Abstract
Imagelike data from quantum systems promises to offer greater insight into the physics of correlated quantum matter. However, the traditional framework of condensed matter physics lacks principled approaches for analyzing such data. Machine learning models are a powerful theoretical tool for analyzing imagelike data including manybody snapshots from quantum simulators. Recently, they have successfully distinguished between simulated snapshots that are indistinguishable from one and two point correlation functions. Thus far, the complexity of these models has inhibited new physical insights from such approaches. Here, we develop a set of nonlinearities for use in a neural network architecture that discovers features in the data which are directly interpretable in terms of physical observables. Applied to simulated snapshots produced by two candidate theories approximating the doped FermiHubbard model, we uncover that the key distinguishing features are fourthorder spincharge correlators. Our approach lends itself well to the construction of simple, versatile, endtoend interpretable architectures, thus paving the way for new physical insights from machine learning studies of experimental and numerical data.
Introduction
There have been growing efforts to adopt data science tools that have proved effective at recognizing everyday objects for objective analysis of imagelike data on quantum matter^{1,2,3,4,5,6}. The key idea is to use the ability of neural networks to express and model functions to learn key features found in the imagelike data in an objective manner. However, there are two central challenges to this approach. First, the “black box” nature of neural networks is particularly problematic when it comes to scientific applications, where it is critical that the outcome of the analysis is based on scientifically correct reasoning^{7}. The second challenge unique to scientific application of supervised machine learning (ML) approaches is the shortage of real training data. Hence the community has generally relied on simulated data for training^{1,3,8}. However, it has not been clear whether the neural networks trained on simulated data properly generalize to experimental data. The path to surmounting both of these issues is to obtain some form of interpretability in our models. To date, most efforts at interpretable ML on scientific data have relied on manual inspection and translation of learned features from training standard architectures^{9,10,11}. Instead, here we propose an approach designed from the groundup to automatically learn information that is meaningful within the framework of physics.
The need for a principled datacentric approach is particularly great and urgent in the case of synthetic matter experiments such as quantum gas microscopy (QGM)^{12,13,14}, ion traps^{15}, and Rydberg atom arrays^{16,17}. While our technique is generally applicable, in this work we focus on QGM, which enables researchers to directly sample from the manybody density matrix of strongly correlated quantum states that are simulated using ultracold atoms. With the quantum simulation of the fermionic Hubbard model finally reaching magnetism^{14} and the strange metal regime^{18,19}, QGM is poised to capture a wealth of information on this famous model that bears many open questions and is closely linked to quantum materials. However, the realspace snapshots QGM measures are a fundamentally new form of data resulting from a direct projective measurement of a manybody density matrix as opposed to a thermal expectation value of observables. While this means richer information is present in a full dataset, little is known about how to efficiently extract all the information. When it comes to the questions regarding the enigmatic underdoped region of the fermionic Hubbard model, the challenge is magnified by the fact that fundamentally different theories can predict QGM data with seemingly subtle differences within standard approaches^{19,20}.
In this work, we develop Correlator Convolutional Neural Networks (CCNNs) as an architecture with a set of nonlinearities which produce features that are directly interpretable in terms of correlation functions in imagelike data (see Fig. 1). Following training of this architecture, we employ regularization path analysis^{21} to rigorously identify the features that are critical in the CCNN’s performance. We apply this powerful combination of CCNNs and regularization path analysis to simulated QGM data of the underdoped FermiHubbard model, as well as additional pedagogical examples in Supplementary Note 2. Following this, we discuss the new insights we gain regarding the hidden signatures of two theories, geometric string theory^{22} and πflux theory^{23,24}.
Results
The Hubbard model of fermionic particles on a lattice is a famous model that bears many open questions and is closely linked to quantum materials such as hightemperature superconductors. The model Hamiltonian is given by
where the first term describes the kinetic energy associated to electrons hopping between lattice sites, and the second term describes an onsite repulsion between electrons. At halffilling, and in the limit U ≫ t, the repulsive Hubbard model maps to the Heisenberg antiferromagnet (AFM)^{25}. However, the behavior of the model as the system is doped away from halffilling is not as wellunderstood. Several candidate theories exist which attempt to describe this regime, including geometric string theory^{22} and πflux theory^{23,24}. These theories are conceptually very distinct, but at low dopings measurements in the occupation basis do not differ enough in simple conventional observables such as staggered magnetization or twopoint correlation functions to fully explain previous ML success^{3} in discrimination (see Supplementary Note 4). Nevertheless, there are more subtle hidden structures involving more than two sites^{20} which are noticeable. In the “frozen spin approximation”^{26}, geometric string theory predicts that the motion of the holes simply displaces spins backwards along the path the hole takes. Hence the propagation of the doped hole will tend to produce a “wake” of parallel line segments of aligned spins in its trail (Fig. 2(a)). Meanwhile, the πflux theory describes a spin liquid of singlet pairs, where it is more difficult to conceive of characteristic structures (Fig. 2(b)).
Current QGM experiments are able to directly simulate the FermiHubbard model, obtaining one or twodimensional occupation snapshots sampled from the thermal density matrix \(\rho \sim {e}^{\beta {\mathcal{H}}}\) prescribed by the model^{14}. However, currently our experiment can only resolve a single spin species at a time, leaving all other sites appearing as empty. This is not a fundamental limitation of QGM experiments and complete spin and charge readout is beginning to become available to select groups^{27,28}. As we aim to learn true spin correlations, in this work we use primarily simulated snapshots at doping δ = 0.09 sampled from the geometric string and πflux theories using MonteCarlo sampling techniques under periodic boundary conditions. In particular, geometric string snapshots are generated by first sampling snapshots from the AFM Heisenberg model, then randomly inserting strings with lengths drawn from the analytic distribution^{20}. πflux snapshots are generated by standard Metropolis sampling from the Gutzwiller projected thermal density matrix given by the associated meanfield Hamiltonian. (See Supplementary Note 1 for further details.)
We point out that in the context of this paper, when referring to two models as different, we do not imply that they are fundamentally distinct, in the sense that they can not be connected smoothly without encountering a singularity in the partition function. Rather, this is a practical question: we have two or more mathematical procedures for generating manybody snapshots based on variational wavefunctions, MonteCarlo sampling, or any other theoretical approach. Our goal is to develop a ML algorithm that separates snapshots based on which procedure they are more likely to come from and, most importantly, the algorithm should provide information about which correlation functions are most important for making these assignments.
To learn how to distinguish these two theories we propose a neural network architecture, CCNNs, schematically shown in Fig. 1. The input to the network is an imagelike map with 3channels {S_{k}(x)∣k = 1, 2, 3}, where S_{1}(x) = n_{↑}(x), S_{2}(x) = n_{↓}(x), S_{3}(x) = n_{hole}(x). Since the models we consider are restricted to the singlyoccupied Hilbert space, this input only takes on values 0 or 1. From this input, the CCNN constructs nonlinear “correlation maps” containing information of local spinhole correlations up to some order N across the snapshot. This operation is parameterized by a set of learnable 3channel filters, {f_{α,k}∣α = 1, ⋯ , M} where M is the number of filters in the model. The maps for the given filter α are defined as:
Here a runs over the convolutional window of the filter α. Traditional convolutional neural networks employ only the first of these operations, alternating with some nonlinear activation function such as \(\tanh\) or \(\,\text{ReLU}\,(x)=\max (0,x)\). The issue with these typical nonlinear functions is that they mix all orders of correlations into the output features, making it difficult to disentangle what exactly traditional networks measure. In contrast, each order of our nonlinear convolutions \({C}_{\alpha }^{(n)}({\bf{x}})\) are specifically designed to learn nsite semilocal correlations in the vicinity of the site x, which appear as patterns in the convolutional filters f_{α}. Note that the sums in Eq. (2) exclude any selfcorrelations to aid interpretability. During training, a CCNN tunes the filters f_{α,k}(a) such that correlators characteristic of the labeled theory are amplified while others are suppressed. To aid interpretation, we force all filters to be positive f_{α,k}(a) ≥0 by taking the absolute value before use on each forward pass. We note that a multisite kernel used in a support vector machine, as introduced in refs. ^{29,30}, could also learn higherorder correlators. However, CCNNs allow highorder correlations to be efficiently parameterized and discovered by leveraging automatic differentiation and the structure of convolutions.
A direct computation of the nonlinear convolutions following Eq. (2) up to order N requires \({\mathcal{O}}({(KP)}^{N})\) operations per site, where P is the number of pixels in the window of the filter and K is the number of species of particles. However, we can use the following recursive formula which we prove in Supplementary Note 3:
where all powers are done pixelwise, and we define \({C}_{\alpha }^{(0)}({\bf{x}})=1\). This improves the computational complexity to \({\mathcal{O}}({N}^{2}KP)\) while also allowing us to leverage existing highlyoptimized GPU convolution implementations. Use of this formula leads to a “cascading” structure to our model similar to^{31}, as seen in Fig. 1. First, the input S is convolved with filters f_{α} to produce the firstorder maps \({C}_{\alpha }^{(1)}\). Using Eq. (3), these firstorder maps can be used to construct second order maps \({C}_{\alpha }^{(2)}\), and onwards until the model is truncated at some order N. Since the Hamiltonians being studied are translationinvariant, we then obtain estimates of correlators from these correlation maps by simple spatial averages to produce \({c}_{\alpha }^{(n)}=\frac{1}{{N}_{\text{sites}}}{\sum }_{{\bf{x}}}{C}_{\alpha }^{(n)}({\bf{x}})\). In addition, we employ an explicit symmetrization procedure to enforce that the model’s predictions are invariant to arbitrary rotations and flips of the input, detailed in Supplementary Note 1. Concatenating these correlator estimates results in an NMdimensional feature vector \({\bf{c}}=\{{c}_{\alpha }^{(n)}\}\).
In the back portion of a CCNN (see Fig. 1), the feature vector c is normalized using a BatchNorm layer^{32}, then used by a logistic classifier which produces the classification output \(\hat{y}({\bf{c}};{\boldsymbol{\beta }},\epsilon )={[1+\exp ({\boldsymbol{\beta }}\cdot {\bf{c}}+\epsilon )]}^{1}\) where \({\boldsymbol{\beta }}=\{{\beta }_{\alpha }^{(n)}\}\) and ϵ are trainable parameters. If \(\hat{y}<0.5\), the snapshot is classified as πflux, and otherwise it is classified as geometric string theory. The \({\beta }_{\alpha }^{(n)}\) coefficients are central to the interpretation of the final architecture, as they directly couple the normalized correlator features \({c}_{\alpha }^{(n)}\) to the output. For training, we use L1 loss in addition to the standard crossentropy loss, i.e.,
where y = {0, 1} is the label of the snapshot, and γ is the L1 regularization strength. The role of the L1 loss is to promote sparsity in the filter patterns by turning off pixels which are unnecessary^{10,11}.
We fix the number of filters M and the maximum order of the nonlinear convolutions N, a hyperparameter specific to CCNN, by systematically observing the training performance. We found that two filters gives sufficient performance while allowing for simple interpretation. Hence we consider two filters, i.e., M = 2 in the rest of the paper. For the maximum order of nonlinear convolution N we found the performance to rapidly increase with increase in N up to N = 4, past which performance plateaus. Hence we fix N = 4 in the rest of the paper. In addition, we limit our investigation to 3 × 3 convolutional filters. With the architecture of the CCNN sofixed we found the performance of this minimalistic model to be comparable with a more complex traditional CNN architecture^{3} (see Supplementary Note 1 for these performance results).
After a CCNN is trained, we fix the convolutional filters f_{α} and move on to a second phase to interpret what it has learned. We first determine which features are the most relevant to the model’s performance by constructing and analyzing regularization paths^{33} to examine the role of the logistic coefficients \({\beta }_{\alpha }^{(n)}\). We apply an L1 regularization loss to these \({\beta }_{\alpha }^{(n)}\) and retrain the back portion of the model (see Fig. 1) using a new loss function:
where λ is the regularization strength. Again, the L1 loss plays a special role in promoting sparsity in the model parameters, but we are now penalizing the use of coefficients \({\beta }_{\alpha }^{(n)}\) and hence the corresponding features \({c}_{\alpha }^{(n)}\). This results in an optimization tradeoff between minimizing the classification loss and attempting to keep \({\beta }_{\alpha }^{(n)}\) at zero, where the relative importance of these terms is tuned by λ. At large λ, the loss is minimized by keeping all \({\beta }_{\alpha }^{(n)}\) at zero, resulting in a 50% classification accuracy due to the model always predicting a single class. As λ is slowly ramped down, eventually the “most important” coefficient \({\beta }_{\alpha }^{(n)}\) will begin to activate, due to the decrease in classification loss surpassing the increase in the activation loss. As these coefficients couple the correlator features \({c}_{\alpha }^{(n)}\) to the prediction output, this process offers clear insight into which features are the most relevant.
We show a typical regularization path analysis in Fig. 3, where the filters f_{α} of a trained model are shown in the inset. The activation of each coefficient \({\beta }_{\alpha }^{(n)}\) is tracked while tuning down the regularization strength λ (increasing 1/λ). The resulting trajectories in Fig. 3(a) show that the 4th order correlator features, \({c}_{1}^{(4)}\) and \({c}_{2}^{(4)}\) are most significant for the CCNN’s decision making since \({\beta }_{1}^{(4)}\) and \({\beta }_{2}^{(4)}\) are the two first coefficients to activate. Furthermore, parallel tracking of the accuracy in Fig. 3(b) shows that the activation of these features results in large jumps in the classification accuracy, comprising almost all of the network’s predictive power. While the details of the paths vary between training runs, we find robust dominance of fourthorder correlations as the first features to be activated to give the majority of the network’s performance.
The regularization path distinguishing the geometric string and πflux ansatzes shown in Fig. 3 is in stark contrast to what happens when the identical architecture is trained to discriminate between a thermally excited antiferromagnetic Heisenberg state and a state with purely random spins (see Supplementary Note 2). In that scenario, the network learns that twopoint correlations \({c}_{\alpha }^{(2)}\) carry the key information for nearperfect classification. In Supplementary Fig. 5, the regularization path shows only \({\beta}_{1}^{(2)}\) activating to achieve full performance, and the learned filter obviously resembles the AFM pattern. Meanwhile, the behavior seen in Fig. 3 evidences that the subtle differences between πflux and geometric string theory instead hinges on fourthorder correlations.
Now that we know fourthorder correlations are the important features, we look at which physical correlators are being measured by the features \({c}_{\alpha }^{(4)}\) by simply inspecting 4pixel patterns made from highintensity pixels from each channel of the learned filters, as we show in Fig. 4. Comparing these patterns with the depiction of the two candidate theories, we can understand why these correlators measured by the two filters are indeed prominent motifs. Specifically, the 2 × 2 correlators in the fourthorder feature of the filter associated to the geometric string theory (Fig. 4(a)) are easily recognizable in the “wake” and the termination of a string. These discovered correlations are in agreement with those examined in ref. ^{28}, which found pronounced spin anticorrelations induced on the spins located on the diagonal adjacent to a mobile chargon. Meanwhile, the 2 × 2 motifs in the filter learned to represent the πflux theory (Fig. 4(b)) are either a single spinflip or a simple placement of a hole into an AFM background. It is evident that this CCNN is learning the fingerprint correlations of geometric string theory, recognizing the πflux theory instead from fluctuations which are uncharacteristic of the string picture. Furthermore, a subset of learned patterns that are not obvious from the simple cartoons can be used as additional markers to detect the states born out of the two theoretical hypotheses in experiment (see Supplementary Note 4 for more detail).
It is important to note that the above insights relied on the fact that our CCNN’s structure can be understood as measuring collections of correlators. Although the regularization path analysis can be applied to any architecture, the typical nonlinear structures of offtheshelf CNNs inhibit direct connections between the dominant filters and physically meaningful information^{34}. In Supplementary Note 5 we present how interpretation of the architecture of ref. ^{3} can be attempted following similar steps as above. Since the fully connected layer contains tens of thousands of parameters, after training we show that we can reduce this layer to a simple spatial averaging to attempt interpretation, with no loss in performance. The reduced architecture with a single “feature” per convolutional filter, similar to the architecture of ref. ^{34}, is trained, after which we fix the filters for the regularization path analysis. We can clearly determine which filters produce the important features, but it is unclear what these features are actually measuring due to the ReLU nonlinearity. However, without any nonlinearity the architecture only achieves close to 50% performance. This failure to enforce simplicity on traditional architectures shows the importance of designing an architecture, which measures physically meaningful information from the outset.
The ML method presented in this paper considers shortrange multipoint correlation functions (up to three lattice sites in both x and y directions), but does not include longrange twopoint correlations needed for identifying spontaneous symmetry breaking. Two considerations motivate this choice: (i) Current experiments with the FermiHubbard model are done in the regime where correlations involving charge degrees of freedom are not expected to exceed a few lattice constants due to thermal fluctuations. (ii) The energy of systems with local interactions, such as the FermiHubbard model, is primarily determined by shortrange correlations. We note, however, that the current method can be extended to include longer range correlations either by expanding the size of the filters used in Eq. (2), or by using dilated convolutions.
To summarize, we proposed a neural network architecture that discovers most relevant multisite correlators as key discriminative features of the state of interest. We then applied this architecture to the supervised learning problem of distinguishing two theoretical hypotheses for the doped Hubbard model: πflux theory and geometric string theory. Employing a regularization path analysis technique on these trained CCNN architectures, we found that foursite correlators deriving from the learned filters hold the key fingerprints of geometric string theory. A subset of these foursite motifs fit into what is expected from the wake of a propagating hole in an antiferromagnetic background. The remaining foursite motifs which go beyond our existing intuition can be used as additional signatures of the two quantum states. As higherorder correlators are beginning to be probed in QGM experiments^{19}, our work demonstrates an automated method for learning high signaltonoise correlators useful for theory hypothesis testing. It will be interesting to extend our analysis to a broader collection of candidate theories, as well as snapshots generated using recently developed finiteT tensor network methods^{35,36} and spinresolved experimental data.
Discussion
The broad implications of CCNNbased ML for analysis and acquisition of imagelike data are threefold. Firstly, CCNN is the first neural network architecture that was explicitly designed for imagelike quantum matter data. Our results showcase how a successful design of ML architecture that is designed with scientific objectives at the forefront can offer new scientific insight.
Secondly, our approach can guide quantum simulator design by revealing necessary discriminating features. In particular, we found that experimental uncertainties on the actual doping level in QGM without spinresolution led the CCNN to focus on the doping level rather than a meaningful hypothesis testing (see Supplementary Note 6). Hence, access to either spin or chargeresolved snapshots, which are just now becoming available^{27,28,37,38}, will be essential. Finally, our results showcase how a targeted “tomography” can be achieved to extract new insights from nearterm quantum systems from quantumclassical hybrid approaches. Full reconstruction of the density matrices from projective measurements is an exponentially difficult task. However, available and nearterm quantum systems are showing great promise as quantum simulators with their design and objectives guided by classical simulations. For such quantum systems, CCNNbased hypothesis testing can offer much needed state characterization in a scalable fashion.
Disclaimer
This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof.
Data availability
All simulated snapshots examined in this work are available publically at Zenodo, ref. ^{39}. All experimental snapshots examined in the Supplementary Notes are available from ref. ^{20}.
Code availability
All code used for training and analysis of the CCNNs is available at Github, ref. ^{40}.
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Acknowledgements
We thank Fabian Grusdt and Andrew Gordon Wilson for insightful discussions during the completion of this work. C.M. acknowledges that this material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research, Department of Energy Computational Science Graduate Fellowship under Award Number DESC0020347. A.B., R.W., K.W., E.D., EA.K. acknowledge support by the National Science Foundation through grant No. OAC1934714. A.B. acknowledges funding by Germany’s Excellence Strategy  EXC2111  390814868.
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C.M. conceived the CCNN architecture, wrote the ML and analysis code, and performed the training experiments and analysis. R.W. and K.Q.W conceived the application of regularization paths, and provided guidance of the ML procedure. A.B. produced the simulated snapshot data. C.C., M.X., G.J., and M.G. produced the experimental data and provided feedback on connections to experiments. C.M., A.B., R.W., K.Q.W., E.D., and EA.K. initiated the project concept, and guided the work. C.M. and E.A.K. wrote the paper, with input and modifications from all authors. E.A.K. led the project.
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Miles, C., Bohrdt, A., Wu, R. et al. Correlator convolutional neural networks as an interpretable architecture for imagelike quantum matter data. Nat Commun 12, 3905 (2021). https://doi.org/10.1038/s4146702123952w
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DOI: https://doi.org/10.1038/s4146702123952w
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