CERN Accelerating science

\justifying Summary of the workflow: the left panel illustrates the graphical representation of the one-dimensional \ac{annni} model (Eq. \ref{eq:H}) and its phase diagram. For the middle panel, we stress the use of \ac{tn} as a tool to get the phase diagram of the model applying finite-size scaling analysis and as a source of the quantum state for the next level. The right panel showcases the \ac{qml} analysis, with input data sourced from the DMRG, to classify different phases of the model.

\justifying Architecture of the \ac{qcnn}: Input state preparation (blue), then alternating Convolution layers (green) and Pooling layers (red) and a Fully Connected layer (dark green) at the end \cite{Monaco_2023}

\justifying Architecture of the Anomaly detection circuit (yellow) and input state preparation (blue). An example architecture for 6 qubits, with half designated \textit{trash} qubits positioned at the center. \cite{Monaco_2023}

\justifying Phase diagram of the quantum one-dimensional \ac{annni} model of Eq. \ref{eq:H}. The dotted line is the exactly-solvable Peschel-Emery (PE) line $h=1/4\kappa - \kappa$ \cite{peschel1981calculation}. Dashed lines represent Ising (I), Kosterlitz-Thouless (KT), and Pokrovsky-Talapov (PT) phase transitions. The spin chain can be seen as a ladder of two spin chains as sketched in the cartoon spin configurations.

\justifying Inverse of the correlation length computed from $C_{i,j}^{zz}$ as a function of $\kappa$ along the horizontal cut at $h=0.5$ obtained on a finite-size system with $N=240$ sites with open boundary conditions. Colors indicate different phases: the ferromagnetic phase (light green), the paramagnetic phase (dark green), the floating phase (light blue), and the antiphase (dark blue) from left to right. Phase transitions according to previous results \cite{peschel1981calculation,suzuki2012quantum,beccaria2007evidence} are also shown by dashed lines: Ising (I) transition, Kosterlitz-Thouless (KT) transition, and Pokrovsky-Talapov (PT) transition. The dotted line is the exactly solvable Peschel-Emery (PE) line \cite{peschel1981calculation}.

\justifying (Up) Magnetization along $x$ (see Eq. \ref{eq:magnetization}) obtained on four finite-size systems ($N=60,120,240,480$) with open boundary conditions along the horizontal cut at $h=0.5$. As expected, magnetization in the ferromagnetic phase is non-zero and in the paramagnetic phase is zero. Note that the phase transition approaches the perturbative analysis prediction as the length of the chain increases. (Down) Binder's cumulant (see Eq. \ref{eq:Binder}) for $\kappa$-values in the surroundings of the critical point extracted form Eq. \ref{eq:I_transition}.

\justifying Friedel oscillation pattern inside the floating phase for $h=0.5$ $\kappa=0.900$ obtained on a finite-size system with $N=240$ sites with open boundary conditions. Green points are \ac{dmrg} data and blue circles are the result of the fit with Eq. \ref{eq:Friedel} (note the very accurate agreement). The uniform part of the spin-density $\overline{\langle\sigma^{z}_{j}\rangle}$ has been subtracted (see Appendix \ref{ap: Luttinger_parameters}).\justifying Extraction of Luttinger liquid parameters from Friedel oscillations profile. This is the Friedel oscillation pattern for $h=0.5$ inside the floating phase obtained on a finite-size system with $N=240$ and open boundary conditions. Green points are \ac{dmrg} data and blue circles are the result of the fit with Eq. \ref{eq:Friedel} (all green points are inside blue circles). Note that the uniform part of the spin density $\overline{\langle\sigma^{z}_{j}\rangle}$ has been subtracted, and the fitting window is restricted to the range $j\in[48,191]$ to avoid edge effects. The root mean squared error (`RMSE') obtained is $\approx 10^{-5}$.

\justifying Incommensurate wave-vector $q$ obtained on a finite-size system with $N=240$ sites with open boundary conditions along the horizontal cut at $h=0.5$. As it is shown, incommensurability doesn't vanish in the paramagnetic phase. Therefore, the \ac{kt} transition is an incommensurate-incommensurate phase transition.

\justifying (a) Luttinger liquid exponent $K$ and (b) incommensurate wave-vector $q$ obtained on a finite-size system with $N=240$ sites with open boundary conditions along the horizontal cut at $h=0.5$. The \ac{kt} transition appears when $K=1/2$ and the \ac{pt} transition arises when incommensurability vanishes. The solid line in (b) is a fit assuming the \ac{pt} critical exponent $\nu=1/2$.

\justifying Entanglement entropy with conformal distance $d_N(n)$ at $h=0.5$ and $\kappa=0.814$ for a finite-size system with $N=240$ sites for open boundary conditions. The obtained value for the central charge in the critical point agrees within $3.6\%$ with the \ac{cft} prediction $c=1$ for a Luttinger liquid.

\justifying Predictions of the \acp{qcnn} trained on the analytical points of the \ac{annni} Spin Model at different system sizes: $N=12$ (left), $N=16$ (middle), and $N=20$ (right). Colors represent Ferromagnetic (light green), Paramagnetic (dark green), Antiphase (dark blue), and Floating Phase (light blue) as a function of the external magnetic field ($h = B/J_1$) and interaction strength ratio ($\kappa=-J_2/J_1$) (refer to eq. \ref{eq:H}).

\justifying Predictions of the \acp{qcnn} trained on the analytical points of the \ac{annni} Spin Model at different system sizes: $N=12$ (left), $N=16$ (middle), and $N=20$ (right). Colors represent Ferromagnetic (light green), Paramagnetic (dark green), Antiphase (dark blue), and Floating Phase (light blue) as a function of the external magnetic field ($h = B/J_1$) and interaction strength ratio ($\kappa=-J_2/J_1$) (refer to eq. \ref{eq:H}).

\justifying Predictions of the \acp{qcnn} trained on the analytical points of the \ac{annni} Spin Model at different system sizes: $N=12$ (left), $N=16$ (middle), and $N=20$ (right). Colors represent Ferromagnetic (light green), Paramagnetic (dark green), Antiphase (dark blue), and Floating Phase (light blue) as a function of the external magnetic field ($h = B/J_1$) and interaction strength ratio ($\kappa=-J_2/J_1$) (refer to eq. \ref{eq:H}).

\justifying Predictions of the \acp{qcnn}, trained on a subset of points from each phase of the \ac{annni} model at various system sizes: $N=12$ (left), $N=16$ (middle), and $N=20$ (right). The color scheme indicates Ferromagnetic (light green), Paramagnetic (dark green), Antiphase (dark blue), and Floating Phase (light blue), as a function of the external magnetic field ($h = B/J_1$) and interaction strength ratio ($\kappa=-J_2/J_1$) (refer to eq. \ref{eq:H}).

\justifying Predictions of the \acp{qcnn}, trained on a subset of points from each phase of the \ac{annni} model at various system sizes: $N=12$ (left), $N=16$ (middle), and $N=20$ (right). The color scheme indicates Ferromagnetic (light green), Paramagnetic (dark green), Antiphase (dark blue), and Floating Phase (light blue), as a function of the external magnetic field ($h = B/J_1$) and interaction strength ratio ($\kappa=-J_2/J_1$) (refer to eq. \ref{eq:H}).

\justifying Predictions of the \acp{qcnn}, trained on a subset of points from each phase of the \ac{annni} model at various system sizes: $N=12$ (left), $N=16$ (middle), and $N=20$ (right). The color scheme indicates Ferromagnetic (light green), Paramagnetic (dark green), Antiphase (dark blue), and Floating Phase (light blue), as a function of the external magnetic field ($h = B/J_1$) and interaction strength ratio ($\kappa=-J_2/J_1$) (refer to eq. \ref{eq:H}).

\justifying Compression Scores $\mathcal{C}$ of the \ac{ad} circuits trained on the $(\kappa, h) = (0, 0)$ point of the \ac{annni} model phase diagram at different system sizes $N$: 6 (left), 12 (middle), and 18 (right). The scores are showcased as a function of the interaction strength ratio ($\kappa = -J_2/J_1$) and the external magnetic field ($h = B/J_1$). Lower compression scores indicate better disentanglement of trash qubits from others, as defined by eq. \ref{eq:compress}.

\justifying Compression Scores $\mathcal{C}$ of the \ac{ad} circuits trained on the $(\kappa, h) = (0, 0)$ point of the \ac{annni} model phase diagram at different system sizes $N$: 6 (left), 12 (middle), and 18 (right). The scores are showcased as a function of the interaction strength ratio ($\kappa = -J_2/J_1$) and the external magnetic field ($h = B/J_1$). Lower compression scores indicate better disentanglement of trash qubits from others, as defined by eq. \ref{eq:compress}.

\justifying Compression Scores $\mathcal{C}$ of the \ac{ad} circuits trained on the $(\kappa, h) = (0, 0)$ point of the \ac{annni} model phase diagram at different system sizes $N$: 6 (left), 12 (middle), and 18 (right). The scores are showcased as a function of the interaction strength ratio ($\kappa = -J_2/J_1$) and the external magnetic field ($h = B/J_1$). Lower compression scores indicate better disentanglement of trash qubits from others, as defined by eq. \ref{eq:compress}.

\justifying Extraction of the correlation length $\xi$ from correlations in the $z$ direction. These transverse correlations are obtained for $h=0.5$ inside the floating phase on a finite-size system with $N=240$ and open boundary conditions. The green points are \ac{dmrg} data and the blue line is the result of the fit with Eq. \ref{eq:OZ_equation}. Oscillations have been removed to fit the main slope of the decay. Note the semi-log scale.