Clustering and enhanced classification using a hybrid quantum autoencoder

QNNs are the extension of ANNs to QML. The precise form of the QNN, however, is quite non-trivial as it would need to take advantage of unique quantum mechanical properties while also retaining the non-linear functional features of classical ANNs [13]. Hence, there are various proposals for QNN designs that claim similar non-linear dissipative dynamics of ANNs, but are yet to present clear quantum advantage [14–17]. In this paper, the implementation of the HQA will use the simplest design of a QNN: a PQC coupled with a specified observable. The choice of QNN, however, is arbitrary to the overall approach of the HQA (as shown in figure 1(c)). Hence a fair comparison of QNN complexity and expressibility—for the construction of the HQA specifically—is left for future work.

PQCs form the basis of hybrid quantum–classical algorithms that optimise a quantum circuit with respect to a problem-dependent cost function [18]. The optimisation is performed classically to determine a better estimate of parameters which define a variational circuit. The optimisation in this work is performed using the parameter shift rule [19], elaborated in appendix B.

Since any quantum circuit can be defined as a gate sequence $\mathcal{U}(\theta )$, the m parameters of the circuit are the set of θ_i which parameterise the unitaries. We define the measurement of a PQC as a function $f:{\mathbb{R}}^{m}\to \mathbb{R}$, mapping the gate parameters to an expectation value,

$$\begin{equation}{f}_{{\rho }_{0}}(\theta ){:=}\mathrm{Tr}\left[\mathcal{U}(\theta ){\rho }_{0}{\mathcal{U}}^{{\dagger}}(\theta )\hat{B}\right]{:=}\mathrm{Tr}\left[\rho (\theta )\hat{B}\right],\end{equation} \tag{ 1 }$$

where $\hat{B}$ is a predetermined observable (most commonly a Pauli-Z) and ρ₀ is an initial arbitrary state of the circuit. For consistency, the state before measurement will have notation, $\rho (\theta )=\mathcal{U}(\theta ){\rho }_{0}{\mathcal{U}}^{{\dagger}}(\theta )$ as evident from the second line in equation (1). It should be noted that this functional form of the variational circuits hides the fact that on real devices, repeated measurements of the circuit are required to obtain this expectation value. As a result, there is a natural statistical uncertainty to the estimated f, determined by the number of samples taken of the circuit. Caution is thus required when constructing algorithms that require arbitrarily large numerical precision of f. These algorithms may have promising results in state-vector simulations, but become computationally infeasible to measure on real quantum devices.

The optimisation of PQCs remain an area of research as it is not clear that employing classical optimisers will result in optimal solutions for quantum cost function landscapes. This has given rise to works suggesting optimisers that are aware of the underlying quantum structure of quantum states [20–22]. Furthermore, there exist limitations of barren plateaus in deep PQC-based algorithms, as initially realised in [23]. Here, the suppression of gradient with increasing depth of circuit, has also been shown to be linked to the expressibility of PQCs [24]. Critically, shallow circuits are not immune either, with an exponential suppression of gradients with increasing number of qubits [25]. In addition to barren plateaus, PQCs are seen to exhibit narrow gorges [25] which is the occurrence of the existence of cost landscape minima in narrow wells that get steeper with increasing depth. The effects are not only seen with an increase in qubits, but arise due to entanglement [26], certain cost functions [25, 27], and noise [28].

These phenomena clearly have implications on QAEs that employ PQCs and have been shown to be more pronounced for global loss functions—such as a fidelity-based loss [25]. Addressing this problem, there has been recent work employing the generalised quantum natural gradient—over the standard gradient—to train fidelity loss-based variational algorithms without barren plateaus [29]. However, this method requires a lower bound to the fidelity against the initial state, that is independent of the number of qubits and is therefore largely dependent on the chosen quantum data set {ρ_in}. As with many variational quantum algorithms, trainability for larger number of qubits is an active area of research. We therefore aim to emphasise the approach of the HQA algorithm, over its exact implementation.

Many recent works involving QAEs build on the structure first proposed by Romero et al [5], using shallow PQCs to compress quantum states. In [30], QNNs are of the form in [31] to construct a QAE that successfully denoises Greenberger–Horne–Zeilinger states which are subject to spin-flip and random unitary noise errors. Recently, there has been further enhancements with a projected-QAE model to improve the recovery of fidelity for an ensemble of pure states [32]. In [33], a QAE is constructed using approximate quantum adders that are obtained with classical genetic algorithms as opposed to more commonly used gradient methods for parameter optimisation.

All such applications work on the process of funnelling quantum states into a lower dimensional Hilbert space. This naturally returns compressed representations that disregard both stochastic noise fluctuations and irrelevant degrees of freedom. It is important to note that the set of states are assumed to have support on a subset of its Hilbert space, $\mathcal{S}\subset \mathcal{H}$. The existence of such support is not guaranteed, but is instead common in many sets of quantum states, due to symmetries inherent to physical processes. For example, in [5] a QAE is classically simulated to show the compression of ground states of the Hubbard model and molecular Hamiltonians.

QAEs have been experimentally realised for the compression of qutrits with photons in [34] and the compression of two-qubit states into two single-qubit states in [35]. Furthermore experimental realisation of the QAE via quantum adders has been shown in [36]. These QAEs are promising, but are nonetheless distinct to the HQA proposed in this paper. Specifically, the HQA aims to generate a classical representation of the $\mathcal{S}$ sub-manifold for classical analysis. Hence, not only can the data be compressed, but the compressed representation is an accessible classical vector that can be analysed.

There are clear limitations on an autoencoder's ability to compress data. On the compression rate of QAEs, there exist not only a fundamental limit due to the degrees of freedom in a dataset, but a quantum limitation related to the von Neumann entropy of the density operator representing the ensemble of training states [5]. In [37], it is further elaborated that the compressibility is related to the eigenvalues of the weighted ensemble density matrix. Crucially, this theoretical limit is intrinsic to all possible compression strategies and QAEs.

This paper proposes a novel variation of the QAE that we have termed a HQA. The hybrid nature of this model arises from the incorporation of both ML, in the form of classical ANNs, as well as QML, through the use of PQC-based QNNs. Figure 2 illustrates the overall design of the model which is a combination of (i) an encoder that takes a quantum state from Hilbert space ${\mathcal{H}}_{2}^{\otimes n}$ to a subset of the real vector space $\mathcal{V}$ of dimension $v=\mathrm{dim}(\mathcal{V})$, and (ii) a decoder that performs the inverse of such an operation. In general, though quantum states are positive semidefinite (trace = 1) operators ρ on ${\mathcal{H}}_{2}^{\otimes n}$, the HQA is equipped to identify only pure states. Hence, we associate the vector |ψ_a ⟩ to the pure state ρ_a := |ψ_a ⟩⟨ψ_a |. Mathematically, the encoder and decoder have the form of a map,

$$\begin{equation}\mathfrak{E}\;:\;{\mathcal{H}}_{2}^{\otimes n}\to \mathcal{V},\quad \text{where}\quad \mathfrak{E}(\vert {\psi }_{\mathrm{i}\mathrm{n}}\rangle )=\xi ,\end{equation} \tag{ 2 }$$

$$\begin{equation}\mathfrak{D}\;:\;\mathcal{V}\to {\mathcal{H}}_{2}^{\otimes n},\quad \text{where}\quad \mathfrak{D}(\xi )=\vert {\psi }_{\text{out}}\rangle ,\end{equation} \tag{ 3 }$$

where $\mathcal{V}={[-1,1]}^{v}$ is termed the latent space and $\xi \in \mathcal{V}$ is referred to as the latent vector—analogous to the terminology used when dealing with classical autoencoders. This vector is in essence a classical representation of the quantum state—a perfect representation if $\mathfrak{D}{\circ}\mathfrak{E}(\vert {\psi }_{\mathrm{i}\mathrm{n}}\rangle )=\vert {\psi }_{\mathrm{i}\mathrm{n}}\rangle$ is achieved. This indicates that the information of state |ψ_in⟩ was preserved in the latent vector to then be recreated without any loss of information. In information theory this is lossless encoding of information into a compressed latent space.

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Though the functional form of the encoder and decoder are defined, the models themselves have not been specified. As seen in figure 2, the encoder $\mathfrak{E}$ is a PQC parameterised by a vector, α. The PQC receives some state |ψ_in⟩, and applies unitary ${\mathcal{U}}_{1}(\alpha )$ on the combined system of the input state with (v − n) ancilla qubits. From this circuit, the Z expectation value of every qubit is measured, to then form the latent vector ξ,

$$\begin{equation}{\mathfrak{E}}^{(i)}(\vert {\psi }_{\mathrm{i}\mathrm{n}}\rangle )={\xi }^{(i)}=\langle \tilde{{\psi }_{\mathrm{i}\mathrm{n}}}\vert {\mathcal{U}}_{1}^{{\dagger}}(\alpha ){\text{Z}}_{i}{\mathcal{U}}_{1}(\alpha )\vert \tilde{{\psi }_{\mathrm{i}\mathrm{n}}}\rangle ,\end{equation} \tag{ 4 }$$

where $\vert \tilde{{\psi }_{\mathrm{i}\mathrm{n}}}\rangle =\vert {\psi }_{\mathrm{i}\mathrm{n}}\rangle \otimes \vert {0\rangle }^{\otimes (v-n)}$, Z_i is the Z-Pauli operator acting on the ith qubit and α parameterises the PQC that will be optimised. The set of ξ = (ξ⁽¹⁾, ..., ξ^(v)) forms the latent vector which is identified as the classical representation of an input quantum state. With inspiration from shadow tomography [11, 12], it is possible to enlarge the latent space with further X and Y expectation values. However, we leave this for future work. Following the encoder, the decoder is extremely similar to the one defined in equation (4). However, this time it is a mapping that returns a quantum state when given a latent vector,

$$\begin{equation}{\mathfrak{D}}_{A}(\xi )={\mathcal{U}}_{2}\left(f(\xi )\right)\vert 0\rangle \vert {\psi }_{0}\rangle ,\end{equation} \tag{ 5 }$$

where $f:\mathcal{V}\to {\mathbb{R}}^{p}$ is the functional from of the ANN that is parameterised by ${w}_{ij}^{(l)}$ (discussed in appendix A). Training changes only the weights of the ANN and not the parameters of the PQC. The PQC parameters are designed to be the output of the ANN. |ψ₀⟩ in equation (5) is an appropriate n-qubit ansatz for the type states involved. In this paper, we simply take |ψ₀⟩ = |0⟩^⊗n , which requires no additional operations for ansatz preparation.

It is important to note, both encoder and decoder have hyper-parameters (such as the number of parameters that define the PQCs, the number of neurons and depth of the ANN, etc) for which one must optimise. From here on we will assume that these hyper-parameters are accounted, realising that there is possible future work in rigorously addressing the exact optimisation for this HQA model.

Now that we have defined the encoder and decoder, the HQA model, $\mathfrak{A}$, is the combination defined as,

$$\begin{equation}\mathfrak{A}:{\mathcal{H}}^{\otimes n}\to {\mathcal{H}}^{\otimes n}\text{,}\;\;\mathfrak{A}\left(\vert {\psi }_{\mathrm{i}\mathrm{n}}\rangle \right)=\mathfrak{D}{\circ}\mathfrak{E}(\vert {\psi }_{\mathrm{i}\mathrm{n}}\rangle ),\end{equation} \tag{ 6 }$$

where we will refer to the output of the HQA as $\mathfrak{A}(\vert {\psi }_{\mathrm{i}\mathrm{n}}\rangle )=\vert {\psi }_{\text{out}}\rangle$. Though the HQA looks as though it is a single run through both the encoder and decoder, there is an implicit sub-routine for the encoder where the PQC must be sampled multiple times to obtain ξ.

Now that the components of the HQA have been pieced together, the model is trained to copy the input such that |ψ_out⟩ ≈ |ψ_in⟩. To do this we find a measure that can identify the distance between quantum states which will be the foundation of the HQA cost function. There are many possible ways in which to construct a sensible loss function, the one which we will be considering is one minus the fidelity $\mathcal{F}$ between the model output and the expected training output. Hence for a chosen training data set of K quantum states, ${\left\{\vert {\psi }_{i}^{\mathrm{i}\mathrm{n}}\rangle \right\}}_{i=1}^{K}$, and the fidelity defined as $\mathcal{F}(\vert \phi \rangle ,\vert \psi \rangle )=\langle \psi \vert \phi \rangle \langle \phi \vert \psi \rangle$, we define a loss,

$$\begin{equation}\mathcal{L}\left({\left\{\vert {\psi }_{i}^{\mathrm{i}\mathrm{n}}\rangle \right\}}_{i=1}^{K}\right)=1-\tilde{\mathcal{F}}\left({\left\{\mathfrak{A}\left(\vert {\psi }_{i}^{\mathrm{i}\mathrm{n}}\rangle \right),\vert {\psi }_{i}^{\mathrm{i}\mathrm{n}}\rangle \right\}}_{i=1}^{K}\right),\end{equation} \tag{ 7 }$$

where

$$\begin{equation}\tilde{\mathcal{F}}\left({\left\{\vert {\phi }_{i}\rangle ,\vert {\psi }_{i}\rangle \right\}}_{i=1}^{K}\right)=\frac{1}{K}\sum\limits _{i=1}^{K}\mathcal{F}\left(\vert {\phi }_{i}\rangle ,\vert {\psi }_{i}\rangle \right),\end{equation} \tag{ 8 }$$

is the average fidelity across all the training instances. In practice, the learner does not calculate the loss over all training instances per iteration, but rather a small batch or even over a single instance. This is because, (i) it is computationally expensive having a loss function that sums over all training instances, and (ii) doing so may result in over-fitting to the training data.

The fidelity between two states is maximum when input states are identical, and minimum when they are orthogonal. Hence, the aim is to maximise the fidelity to achieve |ψ_in⟩ ≈ |ψ_out⟩, and thereby minimise the loss, which lies in the range [0, 1]. This is a natural choice for a loss function, as the fidelity is a common distance measure between quantum states. The fidelity has also successfully been used for the construction of denoising QAEs [30] and is hence a great starting point for the construction of the HQA loss function. Selecting a method to measure the fidelity now becomes a hyper-parameter of the model—for which this paper will use the swap test (discussed in appendix D). However, there are other methods that can be employed, such as the inversion test used in [38].

Using the swap test, we can make an estimate of the training complexity. The sampling complexity per iteration (derivation in appendix E) is given by,

$$\begin{equation}\mathcal{O}\left(\frac{1+{P}_{\mathfrak{E}}}{{\varepsilon }_{\xi }^{2}}+\frac{1+{P}_{\mathfrak{D}}}{{\varepsilon }_{\text{fid}}^{2}}\right)\end{equation} \tag{ 9 }$$

where ${P}_{\mathfrak{E}}=\mathrm{dim}(\alpha )$ is the number of parameters in the encoder and ${P}_{\mathfrak{D}}=\mathrm{dim}(\theta )$ in the decoder, ɛ_ξ = Δξ_i is the uncertainty in each component of the latent vector $\overrightarrow{\xi }$, and ɛ_fid is the uncertainty in the fidelity measurement. The required ɛ_ξ and ɛ_fid is quite non-trivial and in this work, we use infinite precision for the measured expectation values. This non-triviality will become evident when dealing with the application in section 4, where the HQA is fundamentally unable to learn some states due to their stochasticity.

Now that the HQA has been constructed, one can observe the powerful nature of representing states in a classical latent space. Training the HQA gives rise to order in latent space that is created purely through matching the input quantum state to the output. In other words, even though we are not supplying the HQA with information about the states trained directly, the model is able to learn these differences and form patterns in latent space. It is this order that we can exploit to apply ML learning techniques to cluster and classify states in section 4.

The HQA is trained for a training set of quantum states ${\left\{\vert {\psi }_{i}^{\mathrm{i}\mathrm{n}}\rangle \right\}}_{i=1}^{K}$ that have underlying symmetry. In this paper, $\vert {\psi }_{i}^{\mathrm{i}\mathrm{n}}\rangle$ will correspond to a set of distinct amplitude encoded Gaussian states that can be easily analysed.

A Gaussian distribution is defined as,

$$\begin{equation}\mathcal{N}(x;\mu ,\sigma )=\frac{1}{\sigma \sqrt{2\pi }}{e}^{-\frac{1}{2}{\left(\frac{x-\mu }{\sigma }\right)}^{2}},\end{equation} \tag{ 10 }$$

where μ and σ are the mean and standard deviation respectively. Quantising this function for N equally spaced values in the range $\mu \in \left[-\frac{N}{2},\frac{N}{2}\right]$, $\sigma \in \left(0,\frac{N}{3}\right]$, we have $\vert {d}_{i}(\hat{\mu },\hat{\sigma })\vert =\mathcal{N}(i-\lceil \frac{N}{2}\rceil ;\hat{\mu },\hat{\sigma })$ where i ∈ {0, 1, ..., N − 1} and ⌈a⌉ is the ceiling function that returns the smallest integer above or equal to a. Now that we have discrete distribution, ${d}_{i}(\hat{\mu },\hat{\sigma })$, it can be encoded into the N = 2ⁿ amplitudes of a n-qubit quantum state,

$$\begin{equation}\vert {\psi }_{\mathcal{N}}(\hat{\mu },\hat{\sigma })\rangle =\frac{1}{\sqrt{C}}\sum\limits _{i=0}^{N-1}{\hat{d}}_{i}(\hat{\mu },\hat{\sigma })\vert i\rangle ,\end{equation} \tag{ 11 }$$

where $C={\sum }_{i=0}^{N-1}\vert {d}_{i}(\hat{\mu },\hat{\sigma }){\vert }^{2}$ is the normalisation constant. The ability for such encoded states to be variationally encoded has been shown in [39].

With this set of distinct quantum states, the HQA is employed to generate a classical latent space that represents the subset in which these states lie. Automatic differentiation is assumed through this unsupervised model to update the PQC parameters in the encoder and the weights of the ANN that determine the decoder. The PQC architecture of ${\mathcal{U}}_{1}(\alpha )$ and ${\mathcal{U}}_{2}(\theta )$ are shown in appendix C. We take dim (α) = 4v and dim (θ) = 4n, and a feedforward ANN with one hidden layer of size 2v. The training was performed in batches of 2 states using the Adam (γ = 0.1) optimiser [40], for a fixed number of epochs—which is the number of iterations each training sample is used for optimisation.

In figure 3(a), we see that training convergence is robust to latent space dimension, with similar loss evolution for all v. However, in figure 3(b), larger v is seen to decrease the loss in testing. As expected we see greater expressibility of the quantum state with a larger latent space.

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Once trained, to analyse this model we can apply only the trained encoder, $\mathfrak{E}$, to an input quantum state and observe its location in latent space—this latent vector is what we refer to as the classical representation of the input quantum state. An illustration of this process is shown in figure 4. The latent vector can then be obtained for all the amplitude encoded Gaussian states used for training, and then plotted. Figure 5 shows such a plot for n = 5 and a latent size, v = 12 where first two components (or parameters) of the vector are illustrated. This figure shows that Gaussian encoded states are distributed with a pattern distinguishing their mean and standard deviations. In this specific example, we see that an outward radial movement in the presented latent space corresponds to decreasing the standard deviation; and a positive polar rotation corresponds to increasing the mean. Such elegance is evident from patterns in the original set of quantum states; the two degrees of freedom: μ and σ. However, these patterns have the ability to be extremely non-trivial to visualise and this can be seen when we extend the results of figure 5 to  a 3rd dimension as shown in figure 6. Hence this non-triviality suggests the use of ML to learn patterns in latent space.

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It is not necessarily true that patterns will be visible when plotting the first two parameters of latent space. A plot of, say, the 9th and 10th latent parameters shows no patterns at all, as points seem to all congregate on a line or point. This indicates that these latent parameters are not being used to distinguish the quantum states, suggesting that a possible dimensionality reduction is possible for the latent space. This is where one can use principal component analysis (PCA) [6] that will both, allow for a clearer understanding of how the latent space is used, and also transform the space so that the most principal components can be plotted.

Now that we have identified a method of systematically analysing latent space, we can ask the question of whether this pattern would still occur if one simply trained on Gaussians with different mean values. Or similarly, if the model was built on training on Gaussians with only varying standard deviations. The results of doing so are shown in figure 7 where the two most principal components from PCA dimensionality reduction are plotted. The formation of patterns in latent space can only be observed when training has seen the variations in state. For example, in figure 7 where only μ was varied and σ was kept constant, there is order formed distinguishing μ but not σ. The opposite occurs where the training is switched. This suggests the HQA naturally attempts to allocate areas of similar quantum states without the need for additional supervision. In the context of applications, this is extremely useful as one can exploit the latent space location of a particular unknown state in reference to other known states—where similarity was before not necessarily obvious. In general, this means that we can infer information about states by applying ML algorithms to the states in latent space, as will be explored in section 4.

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In classical ML the most common algorithm for clustering is kmeans [41] (also referred to Lloyd's algorithm) which attempts to find clusters in a data set by observing some classical distance measure (an introduction to kmeans is presented in appendix F). The quantum equivalent was first proposed in [42], in the form of a so-called quantum kmeans. Here they are able to produce a quantum state corresponding to the k-clusters with complexity that grows linearly with the number of qubits, n. However, obtaining a classical description becomes exponential as the quantum states need to be measured using QST techniques, such as compressed sensing [7, 8], which requires $\mathcal{O}(k{N}^{2}\enspace \mathrm{log}\enspace N)$ where N = 2ⁿ . In addition, the authors exploit the fact that kmeans can be expressed as a quadratic programing problem which can be solved using a quantum adiabatic algorithm. In [43], an approach is presented with efficient quantum methods of calculating the Euclidean distance. Alternatively, the quantum approximate optimisation algorithm (QAOA) [44] is used in [45], for clustering by association to the maximum cut problem.

Many of the methods in literature suffer from requiring QST techniques to classically obtain clusters, which is not required when using the HQA. We define a method of clustering states using classical representations generated by the HQA, after which classical clustering algorithms are used on a space that is exponentially smaller.

For the classical clustering of latent space, it is required that a more sophisticated method than kmeans is used. There are many advanced clustering methods that allow non-linear clustering [46], but in general such clustering is not natural. In this work we will use Gaussian mixture modelling which is an extension of the kmeans clustering algorithm that has the flexibility to change the importance of certain parameters of the vector [47]—kmeans simply uses a Euclidean measure of distance.

The predicted labels from clustering latent space are shown in figure 9. The results show remarkable agreement with true labels when considering the minor components of the latent vector: 85.1% accuracy for the HQA trained with only smooth states and 84.4% accuracy for the HQA trained with both classes. On the other hand, clustering based on the principal components are seen to amount to guessing the class of state. The reason for this is evident when looking at the principal components. Fitting using all components is seen to identify clusters relating to the mean of the distributions (grouping negative and positive means), which is also a valid clustering outcome from an unsupervised viewpoint. Such a deficiency is attributed, not to a limitation of the algorithm, but rather to the non-uniqueness of the task's solution.

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At this point, one should retrace the steps of this clustering method in the context of an application. It is conceivable that a quantum experiment is conducted that produces quantum states about which the user has no information. This stream of states could hence be used for the training of the HQA. Importantly, however, one should note that a single sample of a quantum state is not sufficient. There is both, the sub-routine for the encoder, as well as the fidelity computation, that requires multiple copies of the state being produced from the experiment. Once a level of convergence has been reached, a clustering algorithm—such as kmeans or Gaussian mixture modelling—could be used on the latent representations of these states to identify possible groups. Finally, one can learn to identify—possibly highly non-trivial—distinctions between quantum states. This was shown in the distinction between smooth and non-smooth states, however more work is required to extend such a method to further applications.

In a similar process to using classical clustering methods on latent space, we now use classical supervised learning models to classify quantum states. Specifically two ML algorithms are used: support vector machine (SVM) [48] and logistic regression (LR) [49]. These are both supervised learning algorithms that are successfully used for classification. The former attempts to obtain a separating hyper-plane splitting the smooth and non-smooth classes, while the latter is a form of binary regression. It is enough for the reader to understand that these are supervised learning algorithms that are fundamentally linear, but where the SVM can be extended to non-linear decision boundaries with the use of, what is called, a kernel. The decision boundaries of these models are shown in figure 10.

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Splitting the 1600 classical data points (seen in figure 8) in a 3:10 testing/training ratio, the SVM and LR models are trained in two ways: (i) on all components of the latent vectors, and (ii) on just the principal components. The accuracy on the test data for all these variations (including kernel) is shown in table 1.

Table 1. Results of the binary classification problem of distinguishing smooth and non-smooth states from their latent space representations. This is done for two HQA models, with (n, v) = (7, 12), that identify different latent vectors: a HQA trained on only smooth and another HQA trained on both. The accuracy is simply defined as the proportion of correct classifications on a separate group of test data (size = 480). Importantly, both the SVM and the LR models are by no means optimised for the performance in this classification problem. Rather, these classical models are used as an illustration of the possibility of classifying states based on their latent space representation. For completeness, the hyper-parameters used for SVM are: C = 1.0 and γ = 2 (Linear), 20 (RBF). Finally, PCA reduction takes the four most principal components from the latent vector. The PCA reduced space accounts for 55% of the variance for HQA trained on both classes and 48% trained only smooth.

In general, both SVMs with non-linear kernels are seen to have a higher classification accuracy than LR. This makes sense, as the distribution of states on latent space was seen to be highly non-linear. More importantly however, from the performance of the SVM, the interesting result is that the HQA trained on only the smooth distributions, is comparable to the HQA trained on both when considering all components of the vector. At the same time, the HQA trained on both classes performs better when considering ML on only the principal components. To understand this behaviour, we realise that the HQA model attempts to separate states in only a few principal components. This means that when both classes were used for training the HQA, this separation was identifiable by the algorithm.

Considering all components, the polynomial kernel SVM with a HQA trained on only smooth distributions has a 0.95 accuracy rate. This occurs as a result of the non-smooth distributions—that were not seen by the HQA—being stored off the learned manifold and into the distinct unutilised regions of the latent space. It was hence easy for both ML algorithms to distinguish between the classes, with even a linear decision boundary from LR achieving an accuracy of 0.87.

These are significant results to keep in mind, however, not necessarily the most natural use of the HQA. If such a classification of quantum states was the main objective given a set of labelled states, one could instead—potentially—train the encoder $\mathfrak{E}$ rather than an entire HQA. However, the power of the HQA comes down to its ability to be trained unsupervised—i.e. without any labelling of the states that are being fed. For example, it is possible to train a HQA with the output states of some quantum experiment without knowing anything about the states themselves. Post-training one would only be required to label a few states and simply perform classical ML to obtain a working quantum state classifier. The reason that this works lies with the ability of the HQA to learn a manifold in which relevant states lie. Such a manifold is far smaller than the space of all states, hence requiring far fewer labelled instances to train. In ML literature, this process is known as semi-supervised learning, where only a portion of instances are labelled but where the unlabelled instances are also able to help with the overall classification process.