Quantum Optical Experiments Modeled by Long Short-Term Memory

Abstract

We demonstrate how machine learning is able to model experiments in quantum physics. Quantum entanglement is a cornerstone for upcoming quantum technologies, such as quantum computation and quantum cryptography. Of particular interest are complex quantum states with more than two particles and a large number of entangled quantum levels. Given such a multiparticle high-dimensional quantum state, it is usually impossible to reconstruct an experimental setup that produces it. To search for interesting experiments, one thus has to randomly create millions of setups on a computer and calculate the respective output states. In this work, we show that machine learning models can provide significant improvement over random search. We demonstrate that a long short-term memory (LSTM) neural network can successfully learn to model quantum experiments by correctly predicting output state characteristics for given setups without the necessity of computing the states themselves. This approach not only allows for faster search, but is also an essential step towards the automated design of multiparticle high-dimensional quantum experiments using generative machine learning models.

1. Introduction

In the past decade, artificial neural networks have been applied to a plethora of scientific disciplines, commercial applications, and every-day tasks with outstanding performance in, e.g., medical diagnosis, self-driving, and board games [1,2]. In contrast to standard feedforward neural networks, long short-term memory (LSTM) [3,4] architectures have recurrent connections, which allow them to process sequential data such as text and speech [5].

Such sequence-processing capabilities can be particularly useful for designing complex quantum experiments, since the final state of quantum particles depends on the sequence of elements, i.e., the experimental setup, these particles pass through. For instance, in quantum optical experiments, photons may traverse a sequence of wave plates, beam splitters, and holographic plates. High-dimensional quantum states are important for multiparticle and multisetting violations of local realist models, as well as for applications in emerging quantum technologies such as quantum communication and error correction in quantum computers [6,7].

Already for three photons and only a few quantum levels, it becomes, in general, infeasible for humans to determine the required setup for a desired final quantum state, which makes automated design procedures for this inverse problem necessary. With the algorithm MELVIN [8], it was shown how to automate this process. MELVIN uses a toolbox of optical elements, randomly generates sequences of these elements, calculates the resulting quantum state, and then checks whether the state is interesting, i.e., maximally entangled and involving many quantum levels. Furthermore, it has learning capability that speeds up the discoery of more complex systems. The setups proposed by MELVIN have been realized in laboratory experiments [9,10]. Numerous variations have been developed since then. For example, genetic algorithms [11,12] coupled with neural networks [13], reinforcement-learning-based search [14], gradient-descent of a continuous experimental space [15,16] or efficient human-interpretable representations [17] and unsupervised deep generative models [18]. See a recent review about these developments [19].

Inspired by these advances, we investigate how LSTM networks can learn quantum optical setups and predict the characteristics of the resulting quantum states, a task whose level of humanly perceived difficulty distinctly goes beyond that of other deep learning tasks, like object recognition or text generation. We train the neural networks using millions of setups generated by MELVIN. The huge amount of data make deep learning approaches the first choice. We use cluster cross validation [20,21] to evaluate the models.

2. Methods

2.1. Target Values

Let us consider a quantum optical experiment using three photons with orbital angular momentum (OAM) [22,23]. The OAM of a photon is characterized by an integer whose size and sign represent the shape and handedness of the photon wavefront, respectively. For instance, after a series of optical elements, a three particle quantum state may have the following form:

$$|\mathsf{\Psi }\rangle =\frac{1}{2}\left(\right|0,0,0\rangle +|1,0,1\rangle +|2,1,0\rangle +|3,1,1\rangle ).$$

(1)

This state represents a physical situation, in which there is a 1/4 chance (modulus square of the amplitude value 1/2) that all three photons have OAM value 0 (first term), and a 1/4 chance that photons 1 and 3 have OAM value 1, while photon 2 has OAM value 0 (second term), and so on, for the two remaining terms.

We are generally interested in two main characteristics of the quantum states: (1) Are they maximally entangled? (2) Are they high-dimensional? The dimensionality of a state is represented by its Schmidt rank vector (SRV) [24,25]. State $|\mathsf{\Psi }\rangle$ is indeed maximally entangled because all terms on the right hand side have the same amplitude value. Its SRV is (4,2,2), as the first photon is four-dimensionally entangled with the other two photons, whereas photons two and three are both only two-dimensionally entangled with the rest.

A setup is labeled “positive” ($y_{\mathrm{E}}=1$) if its output state is maximally entangled and if the setup obeys some further restrictions, e.g., behaves well under multi-pair emission, and otherwise labeled “negative” ($y_{\mathrm{E}}=0$). The target label capturing the state dimensionality is the SRV $y_{\mathrm{SRV}}={(n,m,k)}^{\top }$. We train LSTM networks to directly predict these state characteristics (entanglement and SRV) from a given experimental setup without actually predicting the quantum state itself.

2.2. Loss Function

For learning classification, we use the binary cross-entropy (BCE) loss function in combination with logistic sigmoid output activation, which dates back to Good [26] and is explained, e.g., by Bishop [27], as the negative log-likelihood of a Bernoulli distribution. For regression, it is always possible to reorder the photon labels such that the SRV has entries in non-increasing order. An SRV label is thus represented by 3-tuple $y_{\mathrm{SRV}}={(n,m,k)}^{\top }$ which satisfies $n\ge m\ge k$. Hence, we will subsequently refer to the SRV’s first entry n as the “leading Schmidt rank”.

With slight abuse of notation, we model $n\sim \mathcal{P}\left(\lambda \right)$ as a Poisson-distributed random variable and $m\sim \mathcal{B}(n,p),k\sim \mathcal{B}(m,q)$ as Binomials with ranges $m\in \{1,\cdots n\}$ and $k\in \{1,\cdots ,m\}$ and success probabilities p and q, respectively. The resulting log-likelihood objective (omitting all terms not depending on $\lambda ,p,q$) for a data point x with label ${(n,m,k)}^{\top }$ is

$$\begin{array}{cc}\hfill \ell (\widehat{\lambda },\widehat{p},\widehat{q}\mid x)=& nlog\widehat{\lambda }-\widehat{\lambda }+mlog\widehat{p}+(n-m)log(1-\widehat{p})\hfill \\ \hfill & +klog\widehat{q}+(m-k)log(1-\widehat{q})\hfill \end{array}$$

(2)

where $\widehat{\lambda },\widehat{p},\widehat{q}$ are the network predictions (i.e., functions of x) for the distribution parameters of $n,m,k$ respectively. The Schmidt rank value predictions are $\widehat{n}=\widehat{\lambda }$, $\widehat{m}=\widehat{p}\widehat{\lambda }$, $\widehat{k}=\widehat{p}\widehat{q}\widehat{\lambda }$. To see this, we need to consider the marginals of the joint probability mass function

$$f(n,m,k)=\frac{{\lambda }^n{e}^{-\lambda }}{n!}\left(\genfrac{}{}{0pt}{}{n}{m}\right)p^m{(1-p)}^{n-m}\left(\genfrac{}{}{0pt}{}{m}{k}\right)q^k{(1-q)}^{m-k}.$$

(3)

To obtain the marginal distribution of m, we can first sum over all possible k, which is easy. To sum out n, we first observe that $\left(\genfrac{}{}{0pt}{}{n}{m}\right)=0$ for $n<m$, i.e., the first m terms are zero, and we may write

$$f\left(m\right)=\sum _{n=0}^{\infty }f(n,m)=\sum _{n=0}^{\infty }f(m+n,m)$$

(4)

capturing only non-zero terms. It follows that

$$\begin{array}{cc}\hfill f\left(m\right)& =\sum _{n=0}^{\infty }\frac{{\lambda }^{n+m}{e}^{-\lambda }}{(n+m)!}\left(\genfrac{}{}{0pt}{}{n+m}{m}\right)p^m{(1-p)}^n\hfill \\ \hfill & =e^{-\lambda }{p}^m{\lambda }^m\sum _{n=0}^{\infty }\frac{{\lambda }^n{(1-p)}^n}{(n+m)!}\left(\genfrac{}{}{0pt}{}{n+m}{m}\right)\hfill \\ \hfill & =\frac{e^{-\lambda }{p}^m{\lambda }^m}{m!}\sum _{n=0}^{\infty }\frac{{\lambda }^n{(1-p)}^n}{n!}=\frac{e^{-p\lambda }{\left(p\lambda \right)}^m}{m!},\hfill \end{array}$$

(5)

which is $\mathcal{P}\left(p\lambda \right)$-distributed. Using the same argument for k, we get that the marginal of k is $\mathcal{P}\left(pq\lambda \right)$-distributed. The estimates $\widehat{n},\widehat{m},\widehat{k}$ are obtained by taking the means of their respective marginals.

2.3. Network Architecture

The used sequence processing model is depicted in Figure 1. We train two networks, one for entanglement classification (target $y_{\mathrm{E}}$), and one for SRV regression (target $y_{\mathrm{SRV}}$). The reason why we avoid multitask learning in this context is that we do not want to incorporate correlations between entanglement and SRV into our models. For instance, the SRV (6,6,6) was only observed in non-maximally entangled samples so far, which is a perfect correlation. This would cause a multitask network to automatically label such a sample as negative only because of its SRV. By training separate networks, we lower the risk of incorporating such correlations.

A setup of N elements is being fed into a network by its sequence of individual optical components $x={(x_1,x_2,...,x_N)}^{\top }$, where, in our data, N ranges from 6 to 15. We use an LSTM with 2048 hidden units and a component embedding space with 64 dimensions. The component embedding technique is similar to word embeddings [28].

3. Experiments

3.1. Dataset

The dataset produced by MELVIN consists of 7,853,853 different setups, of which 1,638,233 samples are labeled positive. Each setup consists of a sequence x of optical elements, and the two target values $y_{\mathrm{E}}$ and $y_{\mathrm{SRV}}$. We are interested in whether the trained model is able to extrapolate to unseen SRVs. Therefore, we cluster the data by leading Schmidt rank n. Figure 2 shows the the number of positive and negative samples in the data set for each n.

3.2. Workflow

All samples with $n\ge 9$ are moved to a special extrapolation set consisting of only 1754 setups (gray cell in

Table 1. Cluster cross validation folds (0–8) and extrapolation set (9–12) characterized by leading Schmidt rank n. Samples with $n=0$ and samples with $n=1$ are combined and then split into two folds (0,1) at random.

0,1	2	3	4	5	6	7	8	9–12
0,1

). The remainder of the data, i.e., all samples with $n<9$, are then divided into a training set and a conventional test set, with 20% of the data drawn at random (iid). This workflow is shown in Figure 3.

The test set is used to estimate the conventional generalization error, while the extrapolation set is used to shed light on the ability of the learned model to perform on higher Schmidt rank numbers. If the model extrapolates successfully, we can hope to find experimental setups that lead to new interesting quantum states.

Cluster cross validation (CCV) [20,21] is an evaluation method similar to standard cross validation [29]. Cross validation randomly partitions the training data into several folds, and then uses all but one fold for training and the remaining fold for validation. This is done several times such that every fold plays the role of the validation set exactly once. Instead of partitioning the folds iid, CCV groups them according to a clustering method. Thus, CCV removes similarities between training and validation set and simulates situations in which the withheld folds have not been obtained yet, thereby allowing us to investigate the ability of the network to discover these withheld setups. We use CCV with nine folds (white cells in Table 1). Seven of these folds correspond to the leading Schmidt ranks $2,\cdots ,8$. The samples with $n=1$ (not entangled) and $n=0$ (not even a valid three-photon state) are negative by definition. These samples represent special cases of $y_{\mathrm{E}}=0$ setups, and it is not necessary to generalize to these cases without training on them. Therefore, the 4,300,268 samples with $n<2$ are divided into two folds at random, such that the model will always see some of these special samples while training.

3.3. Results

Let us examine if the LSTM network has learned something about quantum physics. A good model will identify positive setups correctly, while discarding as many negative setups as possible. This behavior is reflected in the metrics true positive rate $\mathrm{TPR}=\mathrm{TP}/(\mathrm{TP}+\mathrm{FN})$ and true negative rate $\mathrm{TNR}=\mathrm{TN}/(\mathrm{TN}+\mathrm{FP})$, with TP, TN, FP, FN the true positives, true negatives, false positives, false negatives, respectively. A metric that quantifies the success rate within the positive predictions is the precision (or positive predictive value), defined as $\mathrm{PPV}=\mathrm{TP}/(\mathrm{TP}+\mathrm{FP})$.

For each withheld CCV fold n, we characterize a setup to be “interesting” when it fulfills the following two criteria: (i) It is classified positive (${\widehat{y}}_{\mathrm{E}}>\tau$) with $\tau$ the classification threshold of the sigmoid output activation. (ii) The SRV prediction ${\widehat{y}}_{\mathrm{SRV}}={(\widehat{n},\widehat{m},\widehat{k})}^{\top }$ is such that there exists a $y_{\mathrm{SRV}}={(n,m,k)}^{\top }$ with ${∥y_{\mathrm{SRV}}-{\widehat{y}}_{\mathrm{SRV}}∥}_2<r$. We call r the SRV radius. We denote samples which are classified as interesting (uninteresting) and indeed positive (negative) as true positives (negatives). Furthermore, we denote samples which are classified as interesting (uninteresting) and indeed negative (positive) as false positives (false negatives).

We employ stochastic gradient descent for training the LSTM network with momentum 0.5 and batch size 128. We sample mini-batches in such a way that positive and negative samples appear equally often in training. For balanced SRV regression, the leading Schmidt rank vector number n is misused as class label. The models were trained using early stopping after 40,000 weight update steps for the entanglement classification network and 14,000 update steps for the SRV regression network. Hyperparameter search was performed in advance on a data set similar to the training set.

Figure 4 shows the TNR, TPR, and rediscovery ratio for sigmoid threshold $\tau =0.5$ and SRV radius $r=3$. The rediscovery ratio is defined as the number of distinct SRVs, for which at least 20% of the samples are rediscovered by our method, i.e., identified as interesting, divided by the number of distinct SRVs in the respective cluster. The TNR for fold (0,1) is 0.9996, and the precision on the extrapolation set 9–12 is 0.659. Error bars in Figure 4 and later in the text are $95\%$ binomial proportion confidence intervals. Model performance depends heavily on parameters $\tau$ and r. Figure 5 shows the “beyond distribution” results for a variety of sigmoid thresholds and SRV radii.

Finally, we evaluate the conventional in-distribution performance in

Table 2. Conventional in-distribution training and test errors. The test set consists of 20 % of the data. Performance on predicting the entanglement is measured using the BCE loss, TNR, and TPR. Performance on predicting the SRV is measured using the SRV loss according to Equation (2), SRV accuracy, and the mean distance between true SRV and predicted SRV.

	Training	Test
BCE loss	10.2	10.4
TNR	0.9271 ± 2.4 $\times {10}^{-4}$	0.9261 ± 3.8 $\times {10}^{-4}$
TPR	0.9469 ± 4.1 $\times {10}^{-4}$	0.9427 ± 6.5 $\times {10}^{-4}$
SRV loss	2.247	2.24
SRV accuracy	0.9382	0.938
SRV mean distance	1.3943	1.4

. These figures are consistent with a clean training procedure.

4. Outlook

Our experiments demonstrate that an LSTM-based neural network can be trained to model certain properties of complex quantum systems. Our approach is not limited to entanglement and Schmidt rank, but may be generalized to employ other objective functions, such as multiparticle transformations, interference and fidelity qualities, and so on.

Another possible next step to expand our approach towards the goal of automated design of multiparticle high-dimensional quantum experiments is the exploitation of generative models. Here, we consider generative adversarial networks (GANs) [30] and beam search [31] as possible approaches.

Generating sequences such as text in adversarial settings has been done using 1D CNNs [32] and LSTMs [33,34]. The LSTM-based approaches employ ideas from reinforcement learning to alleviate the problem of propagating gradients through the softmax outputs of the network. Since our data are, in structure, similar to text, these approaches are directly applicable to our setting.

For beam search, there exist two different ideas, namely a discriminative approach and a generative approach. The discriminative approach incorporates the entire data set (positive and negative samples). The models trained for this work can be used for the discriminative approach, in that one constructs new sequences by maximizing the belief of the network that the outcome will be a positive setup. For the generative approach, the idea is to train a model on the positive samples, only to learn their distribution via next element prediction. On inference, beam search can be used to approximate the most probable sequence given some initial condition [35]. Another option to generate new sequences is to sample from the softmax distribution of the network output at each sequence position, as has been used for text generation models [36,37].

In general, the automated design procedures of experiments have much broader applications beyond quantum optical setups, and can be of importance for many scientific disciplines other than physics.

5. Conclusions

We have shown that an LSTM-based neural network can be trained to successfully predict certain characteristics of high-dimensional multiparticle quantum states from the experimental setup, without any explicit knowledge of quantum mechanics. For humans, the difficulty of analyzing quantum optical experiments goes far beyond that of other deep learning problems like, e.g., image classification. The network performs well even on unseen data beyond the training distribution, proving its extrapolation capabilities. This paves the way for the automated design of complex quantum experiments using generative machine learning models.