Machine learning
Image by Chris Hohmann (MCQST) & Annabelle Bohrdt
In the last two decades, quantum simulation experiments have become a valuable tool in the study of strongly correlated quantum many-body systems. Typical measurements from quantum simulation experiments such as quantum gas microscopes or superconducting qubits consist in quantum snapshots of the system, which provide information of the underlying quantum many-body state with a resolution of a single lattice site. Conventionally -- based on what's accessible in solid state experiments -- one- and two-point correlation functions are evaluated. Quantum snapshots provide however much more information, and it is an exciting new challenge to figure out ways to make the most of that information.
In my research, I apply techniques established in the machine learning community to study quantum many-body systems and to make the most use of all available data, with the goal of an unbiased and interpretable analysis.
Classifying snapshots of the doped Hubbard model
The Fermi-Hubbard model is one of the most paradigmatic models in condensed matter theory. A lot of research has been conducted, in particular on the Fermi-Hubbard model on a two-dimensional square lattice, which is believed to describe the essential physics of high-temperature superconductivity in the cuprate materials. Despite an intense amount of research and its deceiving simplicity, the Fermi-Hubbard model is still not completely understood, and an effective theory capturing its phase diagram remains lacking (for more details on my research on the Fermi-Hubbard model see here!). This is, however, not for a lack of theoretical proposals -- there are, in fact, many. As Steven Kivelson put it (Science 314,2016), "The theoretical problem is so hard that there isn't an obvious criterion for right." So given the ground truth -- in our case the experimental data from a quantum simulation experiment -- how can we decide which effective theoretical description captures it best? In this work, we applied machine learning: we train a neural network to distinguish snapshots -- as they would be measured with a quantum gas microscope -- of two different theories. After training, the network parameters are fixed. The neural network only knows the two options: theory A (in our case a resonating valence bond state as proposed by Nobel laureate Phil Anderson) or theory B (the geometric string theory, see here for more details). Next, we used experimental snapshots as input. The network can only assign one of the two theory labels. The resulting classification thus directly tells us which theory describes the experiment better, taking all available information (and that means in principle arbitrary higher order correlators, which can be extracted from the snapshots, see below) into account!
You can read more about this work in the original article: "Classifying snapshots of the doped Hubbard model with machine learning" -- A. Bohrdt et al., Nature Physics 15 (2019)
Learning thermalization
How can one call a system thermalized? When local observables, say for example the magnetization, have reached their thermal value? But what about two-point correlations? And maybe most importantly: how can we check without any prior knowledge what to search for? In this work, we study thermalization in the one-dimensional Bose-Hubbard model using snapshots from a cold atom experiment. In particular, we train a neural network to distinguish the current time step during the time evolution from the thermal state. If the system is thermalized, the snapshots from the time evolved state will look a lot like the snapshots from the thermal density matrix. This means on the other hand that the network's performance will be bad, since it will be hard to distinguish the two datasets.
We train a network to distinguish time-evolved from thermal for many different evolution times and monitor its performance. In the experimental system, we can thus distinguish between a localized phase, were no thermalization occurs on the accessible time scales, and the thermal phase. In between, there is a critical phase, where higher-order correlation functions become important. We show this by training the correlator convolutional neural network (CCNN, see below for more details!) to distinguish time-evolved from thermal states: with this network architecture, we can not only monitor the performance of the network as a function of time, but also as a function of the order of correlations taken into account. As it turns out, the performance of the network still increases with higher order correlations in the critical phase, whereas only lower order (one- and two-point) correlations are needed in the thermal and localized cases.
For more details, check out the original letter here: "Analyzing Nonequilibrium Quantum States through Snapshots with Artificial Neural Networks" -- A. Bohrdt et al., Phys. Rev. Letters 127 (2021).
Interpretability: the CCNN
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