Machine learning the arrow of time in solid-state spins

TL;DR

This work harnesses machine learning to identify the arrow of time from individual trajectories generated by a programmable ten-qubit quantum processor based on a nitrogen-vacancy center in diamond and shows that a diffusion-based generative model reproduces essential signatures of directional energy flow and entropy production.

Abstract

Understanding the emergence of the thermodynamic arrow of time in microscopic systems is of fundamental importance, particularly given that unitary evolution preserves time-reversal symmetry. While projective measurements introduce temporal irreversibility, identifying this asymmetry from single evolution trajectories in the presence of stochastic fluctuations presents a considerable challenge. Here, we harness machine learning to identify the arrow of time from individual trajectories generated by a programmable ten-qubit quantum processor based on a nitrogen-vacancy center in diamond. We implement quantum circuits that realize unitary evolutions where heat flows from hotter to colder subsystems and their time-reversed counterparts. Projective measurements inserted in these processes induce entropy production, and their outcomes constitute the evolution trajectory. We demonstrate that an unsupervised clustering algorithm autonomously divides the experimental trajectories into two distinct groups without prior knowledge, while a convolutional neural network identifies the temporal direction of these trajectories with approximately 92% accuracy. In addition, we show that a diffusion-based generative model reproduces essential signatures of directional energy flow and entropy production. Our results establish machine learning as a powerful tool for uncovering underlying physical processes from complex experimental data, advancing the interface between quantum thermodynamics and artificial intelligence.

Figures (13)

• Figure 1: Machine learning of the time arrow in the evolution of quantum systems.a, Illustration of time irreversibility induced by projective measurements. Unitary evolution (blue boxes) preserves time-reversal symmetry, whereas the introduction of projective measurements (orange boxes) breaks this symmetry, establishing the arrow of time. b, Schematic of the experimental system. A nitrogen-vacancy center system in diamond consists of a single electron spin and nine nuclear spins of nearby ${}^{13}\text{C}$ atoms. The system operates at approximately $7$ K and in an external magnetic field of approximately $495$ Gauss. A green laser at 532 nm prepares the charge state of the nitrogen-vacancy center. Two 637 nm lasers resonantly drive the transitions shown in the inset for electron spin initialization and readout. Microwave (MW) fields are tailored for coherent rotations of the electron spin and universal control of individual nuclear spins. The laser beams are directed onto the diamond via an optical setup comprising mirrors, beam splitters (BS), wave plates (WPs), and dichroic mirrors (DM) (Supplementary Information). c, Machine learning frameworks for identifying the arrow of time. The input consists of a single evolution trajectory of the ten qubits over five sequential measurements, represented as a $[10 \times 5]$ binary matrix. This data is processed via two distinct pipelines: an unsupervised $k$-means clustering algorithm (upper middle) that autonomously groups trajectories into forward and backward clusters (upper right), and a supervised convolutional neural network (lower middle) that takes the matrix as input and outputs the classification probability for the temporal direction (lower right). d, Architecture of the diffusion model for trajectory generation. The model consists of fully connected layers with sigmoid linear unit activation functions. Inputs include noisy data, class labels (forward or backward), and time embeddings. The network is trained to predict the injected noise. e, Trajectory generation pipeline using the diffusion model. The model synthesizes trajectories consistent with physical data through a stepwise denoising process starting from the Gaussian white noise.
• Figure 1: The elbow method for determining the optimal number of clusters. The elbow method is employed to determine the optimal number of clusters $K$, using the within-cluster sum of squared Euclidean (SSE) distances (the sum of squared Euclidean distances from each sample to its cluster centroid). As $K$ increases, the SSE initially decreases rapidly and then gradually slows down, forming an elbow-like shape. The $K$ at this elbow balances tight intra-cluster grouping and avoids over-clustering. Here, the optimal number of clusters is determined to be $K=2$. The black dotted line serves as a guide to the eye, highlighting the sharp transition in the slope at this optimal value.
• Figure 2: Experimental observation of the arrow of time in quantum thermodynamics.a, Schematic of quantum thermodynamic processes in a ten-spin system. In the forward process, spins are initialized in thermal Gibbs states at different temperatures. They then undergo repeated cycles of unitary evolution U followed by projective measurements, yielding the forward state trajectory $\{\rho^{k}\}_{k=0}^{n}$. The reverse process starts with the final state of the forward process. The system evolves similarly under unitary $U^\dagger$ in each step. The backward trajectory (dashed arrow) is obtained by time-reversing the trajectory of the reverse process. Due to the back-action of the measurements in our setup, the spins are re-prepared after each measurement according to the outcome. b, Quantum circuit of unitary evolution. The unitary $U$ is implemented as a sequence of pairwise unitaries $\mathcal{U}_i$ between the electron spin and the $i$th nuclear spin, followed by Pauli-$Z$ measurements. c, The pairwise unitary $\mathcal{U}_i = \exp\left[-\mathrm{i}\left(X_\text{e} X_i + Y_\text{e} Y_i\right)\Delta t\right]$ with $\hbar=1$ comprises of single- and two-qubit gates. White blocks represent electron spin rotations by an angle of $\pi/2$, and pink blocks denote nuclear spin rotations by an angle of $2\Delta t$. d, Measured energy of the electron spin (blue symbols) and average energy of the nine nuclear spins (red symbols) as functions of the step index for both forward (squares) and backward (diamonds) processes. Error bars represent one standard deviation of the mean over the nine nuclear spins. e, Plots of the system entropy versus step index. The discrepancy between the forward (red circles) and backward (blue triangles) trajectories highlights the irreversibility of the thermodynamic process.
• Figure 3: Identification of the arrow of time by machine learning.a, An unsupervised clustering $k$-means algorithm without label supervision classifies $1,000$ experimental trajectories into two clusters. These clusters correspond to the forward (red squares) and backward (blue circles) trajectories and are separated in the subspace spanned by the first two principal components. b, Supervised learning on experimental data using a convolutional neural network distinguishes forward from backward trajectories with accuracy up to about 92%. The loss function used here is the binary cross-entropy loss function. c, Learning curves for simulated trajectories governed by pure unitary dynamics without projective measurements. The classification accuracy using the convolutional neural network collapses to the random-guessing baseline around $50\%$.
• Figure 4: Thermodynamic trajectories generated by a diffusion-based generative model.a, Energy of the electron spin (blue symbols) and mean energy of the nine nuclear spins (red symbols), averaged over $90,000$ samples synthesized by the diffusion-based generative model, are plotted against the step index for both forward (squares) and backward (diamonds) processes. Error bars represent one standard deviation of the mean over the nine nuclear spins. b, Plots of system entropy calculated from the trajectories generated by the diffusion-based model as a function of step index. Simulated data in b,c is consistent with the experimental observations in Figs. \ref{['fig:2']}d, \ref{['fig:2']}e. c, Fidelities between the generated and experimental states, averaged over all steps for the forward (red squares) and backward (blue diamonds) processes, are plotted as a function of training epochs (Methods). Error bars represent the standard deviation across time steps.