Generalized quantum similarity learning

Given $\mathcal{E}_{l},\mathcal{D}_{k}\subset X\times\tilde{X}$ with cardinality $l,k$ such that pairs $(x_{i},\tilde{x}_{i})\in\mathcal{E}_{l}$ are deemed related ($y_{i}=1$) and $(x_{i},\tilde{x}_{i})\in\mathcal{D}_{k}$ are unrelated ($y_{i}=0$). Then we try to learn a distance function

$$d:X\times\tilde{X}\to\mathbb{R}_{+}$$

such that

$\displaystyle\begin{cases}d(x,\tilde{x})\leq\delta&(x,\tilde{x})\in\mathcal{E}_{l}\\ d(x,\tilde{x})\geq\varepsilon^{-1}&(x,\tilde{x})\in\mathcal{D}_{k}\end{cases}$

(1)

for some small constants $\delta,\varepsilon>0$. Here, $d$ need not satisfy the usual axioms of a distance metric, such as symmetry, which may not even make sense as the inputs come from different spaces. The problem can also be phrased in terms of a similarity measure $\mathcal{S}$, which simply reverses the inequalities: $\mathcal{S}(x,\tilde{x})$ should be large when $x$ and $\tilde{x}$ are similar and small when they are dissimilar.

We consider $d$ of the form

$\displaystyle d(x,\tilde{x}):=\|f_{\theta}(x)-g_{\eta}(\tilde{x})\|_{\mathcal{H}}.$

(2)

where the feature maps $f_{\theta}:X\to\mathcal{H}$, $g_{\eta}:\tilde{X}\to\mathcal{H}$ encode the raw input data in a common Hilbert space $\mathcal{H}$. Concretely, let $H=(\mathbb{C}^{2})^{\otimes n}$ be the Hilbert space of a quantum computer with $n$ qubits, and let $\mathcal{H}=\mathcal{L}(H)$ be the space of complex-linear operators on $H$ equipped with the Hilbert-Schmidt inner product. Following [21], define the feature maps

$\displaystyle\begin{split}f_{\theta}(x)&=U_{\theta}(x)\outerproduct{0^{n}}{0^{n}}U_{\theta}(x)^{\dagger}\\ g_{\eta}(\tilde{x})&=V_{\eta}(\tilde{x})\outerproduct{0^{n}}{0^{n}}V_{\beta}(\tilde{x})^{\dagger}\end{split}$

(3)

where $U_{\theta}(x)$, $V_{\eta}(\tilde{x})$ are parameterized quantum circuits. Then

$\displaystyle d(x,\tilde{x})^{2}=2-\mathcal{S}_{\theta,\eta}(x,\tilde{x})$

(4)

where the similarity measure

$\displaystyle\begin{split}\mathcal{S}_{\theta,\eta}(x,\tilde{x})&=\Tr{f_{\theta}(x)^{\dagger}g_{\eta}(\tilde{x})}\\ &=\absolutevalue{\expectationvalue{U_{\theta}(x)^{\dagger}V_{\eta}(\tilde{x})}{0^{\otimes n}}_{H}}^{2}\end{split}$

(5)

is nothing but the overlap between states $U_{\theta}(x)\ket{0^{n}}$ and $V_{\eta}(\tilde{x})\ket{0^{n}}$. There are multiple methods to efficiently compute Equation 5 using a quantum device. It can be measured experimentally by first preparing the state $U_{\theta}(x)^{\dagger}V_{\eta}(\tilde{x})\ket{0^{n}}$ and then computing the probability that all $n$ qubits measure to $0$ as shown in Figure 2(b). Alternatively, one can prepare the states $U_{\theta}(x)\ket{0^{n}}$, $V_{\eta}(\tilde{x})\ket{0^{n}}$ in two separate $n$-qubit registers and performing a SWAP test with an ancilla qubit. The latter scheme trades circuit depth for width.

In the previous scenario, we formulated the picture where data in each individual space is mapped by a unique embedding to a Hilbert space. The quantum feature maps are hitherto defined by applying unitaries to a standard $n$-qubit state $\outerproduct{0^{n}}{0^{n}}$. We will explore two variations of Equation 5, which by slightly tweaking, generalizes the previous setting.

Consider first a simple variant of the SWAP test method. Recall that in this setup, the feature maps $f_{\theta}(\cdot)$, $g_{\eta}(\cdot)$ act on separate registers of $n$ qubits each. For $m\leq n$, perform the SWAP test only on the first $m$ qubits of each register to obtain

$\displaystyle\mathcal{S}_{\theta,\eta}^{m}(x,\tilde{x}):=\Tr[f_{\theta,m}(x)^{\dagger}g_{\eta,m}(\tilde{x})\bigr{]},$

(6)

where the (mixed) states

	$\displaystyle f_{\theta,m}(x)=\Tr_{n-m}\bigl{[}{f_{\theta}(x)}\bigr{]}$		(7)
	$\displaystyle\ g_{\eta,m}(\tilde{x})=\Tr_{n-m}\bigl{[}g_{\eta}(\tilde{x})\bigr{]}$		(8)

are the partial trace of the original density matrices over the last $n-m$ qubits, such a circuit is shown in Figure 2(d). When $m<n$, $f_{\theta,m}$ encodes classical data into an $m$-qubit system by applying a possibly non-unitary CPTP map to the initial state $\outerproduct{0^{m}}{0^{m}}$.

The second tweak comes by slightly changing modifying how Equation 5 is measured without SWAP test. An alternative for calculating expectation value, as mentioned previously, is to apply $U_{\theta}(x)^{\dagger}V_{\eta}(\tilde{x})$ to $\ket{0^{n}}$ and measure the probability that all $n$ qubits return $0$. Instead, we will now only inspect the first $m$ qubits to obtain another modified similarity

	$\displaystyle\tilde{\mathcal{S}}_{\theta,\eta}^{m}(x,\tilde{x})$	$\displaystyle=\sum_{i\in\{0,1\}^{n-m}}\absolutevalue{\innerproduct{0^{m}i}{\phi_{x,\tilde{x}}^{\theta,\eta}}}^{2}$		(9)
		$\displaystyle=\Tr{\left\{\outerproduct{0^{m}}{0^{m}}\left(\Tr_{n-m}\left[\ket{\phi^{\theta,\eta}_{x,\tilde{x}}}\bra{\phi^{\theta,\eta}_{x,\tilde{x}}}\right]\right)\right\}}$		(10)

where $\ket{\phi^{\theta,\eta}_{x,\tilde{x}}}=U_{\theta}(x)^{\dagger}V_{\eta}(\tilde{x})\ket{0^{n}}$. This circuit is shown in Figure 2(c).

One way to think of this quantity is to regard the correspondence

$$(x,\tilde{x})\mapsto U_{\theta}(x)^{\dagger}V_{\eta}(\tilde{x})\ket{0^{n}}$$

as an embedding of pairs $(x,\tilde{x})$. One might more suggestively write$U_{\theta}(x)^{\dagger}V_{\eta}(\tilde{x})=\tilde{U}_{\alpha}(x,\tilde{x})$ for some effective $\tilde{U}$ as shown in Figure 3. Ideally, all similar pairs $(x,\tilde{x})\in\mathcal{E}_{l}$ should align perfectly with $\ket{0^{n}}$ and dissimilar pairs $(x,\tilde{x})\in\mathcal{D}_{k}$ should be orthogonal to $\ket{0^{n}}$. Here, we do not lose generality by mapping it to $\ket{0^{n}}$ as any other arbitrary state can be mapped back to $\ket{0^{n}}$ by a unitary which then can be absorbed into the embedding parameterized unitary. Measuring only some of the qubits effectively takes a partial trace over the remaining ones. Note that unlike the previous formulation, $\tilde{\mathcal{S}}^{m}_{\theta,\eta}$ is need not be symmetric in $x$ and $\tilde{x}$, even when $X=\tilde{X}$.

It is crucial to understand what it means for a pairwise functional maps $\mathcal{S}$ to be a “good similarity function” for a given learning problem.

Balcan et al. [22] proposed a new learning theory for similarity functions. This framework aims to generalize the learning techniques of kernels by relaxing some of the constraints we are currently interested in. They give intuitive and sufficient conditions for a similarity function to guarantee performance. Essentially, a similarity function ($\mathcal{S}$) is $(\epsilon,\gamma,\tau)$-good if a $1-\epsilon$ proportion of examples are on average $2\gamma$ more similar to examples of the same class than to examples of the opposite class for a $\tau$ proportion of examples. This is explicitly proved for a general $\mathcal{S}$ which need not be a metric that satisfies positive semi-definiteness or symmetry. Given a $(\epsilon,\gamma,\tau)$-good $\mathcal{S}$, it can be shown that $\mathcal{S}$ can be used to build a linear separator in an explicit projection space that has a margin $\gamma$ and error arbitrarily close to $\epsilon$.

Formally defined in Definition II.1, it can also be shown that the set of similarity maps defined by this is strictly larger than the set of metric maps and also includes non-positive semi-definite kernels and asymmetric maps. With the above definition, Ref[22] proved the following theorem

Theorem II.1 states that, if enough data is available for the problem $P$, then for a $(\epsilon,\gamma,\tau)$-good similarity function, with high probability there exists a low-error (arbitrarily close to $\epsilon$) linear separator in mapped $\phi$-space. This framework of $(\epsilon,\gamma,\tau)$-good functions opens the door for us to evaluate the performance that one can expect a global linear separator to have, depending on how well a similarity function satisfies Definition II.1. We will see in subsection III.2 that this method, at least numerically, can separate the classes as needed by definition.

By taking $X=\tilde{X}$ in metric learning setting, we reduce to the framework of classical metric learning where the sought-after $d$ is assumed to be a distance (pseudo-)metric in the usual sense. Thus, $d$ should satisfy the following properties $d(x_{1},x_{2})\geq 0$ with equality when $x_{1}=x_{2}$; $d(x_{1},x_{2})=d(x_{2},x_{1})$, and $d(x_{1},x_{3})\leq d(x_{1},x_{2})+d(x_{2},x_{3})$ for all triples $x_{1},x_{2},x_{3}\in X$. One way to enforce these constraints is to seek a distance function of the form

$$d_{\theta}(x_{1},x_{2})=\|f_{\theta}(x_{1})-f_{\theta}(x_{2})\|_{Z}$$

where $f_{\theta}$ is some neural network mapping $X$ into some latent Euclidean space $Z$. This construction, depicted in Figure 4, is called a Siamese neural network [20]. By taking $f_{\theta}(x)=\outerproduct{\phi_{\theta}(x)}{\phi_{\theta}(x)}$ as a quantum feature map as in the previous subsection, we have

$\displaystyle d_{\theta}(x_{1},x_{2})^{2}=2-2K_{\theta}(x_{1},x_{2})$

(11)

where $K_{\theta}(x_{1},x_{2})=\absolutevalue{\innerproduct{\phi_{\theta}(x_{1})}{\phi_{\theta}(x_{2})}}^{2}$ is a parameterized quantum kernel[23, 24]. As noted in Ref.[24, Appendix A], the training criterion Equation 1 implicitly guides the latent space embeddings $f_{\theta}$ to separate dissimilar pairs while compressing similar pairs together.

Next, in case of the similarity $\mathcal{S}^{m}$, as hinted in subsection II.2, when $m<n$, $f_{\theta,m}$ is a non-unitary CPTP map. This thus enriches the family of feature maps at our disposal compared to the previous case. Intuitively this can be understood as following, we prepare two quantum states that map classical data $x$ and $\tilde{x}$ to their respective states $\ket{x},\ket{\tilde{x}}$. In the previous multi-space metric learning case, we required that (dis)similar elements be mapped (far)close to each other in the Hilbert space $\mathcal{H}^{n}$ with dimension $n$. In contrast, similarity defined in Equation 6 requires that dis)similar gets mapped (far)close to each other in the Hilbert space $\mathcal{H}^{m}\subset\mathcal{H}^{n}$. This is similar to the projective SWAP test used in Ref[25].

We will now offer some heuristics for the more general similarity measure $\mathcal{\tilde{S}}^{m}$. It is easier to cluster points together in lower-dimensional spaces, which may help when similar raw data are far apart in their ”native” spaces $X,\tilde{X}$. The trade-off is that separating points becomes hard. The number of qubits measured changes the relative difficulties of satisfying the training constraints.

Precisely, suppose one measures $m$ of the $n$ qubits. Then if $(x,\tilde{x})\in\mathcal{E}_{l}$, the possible embeddings $\ket{\phi^{\theta,\eta}_{x,\tilde{x}}}$ that satisfy the constraint $\mathcal{S}^{m}(x,\tilde{x})=1$ live in a subspace of dimension $d_{\mathcal{E}_{l}}=2^{n-m}$. This space is spanned by all basis states $\{\ket{0i},\ i\in\{0,1\}^{n-m}\}$. On the other hand, if $(x,\tilde{x})\in\mathcal{D}_{k}$, the states $\ket{\phi^{\theta,\eta}_{x,\tilde{x}}}$ that satisfy $\mathcal{S}^{m}(x,\tilde{x})=0$ constitute a subspace of dimension $d_{\mathcal{D}_{k}}=2^{n}-2^{n-m}$. The ratio $\zeta=\frac{d_{\mathcal{E}_{l}}}{d_{\mathcal{D}_{k}}}=\frac{1}{2^{m}-1}$ describes the relative difficulties of the training constraints. In the scenario where $m=1$, we have $\zeta=1$, i.e we provide equal volumetric space in the model to place similar and dissimilar data points. In contrast, when $m=n$, $\zeta=\frac{1}{2^{n}-1}$ where we provide larger space for dissimilar points to live. This gives us the ability to tune the model space we are working with based on the data provided, for instance for an highly imbalanced data set (when $\absolutevalue{l-k}>>0$), one can choose $m$ based on if $\frac{l}{k}\approx\zeta$.

We illustrate these considerations with a toy example. Consider the following simple two-qubit circuit – $U(x,\tilde{x})=R_{y}(x,1)CNOT(0,1)R_{x}(\tilde{x},1)$ where $R_{x}(a,b)$ and $R_{y}(a,b)$ are the Pauli $X$ and $Y$ rotations by angle $a$ on qubit $b$ – and the state $\ket{\phi^{\theta,\eta}_{x,\tilde{x}}}=\ket{\phi_{x,\tilde{x}}}=U(x,\tilde{x})\ket{0^{\otimes 2}}$. $U$ can be interpreted as a embedding of $x,\tilde{x}$ in a Hilbert space for a particular choice of parameters. By traversing $0<x,\tilde{x}\leq 2\pi$, we can trivially look at the entire embedding space accessible by the constant ansatz $U$. We can thus look at the number of pairs of similar and dissimilar points that are accommodated by this ansatz based on the measure $\mathcal{S}^{m}$ for various $m$ (in this case $m\in\{1,2\}$), both of which is shown in Figure 5(a) as circuits. As shown in appendix A one gets

	$\displaystyle\mathcal{S}^{2}(x,\tilde{x})$	$\displaystyle=\cos^{2}{\left(\frac{x}{2}\right)}\cos^{2}{\left(\frac{\tilde{x}}{2}\right)},$		(12)
	$\displaystyle\mathcal{S}^{1}(x,\tilde{x})$	$\displaystyle=\frac{\cos{\left(x-\tilde{x}\right)}+\cos{\left(x+\tilde{x}\right)}}{4}+\frac{1}{2}.$		(13)

In Figure 5(a), we plot $\mathcal{S}^{1}$ and $\mathcal{S}^{2}$ for all values of $x,\tilde{x}$. Intuitively, $\mathcal{S}$’s closeness to a value of ($0$)$1$ indicates the paired points $x,\tilde{x}$ are (dis)similar to each other. We see that in the case of $\mathcal{S}^{2}$ we have exactly 4 points in the space that are maximally similar to each other. In contrast, a huge set of points with an almost flat region have a similarity measure of $0$. Comparatively, in $\mathcal{S}^{1}$, we have lifted a majority of this flat region to support more points that are similar to each other. To better understand this, we also look at the Density of States (DoS) of $\mathcal{S}$ for both the cases. The following defines this,

$\displaystyle D(\mathcal{S}):=\int\frac{\mathrm{d}x\mathrm{d}\tilde{x}}{(2\pi)^{2}}\cdot\delta(\mathcal{S}-\mathcal{S}(x,\tilde{x})).$

(14)

$D(\mathcal{S})$ essentially counts the number of points inside each fundamental unit of similarity. Figure 5(b) shows the DOS, where we see a peak close to $0$, for $\mathcal{S}^{2}$ indicating that this measure allows for a high imbalance between similar and dissimilar points. In the case of $\mathcal{S}^{1}$, as discussed previously, we have a perfectly even split of volume between similar and dissimilar points with mid (dis)similarity measure being $0.5$.

Let us now place the above problem in the setting of retrieval problem to gauge the ability of the respective family of similarity measures. Given two subspaces $X,\tilde{X}\in(0,2\pi]$, and $x_{s},x_{d}\in X$, we strive to find some point $\tilde{X}$ that is most similar to $x_{s}$ but most dissimilar to $x_{d}$. This problem can be formulated as minimizing

$\displaystyle\min_{\tilde{x}}\mathcal{L}_{\mathcal{S}}(\tilde{x},x_{s},x_{d})$

(15)

where

$\displaystyle\mathcal{L}_{\mathcal{S}}(\tilde{x},x_{s},x_{d})=\frac{1}{2}\left[(1-\mathcal{S}(x_{s},\tilde{x}))^{2}+\mathcal{S}(x_{d},\tilde{x})^{2}\right]^{\frac{1}{2}},$

(16)

for some similarity measure $\mathcal{S}$ and loss function $\mathcal{L}$. $\mathcal{L}(\tilde{x},\cdot,\cdot)$ measures the loss of how well the similarity measure has performed. As a first illustration, we pick $x_{s}=0.3$ and $x_{d}=0.5$ and then calculate $\mathcal{L}(\tilde{x},x_{s}=0.3,x_{d}=0.5)$ using the same embedding $U$. This is shown in Figure 6(a), where the dotted lines use the measure $\mathcal{S}^{m=2}$ while the red line uses $\mathcal{S}^{m=1}$. Since the quantum embedding map is the same for both the cases, they both have the same optimal $\tilde{x}\approx\frac{\pi}{2}$. We see that the loss function, which can be used as a surrogate to quantify how good the similarity measure is in separating the optimal $\tilde{x}$, is much lower/deeper in the case of $m=1$. Thus partial measurement of $m=1$ is able to attain a better separation than $m=2$. To complete the analysis, we calculate the quantity

$\displaystyle\lambda(x_{s},x_{d})=\mathcal{L}_{\mathcal{S}^{1}}(\tilde{x}_{x_{s},x_{d}}^{*},x_{s},x_{d})-\mathcal{L}_{\mathcal{S}^{2}}(\tilde{x}_{x_{s},x_{d}}^{*},x_{s},x_{d}),$

(17)

where $\tilde{x}_{x_{s},x_{d}}^{*}$ is the optimal $\tilde{x}$ for the pair $(x_{s},x_{d})$. $\lambda(x_{s},x_{d})$ quantifies the (dis)advantage one gets by using $\mathcal{S}^{1}$ instead of $\mathcal{S}^{2}$. Figure 6(b) plots $\lambda$, where one sees that at worst case, we do not get any larger separation/better performance using partial measurement. But there are cases (as seen in Figure 6(a) and indicated by non zero value in (b)), where the (dis)similarity of $\tilde{x}$ is better in partial measurement.

Classification tasks and the notion of similarity are deeply connected. Suppose we are given $m$ clusters of data

$$C_{1}=\{x_{1,i}\}_{i},\ C_{2}=\{x_{2,i}\}_{i},\dots,C_{m}=\{x_{m,i}\}_{i}\subset\mathbb{R}^{d}.$$

To classify a new data point $x\in\mathbb{R}^{d}$, we want to determine the class “most similar” to $x$ - in other words, a fidelity classifier[23, Appendix C]. Here, maximizing the separation of embedded data from different classes amounts to minimizing the empirical risk of the classifier for a linear loss function. An alternative method, which we do not pursue here, is to use the trained kernel in a support vector machine classifier as proposed by Hubregtsen et al [24]. Instead, it suffices to compare $x$ with a representative sample from each class. While the usual distance in $\mathbb{R}^{d}$ provides a naive notion of similarity, distance measures tailored to the data may perform better if the classes are irregularly shaped. In Figure 9, we show our GQSim being used for $\mathcal{X}=\mathcal{\tilde{X}}=\mathbb{R}^{2}$ where (a) is multi-class classification and (b) is a more complex structure for similarity measure. In (a), we compare each point in the the space to single sample from each train class to do a one shot classification while in (b) we compute the similarity and normalize it to 1 for each point in space to get the probability of them being similar to yellow or purple family.

The image example discussed in subsection III.2 is a toy model of the following classical problem: given $m$ clusters of images and crude information about the clusters, one can use this information to encode the images in a different subspace $\tilde{X}$. This information is termed side information, and it appears naturally in many applications. For example, when classifying images containing {house, dog, cat}, we know that even though they belong in discrete categories, the distance between categories dog-cat is closer than dog/cat-house. Another example of side information is categorizing the rating of a given movie in {1,2,3}. Even though each movie has a category, one has the information that the categories are ordered i.e $1<2<3$. This information can be encoded in $\tilde{X}$ space by manually choosing clusters for $\{1,2,3\}\to\{c_{1},c_{2},c_{3}\}$, say in $\mathbb{R}^{1}$, such that center of $c_{1}<c_{2}<c_{3}$. Thus, now learning the similarity between the movie space and $\tilde{X}$ space and using this similarity to categorize movies has more implied structure built into it than directly classifying them.

Link prediction[4] is an important aspect of network analysis and an area of key research. Graph completion problem[28] is a subset of link prediction, where it is assumed that only a small sample of a large network (e.g., a complete or partially observed sub-graph of a social graph) is observed, and one would like to infer the unobserved part of the network. To formalize the setting, we assume there is a true uni directed unweighted graph $\mathcal{G}=(\mathcal{V},\mathcal{E})$ on $n=|\mathcal{V}|$ distinguishable nodes with its own attributes $\mathbf{v}_{i}$ for $1\leq i\leq n$ with an adjacency matrix $\mathbf{A}\in\{0,1\}^{n\times n}$ where $0$ denotes an absent edge and $1$ denotes a connected edge. We then assume that only a partially observed adjacency sub-matrix $\mathbf{O}\in\{0,1\}^{m\times m}$ with $1\leq m\leq n$ induced by a sample sub-graph $\mathcal{G}^{\prime}=(\mathcal{V},\mathcal{E}^{\prime})$ with $\mathcal{E}^{\prime}\subset\mathcal{E}$, of original graph is given. We are now interested in predicting the complete adjacency matrix $\mathbf{A}$ based on the partially observed sub-matrix $\mathbf{O}$ using the learned similarity measures of the node attributes $\mathbf{v}_{i}$. For example, in a social community network, the nodes might represent the individuals, and the link of nodes represents the relations between the individuals. The attributes for each node could include the person’s metadata like interest, age, location, occupation, etc. The more similar attributes of the two nodes are, the higher probability they have a relationship and are linked in the network.

As an example, in Figure 10 we will associate attributes of each node with a $d=2$ dimensional data coming from $k=2$ unique distributions $P_{k}$. Edges are then connected ($1$) to nodes having attributes from the same distribution and not connected $(0)$ if their attributes correspond to a different distribution. This defines the real hidden graph $\mathbf{A}$. After this, we randomly select $\approx 10\%$ of the edges to create the sub-graph $\mathbf{O}$. This is then fed into the algorithm to learn the similarity between the nodes, which is then used to complete the graph. Since we have chosen an uni-directed graph $\mathcal{G}$, connection between $(v_{i},v_{j})\in\mathcal{V}$ is going to be the same between $(v_{j},v_{i})$. This symmetry is forced in the model by setting $f_{\theta}=g_{\eta}$ of the learning model, i.e. we choose to embed both comparing points with the same embedding. Clearly, for general directed multigraph, one needs to go beyond symmetry and have the property that $(v_{i},v_{j})\neq(v_{j},v_{i})$. Moreover, in the case of a directed multigraph, one needs to allow for nontrivial $(v_{i},v_{i})$ to indicate loops in the graph potentially. For these cases, one may opt to use the generalized similarity learner to embed the points in different embedding and trace out parts of the Hilbert space.

The figure shows the graph nodes being plotted in a 2D space, with each node placed corresponding to its attribute. Red lines indicate the connected nodes, while dotted lines indicate no connection. Missing lines in the left figure correspond to the lack of information. The right figure shows the predicted complete matrix. As seen from the figure, the points from the same spatial cluster are connected while the nodes with attributes belonging to different $P_{k}$ are disconnected. In the experiments where we have $P_{k}$ as Gaussian with random spread $\in[0.5,1.5]$, a reduced graph with edges as little as $1\%$ reproduces the final graph with $100\%$ accuracy.

Given a learned similarity measure between two different spaces, one can now use support points to generate new data in the other space. More concretely, given a learned similarity measure $\mathcal{S}$, and a support point $x_{s}\in X$, one can generate an unseen data point $\tilde{x}_{s}$ in $\tilde{X}$ using the following optimization problem

$\displaystyle\min_{\tilde{x}}1-\mathcal{S}(x_{s},\tilde{x}).$

(20)

Such tasks occur naturally in many scenarios; for example, in the case of language translation between two languages, one can generate new unseen similar sentences in another language based on learned similarity measure. Despite being an optimization problem in feature space, this is efficient in our quantum case. This is because features are directly encoded as parameters in our PQCs and thus can be efficiently differentiated[29]. To illustrate this, we again invoke the example discussed in subsection III.2. Given an image, we solve Equation 20 to find the closest point in $\mathbb{R}^{2}$ for a given image of type “right”. We show this in Figure 11, where we plot the cost function of this minimization problem and the minimization steps.

The authors thank Jack Baker for insightful discussions.