out of equilibrium \cite{compression3} and helps to solve other interesting problems \cite{compression4, compression5}. Importantly, an effective implementation of the compression algorithm assumes the possibility to generate large amount of data and the data itself should fit a dictionary with discrete set of items.

\begin{figure}[t]
\centering
\includegraphics[width=1.0\columnwidth]{fig4.png}
\caption{\label{fig:d_avg} Comparison of the $\mathcal{D}_{\rm xt}$ (purple triangles), $\mathcal{D}_{\rm x}$ (orange squares) and $\mathcal{D}_{\rm t}$ (green circles) behaviour in case of CLG problem on 1D chain with $L=64$, $T=1024$. The results are averaged over 100 independent starting configurations at each density.}
\end{figure}

Here we compare $\mathcal{D}_{\rm xt}$ with CID in case of small chains with up to $L = 256$ sites. As can be seen from Fig.~\ref{fig:1d_gas}, the maximum value of dissimilarity is not sensitive to the size of the considered system. In turn, CID is not only exhibit strong fluctuations, but also lies in a significantly different range depending on $L$. Here we used LZMA compression algorithm as it is one of the extensions of LZ77 used in Ref.~\onlinecite{clg_cid} and shows the best results among other standard techniques. When decreasing the system size the amplitude of $\mathcal{D}_{\rm xt}$ drops a bit for $\rho > 0.5$. This is due to the rearrangement of the partial contributions to dissimilarity from different scales $\mathcal{D}_{\rm xt}^k$ (see Fig.~\ref{fig:partial_lg} in Appendix A) and the fact that we do not consider $k = 0$ step. It should be mentioned, that to get smooth graph with zero value at $\rho = 0.5$ we still have to consider sufficient number of time steps such that the dynamics at the critical point stops.   

The deeper analysis of the dissimilarity (see Fig.~\ref{fig:d_avg}) shows that the behaviour of $\mathcal{D}_{\rm xt}$ is similar to the fully spatial case $\mathcal{D}_{\rm x}$. Moreover, the slight differences occur only in active phase when the temporal part is high enough. Therefore, spatial patterns in digital representation provide most of meaningful information about the system. Nevertheless, having only the $\mathcal{D}_{\rm t}$ one can still define a phase transition analysing the derivative behaviour, as it was proposed in Ref.~\onlinecite{PNASComplexity}.

\subsection*{Patterns of discrete time crystal}

The main focus in our paper is on the classification of dynamical phases realized in driven non-equilibrium systems of qubits. More specifically, we are interested in exploring  the properties of a periodically driven quantum system hosting the discrete time crystal (DTC) state. As it was shown in the previous works\cite{DTC1,DTC2,DTC3} DTC is formed as a result of  breaking discrete time translation symmetry in a Floquet system with time-periodic Hamiltonian and retaining memory of the initial state indefinitely. Experimentally, it can be realized in systems of superconducting qubits \cite{DTC_IBM, DTC_Google, Khemani}, but other platforms, for instance quantum-optical systems \cite{Fedorov} are also promising candidates for observing this non-trivial state. The main signature of DTC, the spatio-temporal order can be detected and probed with the combination of temporal autocorrelators and qubit-qubit spatial correlation functions. As it was demonstrated theoretically \cite{Khemani} and experimentally\cite{DTC_IBM, DTC_Google}, these correlation functions can be defined using the results (bitstrings) of the numerous projective measurements performed in a number of consecutive Floquet cycles during the evolution. In this work, by using the developed spatio-temporal dissimilarity measure we show that one can perform a complete low-level characterization of DTC with a limited number of bitstrings, directly without calculating correlation functions.

    \begin{figure}
		\centering
		\includegraphics[width=0.45\textwidth]{fig5.png}
		\caption{\label{fig:DTCcircuitInUniversalGates} Example of DTC quantum circuit for 4-qubit system where single-qubit rotational gates ($X_\epsilon=R_x((1-\epsilon)\pi)$, $Z_{J}=R_z(J_i)$, $Z_h=R_z(h_i)$) and CNOT-gate were used.}
	\end{figure}
    
Time evolution of the quantum system needed to stabilize non-equilibrium discrete time crystal state is described by the Floquet unitary $U_{\rm F}$ \cite{DTC_Google}:
\begin{multline} 
\label{U_F}
        U_{\rm F} = 
        e^{ \scalebox{1.0}{$-\frac{i}{2} \left(1-\epsilon\right) \pi \sum\limits_{i} X_i$} }   \\
        \cdot e^{\scalebox{1.0}{$-\frac{i}{2} \sum\limits_{i<j} J_{ij} Z_i Z_{j} -\frac{i}{2} \sum\limits_{i} h_i Z_i $ }}
\end{multline}
which is defined in term of two unitaries.
The first one corresponds to the imperfect global spin-flip (imperfect $\pi$-pulse). Here $X_i$ is the Pauli X-gate acting on the $i$-th qubit, $\epsilon\in[0;0.5]$ is the spin-flip imperfection parameter. The second unitary represents a combination of a random nearest-neighbor Ising interactions, $J_{ij}$ and a random longitudinal field, $h_i$. Here $Z_i$ is the Pauli Z-gate on the $i$-th qubit. Following to Ref.~\onlinecite{DTC_Google} we use a trotterized time evolution to simulate the DTC with a quantum simulator. 

In Fig.~\ref{fig:DTCcircuitInUniversalGates} we show a fragment of the quantum circuit that corresponds to one trotter cycle for a 4-qubit system driven with Floquet unitary described above. It comprises rotation single-qubit gates and entangled CNOT gates. To construct a complete quantum circuit that imitates periodically driven quantum system for a given number of time steps we randomly choose  nearest neighbors interaction parameters $J_{ij}$ and local magnetic fields $h_i$ from the ranges $\left[-0.75\pi;-0.25\pi\right]$ and $\left[-\pi;\pi\right]$, respectively. These parameters are fixed for each quantum circuit instance independent on its depth. Wherein the imperfect spin-flip, $\epsilon$ plays a role of a global parameter whose value allows to choose a particular non-equilibrium phase. To be more concrete, $\epsilon$ is varied within [0, 0.5]. In our work we perform simulations of the DTC on the 16-qubit one-dimensional system with open boundaries.

\begin{figure}[!t]
	\centering
	\includegraphics[width=0.45\textwidth]{fig6.pdf}
	\caption{Examples of bitstring evolution within the time window [0, 63] for particular circuits constructed with five representative values for the $\epsilon$ parameter. \label{bitstrings}}
\end{figure}

While the previous works devoted to the DTC were mainly focused on the analysis of different forms of the spin polarization, here we show that the non-trivial state can be fully described at a more basic level of bitstrings. From 
Fig.~\ref{bitstrings} it follows that there are two different types of bitstring temporal evolution depending on the parameter of imperfect rotation. At small values of $\epsilon$ one can observe period-doubled oscillations between initial bitstring and its flipped counterpart. At some times the projective measurement shows slightly modified bitstrings that differ from the initial one by a few bits. The latter means that quantum state becomes slightly delocalized in the Hilbert space with respect to the initial trivial state. At the same time, there are nearly perfect oscillations at the level spin polarization \cite{Khemani} within the DTC phase, when each spin is fully flipped. Thus, although from the point of view of structure, the quantum state changes during the evolution due to disorder, nevertheless it reproduces the period-doubled dynamics in terms of the correlation functions local in space, which is a signature of DTC.   

As $\epsilon$ increases, bit-string patterns are becoming more and more diverse over time, which means that different basis functions are involved and the delocalization of the quantum state increases. For $\epsilon = 0.3$ we observe a complete loss of information about the initial configuration.   

\begin{figure}[!b]
	\centering
	\includegraphics[width=0.40\textwidth]{fig7.pdf}
	\caption{ Probability distribution of the Hamming distances $P_n(d)$ computed between the initial state and bitstrings measured at time steps 62 (blue) and 63 (orange) of the Floquet evolution. The histograms are obtained using 512 independent circuits.}
	\label{Hamming}
\end{figure}

To quantify the bitstring dynamics described above we employ the Hamming distance that measures the number of bits by which one bitstring differs from another and is defined as $d$. Firstly, the probability distribution of Hamming distances, $P_{n} (d)$ between the initial and time-evolved states is calculated. However, in contrast to  Ref.~\onlinecite{Khemani} for our analysis we use a limited number of measurements, which doesn't allow to reconstruct the complete probability of the quantum system at each time step. Dependence of $P_{n}(d)$ on the parameter of imperfect rotation presented in Fig.~\ref{Hamming} unveils two boundary cases. The first one is observed at small $\epsilon$ and represents a signature of the DTC phase according to the results of Ref.~\onlinecite{Khemani}. The probability distributions of the Hamming distances calculated between the initial state and two consecutive states obtained after a moderate number of time-evolution steps are localized at $d = 0$ and $d = 16$.  Another distinct regime is revealed for $\epsilon \ge 0.23$. The calculated $P(d)$ is centered at $d = 8$ for even and odd time steps, which indicates the thermal phase. While DTC and thermal regimes of dynamics can be identified with the probability distribution of the Hamming distance, the transition between them on the level of the calculated $P(d)$ is smooth and doesn't show any feature that can be associated to the critical point. The presented distributions are not symmetrical with respect to $d = \frac{L}{2}$, since the number of bitstrings we used is much smaller than the total size of the Hilbert space for 16-qubit system. In Appendix B we complete the picture of the non-equilibrium dynamics in the considered quantum system by analyzing the behaviour of the Shannon and von Neumann entropies.

As it was shown previously~\cite{Khemani}, the transition point between DTC and thermal phases can be defined with the Edwards-Anderson spin glass order parameter \cite{Anderson}, which represents a total spin-spin correlation function averaged over several time steps and different instances of disorder. In our work to explore a dynamical order and define critical value of $\epsilon$ we use the developed $\mathcal{D}_{\rm xt}$ measure that allows to define the critical point directly from the spatio-temporal representation of bitstrings. The key moment of our scheme is that at each time step of the quantum evolution we perform only one measurement of all the qubits and the resulting bitstrings obtained from several consecutive time steps are used to form qubit-time grid. For each value of $\epsilon$ the dissimilarity $\mathcal{D}_{\rm xt}$ was averaged over 512 realization of disorder (circuit instances) each of which assumes a random initial state. We have found that the minimal number of the time cycles required for calculating spatio-temporal dissimilarity in the case of DTC is equal to 16. Moreover, an increase in this number does not change the $\mathcal{D}_{\rm xt}$ estimates. In Appendix C we discuss different averaging schemes with a fixed disorder or a fixed initial state. 

The resulting dissimilarities are given in Fig.~\ref{dissimilarity}. One can clearly distinguish between discrete time crystal and thermal phases by analyzing the behaviour of the calculated $\mathcal{D}_{\rm xt} (\epsilon)$. In the DTC region the dissimilarity demonstrates a rapid growth with increasing $\epsilon$, which can be explained by the fact that the target wave function becomes more delocalized in the Hilbert space. In turn, in the thermal phase $\mathcal{D}_{\rm xt}$ is independent on this parameter and shows nearly zero slope. The critical value of the imperfect rotation parameter can be found as a crossing point of the corresponding linear fits \cite{glass1} of the results deep inside the DTC and thermal phases. Dependence of the $\epsilon_c$ on the system's size presented in Fig.\ref{dissimilarity} bottom shows the stabilization of the critical value for $L \geq 16$. The obtained $\epsilon_c\approx0.16$ is in good agreement with theoretical estimations based on the Edwards-Anderson order parameter\cite{Khemani} and lies within the range of experimental estimates~\cite{DTC_Google}.

The results presented in Fig.\ref{dissimilarity} were obtained for random initial states of zero entanglement, which is a standard setup widely used in the literature devoted to the DTC problem. In Appendix D we discuss the simulations of the DTC phase with different entangled initial states.

As it was shown in the classical case for one-dimensional CLG the exploration of the non-equilibrium critical behaviour with spatio-temporal dissimilarity can be successfully performed using much less data about the system in comparison with other measures. Thus, in the quantum case it is instructive to compare the total number of measurements required for characterization of DTC with $\mathcal{D}_{\rm xt}$ and standard order parameter of the spin glass state used in previous works \cite{Khemani}. To get one point on the graph presented in Fig.~\ref{dissimilarity} we use 8192 measurements (number of time steps $\times$ number of disorders). In turn, to estimate the glass order parameter for the given $\epsilon$ in the real experiment the authors of Ref.~\onlinecite{DTC_Google} have performed 1600000 measurements which is mainly caused by noise and gates imperfections. In contrast, theoretical results on the correlation functions presented in the same work for the system of 16 qubits were obtained with 40000 measurements, which is still considerably larger than  those needed to calculate the dissimilarity. 

\begin{figure}
	\centering
	\includegraphics[width=0.45\textwidth]{fig8.pdf}
	\caption{ (Top) Dependence of the spatio-temporal dissimilarity on the critical imperfect rotation parameter. Blue squares and red circles indicate the internal regions of the DTC and thermal phases, respectively. Dashed blue and red lines show the linear fittings for the corresponding points. Green dash-dotted line denotes the value of $\epsilon_c$ corresponding to the DTC-thermal transition. (Bottom) Dependence of the $\epsilon_c$ on the system size $L$. These results were obtained with 4096 ($L$=8), 6144 ($L$=12), 8192 ($L$=16), 4000 ($L$=20) and 4800 ($L$ =24) measurements. } \label{dissimilarity}
\end{figure}

\section*{Discussion and Outlook}
We propose a flexible and simple approach for diagnosing non-equilibrium dynamics in classical and quantum systems. It allows identifying different dynamical regimes including hidden order out of equilibrium and accurate detecting critical parameters. Taking a time-resolved digital representation of a dynamical process as input, our method quantifies the variety or dissimilarity of spatio-temporal patterns by representing it as a single number. Thus, distinguishing different parts of the same process or different dynamical processes boils down to comparison of the corresponding numbers which considerably facilitates and accelerates characterization of non-equilibrium systems and constructing phase diagrams. 

By using the proposed spatio-temporal dissimilarity measure we have analyzed non-equilibrium dynamics of one-dimensional classical and quantum models. In classical case we show that  $\mathcal{D}_{\rm xt}$ reveals a hidden order in one-dimensional conserved lattice gas model for much smaller system's sizes than in the case of the compression measure. In turn, non-trivial dynamically driven discrete time crystal phase in a non-equilibrium quantum system can be fully described on the level of bitstrings without calculating standard correlation functions, which also substantially reduces the total number of the measurements.   

In this work to extract information on a non-equilibrium one-dimensional system we explore two-dimensional spatio-temporal grid formed with digital representations of the system in question measured at different times. To consider higher dimensional physical systems, two different strategies can be adopted. The first one assumes using one-dimensional representation for spatial degrees of freedom of the high-dimensional system at each time step. As it was shown previously, in Ref.~\onlinecite{Dissimilarity} such a dimensional reduction in space still allows an accurate description of equilibrium phase transitions in magnetic systems, however, it assumes testing different numeration schemes to order spatial degree of freedom and to mimic the topology of the initial system in one-dimension representation. Another possibility is to use high-dimensional spatio-temporal filters when calculating the dissimilarity, $\mathcal{D}_{\rm xt}$. 

The dissimilarity method supports processing digital representations of completely different dynamical processes. For instance, in this work we are concentrated on the characterization of the non-equilibrium quantum systems for which measurement's outputs are given in the binary format, bitstrings that consist of 0 and 1 (-1 and 1 in our calculations). Vector fields with floating-point type components are another important example of a digital representation that can be handled with dissimilarity\cite{PNASComplexity}, which paves the way to characterize the dynamics of off- and in-lattice models. From the perspective of real experiments one can calculate spatio-temporal dissimilarity directly for a stream of experimental images in RGB or other formats without mapping onto a mathematical model. This can be rationalized in non-scientific applications, for instance as a pre-processing scheme in automatic video content classification systems.

\section*{Acknowledgements}
The research funding from the Ministry of Science and Higher Education of the Russian Federation (Ural Federal University Program of Development within the Priority-2030 Program) is gratefully acknowledged. Quantum simulations were performed on the Uran supercomputer at the IMM UB RAS. 

\section*{Appendix A: Partial dissimilarity for CLG}\label{AppDK}
\begin{figure*}[t]
\centering
\includegraphics[width=.85\textwidth]{fig9.png}
\caption{\label{fig:partial_lg} The behavior of $\mathcal{D}_{\rm xt}^k$ for configurations corresponding to the maximum $\mathcal{D}_{\rm xt}$ in adsorbing (green squares) and active (orange circles) phases for $L=8$ (a) and $L=256$ (b). (c) Dependency of $\mathcal{D}_{\rm xt}^0$ on the density of particles in the system for $L=8$ (red diamonds) and $L=256$ (purple triangles).}
\end{figure*}

As one can see from Fig.~\ref{fig:1d_gas}, the total $\mathcal{D}_{\rm xt}$ for the CLG problem exhibits a symmetric behaviour close to the one presented in Ref.~\onlinecite{clg_cid} for the thermodynamic limit. However, unlike the CID, which is just a single value, we can still define the phase of a considered configuration by analyzing its partial spatio-temporal dissimilarities. According to Fig.~\ref{fig:partial_lg}, configurations corresponding to the active phase have monotonously decreasing $\mathcal{D}_{\rm xt}^k$ profiles while the ones from adsorbing phase have more complex $\mathcal{D}_{\rm xt}^k$ behaviour. Moreover, the $k=0$ step can be used as an indicator of how close we are to the phase transition point.  

\section*{Appendix B: Analysis of the DTC entanglement entropy}
Hamming distance analysis as well as dissimilarity consideration presented in the main text are based on the results of a limited number of measurements, which is clearly not enough to reconstruct the true probabilities of the basis functions of the quantum system at each cycle of the Floquet unitary evolution. To complete the picture of non-equilibrium dynamics of the quantum system in question, we analyze time-dependent behaviour of the information theory measures, Shannon entropy and von Neumann entanglement entropy. For a given quantum state the Shannon entropy is defined as
\begin{eqnarray}
H(X) = -\sum_{n} p_n \log_2 p_n,
\end{eqnarray}
where $p_n$ is the probability of the $n$th basis function. The obtained time dependence of $H(X)$ are visualized in Fig.\ref{fig:EntangledStates} left. One can see that the Shannon entropy calculated at different $\epsilon$ demonstrates a fast growth and saturation for time steps $t > 20$. It corresponds to a delocalization of the quantum state in the Hilbert space. For instance, in the case of $\epsilon = 0.03$ such a delocalization can be qualified as moderate. However, from the Shannon entropy alone calculated in single $\sigma^z$ basis it is not clear whether the initial trivial state transforms into an entangled one during the evolution or it remains trivial. To clarify this point we have calculated the von Neumann entropy, $S_N = -{\rm Tr} \rho_{\rm A} \log_2 \rho_{\rm A}$, where $\rho_{\rm A} = {\rm Tr}_{\rm B} \rho_{\rm AB}$ is the reduced density matrix for a half-system biparition into parts A and B. It is found that for $\epsilon = 0.03$ the quantum system remains non-entangled during the whole evolution time. For larger values of $\epsilon$ non-equilibrium dynamics creates entanglement between spatial subsystems. The thermal phase is characterized by a very rapid growth of the von Neumann entropy with time cycles. For $\epsilon=0.30$ the Shannon entropy and the von Neumann entropy reach the values of 15.35 and 7.22, respectively. It means that at this $\epsilon$ the wave function of the system is close to the Haar random state that is characterized by the ''Page value'' $\log_2 2^{\frac{L}{2}} - \frac{1}{2}$ for von Neumann entropy \cite{Page} and $\log_2 2^{L} - 1 + \gamma$ for the Shannon entropy (here $\gamma$ is Euler constant) \cite{supremacy}. Similar to the Shannon entropy we cannot discriminate DTC-thermal transition analyzing the calculated time dependencies of $S_N$.  

	\begin{figure}
		\centering
		\includegraphics[width=0.45\textwidth]{fig10.pdf}
		\caption{\label{fig:EntangledStates} Time evolution of the Shannon entropy $H(X)$ (left) and entanglement entropy $S_N$ (right) calculated at different values of the $\epsilon$ parameter.}
	\end{figure}


\section*{Appendix C: Averaging DTC data}

The results of the dissimilarity calculations used for constructing the phase diagram, Fig.\ref{dissimilarity} in the main text are averaged over initial states as well as disorder instances. In this section we perform a separate consideration of these two kinds of averaging. Fig.\ref{average} A gives the spatio-temporal dissimilarities calculated with fixed initial states of three types and only averaged over disorders. More specifically we consider the polarized state $\ket{\Psi_{\rm polar}} = \ket{0000..00}$, random wave function $\Psi_{\rm R}$ for which the state of each qubit is randomly chosen between 0 and 1, and the N\'eel state $\ket{\Psi_{\rm Neel}} = \ket{0101..01}$  One can see that each initial state results in a distinct dependence $\mathcal{D}_{\rm xt}(\epsilon)$. The critical $\epsilon$ parameters are slightly different within this series of experiments and vary from 0.15 to 0.20.     

 	\begin{figure}
		\centering
		\includegraphics[width=0.45\textwidth]{fig11.pdf}
		\caption{\label{average} (A) Spatio-temporal dissimilarities calculated with different initial wave functions: N\'eel (green), random (orange) and polarized (blue) states. In each case $\mathcal{D}_{\rm xt}$ is averaged over 512 disorder instances. (B) Examples of the $\mathcal{D}_{\rm xt} (\epsilon)$ functions calculated for three different instances of disorder. Each curve is averaged over 512 trivial initial states.}
	\end{figure}
In turn, comparison of the dissimilarities obtained with different disorders and averaged over 1024 random initial states are presented in Fig.\ref{average} B. Variation of disorder gives rise to a slight difference in behaviour between the calculated dissimilarities, which also affect the estimation of the critical point in each case. Based on these results we can conclude that the spatio-temporal dissimilarity measure demonstrates a rather robust behaviour for different choices of the averaging data procedure.  

\section*{Appendix D: Initial entangled states for DTC}
The use of the time-dependent spin polarization of qubits as indicator of the dynamical state of the system is not well defined if the initial state of the quantum system is non-trivial and characterized by non-zero entanglement. In this case the initial zero averaged magnetization of individual qubits can challenge probing dynamical properties of the quantum system. A possible solution of such a problem was recently demonstrated in Ref.~\onlinecite{DTC_Google}, where the authors used an ancilla qubit to confirm the DTC-MBL dynamics of the quantum systems initially prepared in the highly-entangled Haar-random state with zero magnetization. 
    
Our dissimilarity approach is suitable for extracting useful information on the unitary dynamics of the zero-magnetization entangled quantum states without introducing additional ancilla-qubits. To demonstrate this we use Dicke and Schr\"odinger cat
wave functions of different entanglement as initial states for unitary dynamics with the DTC protocol. The Dicke quantum state is defined as
\begin{eqnarray}
|\Psi_D \rangle = \frac{1}{\sqrt{C^N_D}} \sum_{j} P_{j}(|0\rangle^{\otimes N-D} \otimes |1\rangle^{\otimes D}),
\label{eq:Dicke_wf}
\end{eqnarray}
where the sum goes over all possible permutations of qubits. The parameter $D$ controls the number of 1's in each basis function contributing to $\Psi_D$. 

In turn, the Schr\"odinger cat state is controlled with the angle $\theta$,
\begin{eqnarray}
|\Psi_{\theta} \rangle = {\rm cos}(\frac{\theta}{2}) |0\rangle^{\otimes N} + {\rm sin}(\frac{\theta}{2}) |1\rangle^{\otimes N}.
\end{eqnarray} 
For $\theta = 0$ or $\pi$ one deals with trivial states discussed in the main text. 
Comparing the structures of the Dicke and cat states one can consider them as two limiting examples of the localized wave functions. Such a difference in the basis functions contributing to the quantum state probability manifests itself not only at the level of entanglement entropy, but also in the partial dissimilarities calculated in Ref.\onlinecite{Dissimilarity}.

As representative examples of the entangled quantum states that are characterized by zero local qubit polarization  we consider the Dicke state with $D= 8$ and Schr\"odinger cat with $\theta = \frac{\pi}{2}$ that are used as initial ones in the DTC protocol. Fig.\ref{entangled} illustrates the obtained $\mathcal{D}_{\rm xt}$ as functions of $\epsilon$. In contrast to non-equilibrium dynamics initialized with different trivial wave functions (presented in the main text), using the initial entangled states results in a more diverse behaviour of the $\mathcal{D}_{\rm xt} (\epsilon)$ functions  within the DTC part of the phase diagram. 

\begin{figure}[!hb]
	\centering
	\includegraphics[width=0.45\textwidth]{fig12.pdf}
	\caption{\label{entangled} Comparison of the spatio-temporal dissimilarities calculated with different initial entangled states within DTC protocol. Each curve is obtained with averaging over 512 realizations of disorder. }
\end{figure}

The main peculiarity of the spatio-temporal dissimilarity for the Schr\"odinger cat state is a large standard deviation at small values of the imperfect rotation parameter. The deviation gradually decreases as approaching to $\epsilon$ = 0.25. Such strong fluctuations of the calculated dissimilarities are mainly due to smallness of the space-time grid (16$\times$16) used in our study. As the result, each of 1024 circuits produces the $B^{0}_{L \times T}$ matrix that contains a distinct sequence of 0000..00 and 1111..11 bitstrings. It can be rationalized by considering the bitstings examples presented in Fig.\ref{cat_bitstrings}. For $\epsilon = 0.01$ one could randomly choose a fragment of 16 consecutive bitstrings. Each of such fragments would be characterized by different number of ''0'' and ''1'' bits, which explains fluctuations of the polarization and dissimilarity.     

\begin{figure}[!h]
	\centering
	\includegraphics[width=0.45\textwidth]{fig13.pdf}
	\caption{Examples of bit-string sequence obtained in a time window [0, 63] with Schr\"odinger cat state $\Psi_{\theta=\pi/2}$ as initial state of the qubits system. \label{cat_bitstrings}}
\end{figure}

In turn, the bitstrings corresponding to the Dicke state constitutes of the same number of the ''0'' and ''1'' bits, which together with the large number of the basis functions that contribute to $\Psi_{D}$ (Eq.\ref{eq:Dicke_wf}) results in nearly constant profile of the mean spatio-temporal dissimilarity and the corresponding standard deviation within the working area. A zoom view (inset of Fig.\ref{entangled}) revealed an oscillating dissimilarity that, nevertheless, reproduces the main trends of the $\mathcal{D}_{\rm xt} (\epsilon)$ function calculated for the trivial initial state.  


