Universal algorithms for quantum data learning

Data analysis is a central element of the contemporary world, where data is abound but mostly featureless. The overall aim is to extract information from structured or unstructured data that generally comes from an unknown stochastic process. This information may be about the generating process or the data itself, e.g., a feature shared by all data instances or a global property of the analyzed dataset.

For instance, a dataset $\mathcal{D}_C=\{x_i\}$ may allow us to infer some property of an unknown underlying distribution p_X under the assumption of independent and identically distributed (iid) sampling; dropping this assumption, we may instead aim to identify some geometric property of $\mathcal{D}_C$; or if data points x_i come with labels y_i , we may use the set $\{(x_i,y_i)\}$ to learn how to attach a label $y'$ to an unlabeled data point $x'$. The inference algorithms in these examples all try to learn some property of a given dataset, and their design is made with universality in mind: the analysis may only partially depend on certain generic features of the data, but not on how the data looks like in a particular instance, and therefore it works for any possible input that has these features.

In the emerging world of quantum technologies, one can conceive scenarios where not only the processing of information is done by exploiting quantum resources but also data itself is quantum. It is only natural to ask: what sort of information can we learn from a quantum dataset, and how do we extract it? In this Perspective, we review under a common framework a series of works that deal with these questions and share two important features: they are universal algorithms in the above sense, and they use symmetries present in the dataset to their advantage.

First, we need to define precisely what we mean by quantum data. We can think of a quantum particle in some state $\rho_i$ as the natural generalization of a sample x_i from a classical probability distribution p_X , and hence a quantum dataset would be a collection of quantum states $\mathcal{D}_Q=\{\rho_i\}$. Importantly, we assume to have access to the quantum particles but not the density matrices representing their states, which are unknown to us. The states $\rho_i$ are the result of an inaccessible quantum process, in the same way as we do not control the generating processes of classical data samples x_i .

The types of properties that we may aspire to learn from a quantum dataset crucially depend on the assumed structure of $\mathcal{D}_Q$. For instance, if the data consists of N identical copies of an unknown pure qubit state, e.g., $\mathcal{D}_Q = \rho^{\otimes N}$ with $\rho=|{\psi}\rangle\langle{\psi}|$, we can try to learn the direction the state is pointing at; this is an individual property of every element in $\mathcal{D}_Q$. However, if $\mathcal{D}_Q = \rho_0^{\otimes n}\otimes\rho_1^{\otimes N-n}$ with $\rho_0 \neq \rho_1$, the above property becomes ill-defined. Instead, if we know n, we may learn a relational property of the components of the dataset, e.g., the overlap $tr \rho_0\rho_1$. As a last example, in the case we do not even know n in the dataset above, we can aim at learning it, that is, how to split $\mathcal{D}_Q$ into different subsets of identical copies. This would be a global property of the set.

A powerful way to address this kind of problems is to consider the symmetries of $\mathcal{D}_Q$ under which the property of interest remains invariant. These symmetries are induced by the structure of the dataset, e.g., in the first example above, permuting the elements of $\mathcal{D}_Q$ does not alter the direction encoded in ρ. The symmetries of the properties determine that the optimal measurement can be chosen to respect the symmetries, simplifying both the design of the measurement and the analysis of the performance. Furthermore, measurements can be chosen to be sensitive only to changes in the property of interest and not depend explicitly on any other information contained in an instance of $\mathcal{D}_Q$, e.g., on the particular states $\rho_0$, $\rho_1$ for any of the problems outlined above. This type of quantum learning algorithms are thus universal, agnostic to the input data as long as $\mathcal{D}_Q$ fulfils the symmetries prescribed by the problem at hand, and are consistent with the described notion of quantum data where the actual density matrix of $\mathcal{D}_Q$ is unknown to the experimenter.

Universal algorithms are generally optimized for two kinds of figures of merit: worst-case and average-case. In the worst-case approach, the interest is usually in quantifying the smallest dataset size (sample complexity) such that the inference task is guaranteed to be successfully completed with high probability even with the worst possible input, and expressing this minimal size in terms of the extensive parameters of the problem, such as the dimension of the quantum systems involved [1]. In the average case approach, the interest is in quantifying the average performance over all admissible datasets of a given size, with a prior probability distribution over inputs reflecting the symmetries of the property of interest and our prior knowledge about the parameter we want to infer (e.g., [2]) ¹ . The latter approach often allows us to work out explicit expressions for the optimal value of the figure of merit and to compute its asymptotics for large datasets, while the former usually reports only the asymptotic scaling of the sample complexity, but has the benefit of not having to specify any prior probability distribution. Both approaches give interesting information about the performance of optimal tasks. While primitive tasks like tomography and identity testing are studied in depth in the worst-case scenario (for a review of these results, see [3]), problems with more structure such as supervised and unsupervised classification of quantum states and change point detection (described below in detail) have been analyzed mostly in the average-case asymptotic setting. This is also usually more natural since the task itself may not always be realizable with arbitrarily high probability of success as the size of the dataset increases, so that there is not an immediate notion of sample complexity. In this Perspective we want to highlight the importance of these structured problems as generalizations of learning tasks to quantum data, therefore we will see average-case results more in depth. It would be nonetheless interesting and relevant to formulate adequate worst-case questions for the same problems.

In order to illustrate the strength of symmetry principles in this context, we begin with the primitive example of estimating an unknown pure state given N copies of it. We proceed with a more structured dataset in the case of programmable discrimination of pure qubits, followed by a formal description of the underlying techniques and the abstract general structure of the problem. We then review more complex results that share the same approach, and we end the paper with a discussion.

State estimation [2] is an essential primitive in quantum inference where given n copies of an unknown state $|{\psi}\rangle$ of dimension d a suitable measurement strategy is devised in order to infer the state of the system. Given a measurement characterized by a positive operator-valued measure (POVM) $E=\{E_{\alpha}\geq 0, \sum_{\alpha} E_{\alpha}=\mathbb{1}\}$, an estimate $|{\phi_\alpha}\rangle$ needs to be produced for each of the outcomes α such that it resembles as much as possible the input state $|{\psi}\rangle$, as measured by the fidelity $f(\phi_{\alpha},\psi)=|\langle{\phi_{\alpha}}{\psi}\rangle|^{2}$. In such a setting it is quite natural to designate a prior distribution for our dataset: assuming that all pure states $|{\psi}\rangle$ are equally likely singles out the unitarily invariant measure $\mathrm{d}\psi$. Once equipped with a prior distribution of the data we can assess the performance of a given strategy by the average fidelity over all input states and all measurement outputs. The optimal strategy is given by the following Bayesian optimization:

$$\begin{equation} F_{\mathrm{ave}}:=\langle{f}\rangle=\max_{E} \int \mathrm{d}\psi\sum_{\alpha}f(\phi_{\alpha},\psi) \mathrm{tr}\left(E_{\alpha}|{\psi}\rangle\langle{\psi}|^{\otimes n}\right). \end{equation} \tag{ 1 }$$

We note here that in general there are no established criteria to designate a prior distribution on the input data —this is the main caveat of such Bayesian formulations. Alternatively, one can define a worst-case figure of merit for $F_{\mathrm{wc}}= \min_{\psi} \max_{E} \sum_{\alpha}f(\phi_{\alpha},\psi){\rm tr}\left(E_{\alpha}|{\psi}\rangle\langle{\psi}^{\otimes n}\right)$, which does not require to fix a prior.

Next we show how the symmetries readily lead to elegant solutions to the optimization problem. From eq. (1) and writing $\psi=|{\psi}\rangle\langle{\psi}|$,

$$\begin{align} F_{\mathrm{ave}} &= \max_{E} \int\mathrm{d}\psi\sum_{\alpha}\mathrm{tr}\left[(\phi_{\alpha}\otimes E_{\alpha}){\psi}^{\otimes n+1}\right] \end{align} \tag{ 2 }$$

$$\begin{align} &\quad=\frac{1}{d_{n+1}}\max_{E} \sum_{\alpha}\mathrm{tr}[(\phi_{\alpha}\otimes E_{\alpha})\mathbb{1}_{\mathrm{sym}}^{(n+1)}] \end{align} \tag{ 3 }$$

$$\begin{align} &\quad\leq \frac{1}{d_{n+1}}\max_{E} \sum_{\alpha}\mathrm{tr}\left[(\phi_{\alpha}\otimes E_{\alpha})\right] \end{align} \tag{ 4 }$$

$$\begin{align} &\quad=\frac{1}{d_{n+1}}\mathrm{tr}\left(\sum_{\alpha} E_{\alpha}\right)= \frac{d_{n}}{d_{n+1}}=\frac{n+1}{n+d} \,, \end{align} \tag{ 5 }$$

where in (3) we have used that $\int\mathrm{d}\psi \psi^{\otimes n}$ is the unique trace one operator with support in the fully symmetric subspace ² of n qudits that is invariant under rigid $U^{\otimes n}$ unitary transformations, i.e., the projector $\mathbb{1}_{\mathrm{sym}}^{(n)}$ normalized by its dimension $d_{n}=\binom{n+d+1}{n}$ (see Schur lemma [4]); in (5) we have also used that the input data lies on the symmetric subspace and hence without loss of generality we can take $\sum_{\alpha} E_{\alpha}=\mathbb{1}_{\mathrm{sym}}^{(n)}$.

On the other hand it is straightforward to check that the inequality in (4) can be attained by picking a covariant continuous POVM with elements $E_{\phi}=d_{n} \phi^{\otimes n}$, which fulfill $\int \mathrm{d}\phi E_{\phi}=\mathbb{1}_{\mathrm{sym}}^{(n)}$, and a corresponding estimate ϕ. Note that this optimal covariant strategy returns an equally good estimate for each possible input state, hence the average fidelity coincides with the worst-case fidelity $F_{\mathrm{wc}}$.

Programmable discriminators are universal devices capable of performing binary classification of an unknown quantum state when the information about the two possible classes is provided as quantum data. This is in contrast to the standard state discrimination scenario, where we are provided with the classical description of the possible states of the system. In their simplest form, they take as input a quantum dataset of the form $\mathcal{D}_Q=|{\psi_0}\rangle_A\otimes|{\psi_i}\rangle_B\otimes|{\psi_1}\rangle_C$, where $i=0,1$ and $|{\psi_{0,1}}\rangle$ are unknown qubit states. Registers A, C are termed program ports, and take the two different "template" quantum states. The register B is called data port, and the goal of the device is to identify the value of the label i.

We demand such device to be universal, i.e., that it works for any pair of program states $|{\psi_{0,1}}\rangle$. Choosing an average case approach, we say that a programmable device performs optimally if the average error rate in the label identification with respect to a uniform distribution of pairs of states is the minimal one. The key observation that exploits the symmetries of $\mathcal{D}_Q$ and allows to solve the problem comes from the following consideration: if the data state is $|{\psi_0}\rangle$, the effective state entering the device is

$$\begin{eqnarray} &&\sigma_0 = \int \mathrm{d}\psi_0 \mathrm{d}\psi_1 \psi_0^{\otimes 2}\otimes \psi_1 = \frac{1}{6}\mathbb{1}^{AB}_{\rm sym}\otimes\mathbb{1}^C, \end{eqnarray} \tag{ 6 }$$

where we have applied Schur lemma [4] as in deriving eq. (3), and we have used that $d_2 d_1 =6$ for qubits. Here $\mathbb{1}^{AB}_{\rm sym}$ is the projector onto the fully symmetric space of qubit systems A and B, and $\mathbb{1}^C$ is the identity matrix in the qubit system C. Likewise, if the data state is $|{\psi_1}\rangle$ one has $\sigma_1= \frac{1}{6}\mathbb{1}^A\otimes 1^{BC}_{\rm sym}$.

Notice that the spectrum of both matrices $\sigma_0$ and $\sigma_1$ is identical and that the basis elements of their support simply differ in the way the three qubits are coupled, hence this information is the only one that can be used to distinguish the two effective states. The states are diagonal in the total angular momentum basis $|{JM;q_i}\rangle$, where q_i tags the two different ways the qubits are coupled, i.e., $q_0=\{AB, ABC\}$ and $q_1=\{BC, ABC\}$, to arrive to $J=3/2,1/2$. Furthermore, we note that the overlap between the two bases fulfils $\langle{JM;q_{0}|J'M';q_{1}}\rangle=C_J \delta_{JJ'}\delta_{M M'}$, that is, the relevant quantum numbers for distinguishing $\sigma_0$ from $\sigma_1$ are J and q_i . The numbers C_J are called Racah coefficients [5], which in this example take the values $C_{3/2} = 1$ and $C_{1/2}=1/2$.

The optimal classification procedure is then provided by first measuring J, and, upon having obtained a specific result with probability $p_J=\mu_J/6$, with $\mu_J$ the dimension of the subspace J, a subsequent Helstrom measurement [6] that optimally distinguishes the pure states $|{JM,q_0}\rangle$ and $|{JM,q_1}\rangle$ and has error probability $P_e^J=[1-\sqrt{1-C_J^2}]/2$. The overall minimum error probability reads

$$\begin{align} P_e =\sum_{J=1/2}^{3/2} p_{J} P_e^J=\dfrac{1}{2}\left(1-\dfrac{1}{2\sqrt{3}}\right) \,. \end{align} \tag{ 7 }$$

In contrast to the previous example of state estimation, where the covariant continuous POVM is sensitive to the quantum number M encoding information about the direction where $|{\psi}\rangle$ is pointing at, here this information is irrelevant. Observe that eq. (7) is solely a function of invariant quantities such as dimensions and overlaps C_J . In spite of the little information we had about the states (not even its classical description) only roughly one third of the times (on average) the data state will be wrongly classified.

The previous two examples demonstrate how a clever use of the inherent symmetries possessed by the quantum data sample leads to elegant, closed form expressions for assessing the average performance of the requisite task. We now introduce the mathematical framework that such symmetries impose on the state space of a quantum dataset, and show how such framework can be generally exploited to significantly constrain the search for the optimal measurement [7].

Let $\mathcal{H}_d$ be a d-dimensional Hilbert space, and let us assume that $\mathcal{D}_Q$ is an element of the space of linear operators over the n-fold tensor product of $\mathcal{H}_d$, $\mathcal{D}_Q\in\mathcal{L}(\mathcal{H}_d^{\otimes N})$. The tensor product space $\mathcal{H}_d^{\otimes N}$ naturally carries the action of two fundamental symmetry groups: that of the special unitary group in d dimensions SU(d), as well as that of the permutation group of N elements, S_N . The action of these two groups on any state $\mathcal{D}_Q$ is mediated through their respective unitary representations, $\{U^{\otimes N}_g\in {GL}(d), \forall\, g\in {SU}(d)\}$, $\{V_\tau\in {GL}(d), \forall\, \tau\in {S}_N\}$, whose effect on the orthonormal basis $\{|{i_1,\ldots,i_N}\rangle:=|{i_1}\rangle\otimes\ldots\otimes|{i_N}\rangle\}$ of $\mathcal{H}^{\otimes N}_d$ is given by

$$\begin{align} \begin{array}{rcl} U^{\otimes N}_g|{i_1,\ldots,i_N}\rangle&=& U_g|{i_1}\rangle\otimes\ldots\otimes U_g|{i_N}\rangle,\\ V_\tau|{i_1,\ldots,i_N}\rangle&=&|{i_{\tau^{-1}(1)}}\rangle\otimes\ldots\otimes|{i_{\tau^{-1}(N)}}\rangle. \end{array} \end{align} \tag{ 8 }$$

Using Schur lemmas [4], the representations U_g and $V_\tau$ in eq. (8) can be decomposed into a direct sum of irreducible representations (irreps), $U_g^{(\lambda)},\, V_\tau^{(y)}$, as follows:

$$\begin{eqnarray} \begin{array}{rcl} U^{\otimes N}_g &=& \displaystyle\bigoplus_{\lambda} \, U^{(\lambda)}_g\otimes \mathbb{1}^{(\lambda)}\quad \forall\, g\in \textit{SU}(d)\,,\\ V_\tau &=& \displaystyle\bigoplus_{y}\, V^{(y)}_\tau\otimes \mathbb{1}^{(y)} \quad \forall\, \tau\in\textit{S}_N\,, \end{array} \end{eqnarray} \tag{ 9 }$$

where $\lambda, y$ label inequivalent irreps of $\textit{SU}(d)$ and S_N , and $\mathbb{1}^{(\lambda)}, \mathbb{1}^{(y)}$ have dimensions $\nu_\lambda, \mu_y$ corresponding to the number of times the irreps $U_g^{(\lambda)},\, V_\tau^{(y)}$ appear in the decomposition of the corresponding representations. For the symmetric group the irrep label y denotes the ordered, integer partitions of N into at most d parts, i.e.,

$$\begin{equation} y:=\left\{(y_1,\ldots, y_d) \mid y_1\geq \ldots\geq y_d~~\mathrm{and}~~\sum_{k=1}^d y_k=N\right\}. \end{equation} \tag{ 10 }$$

The irrep label λ of $\textit{SU}(d)$, on the other hand, has a rather more complicated interpretation ³ . For the case where d = 2, λ labels the total angular momentum quantum number.

The irrep labels λ and y appear, at first sight, unrelated, but the irreps with non-zero multiplicity in the decomposition are in fact in one-to-one correspondence due to a important result in representation theory known as Schur-Weyl duality [8]. Noting that $[U^{\otimes N}_g, V_\tau]=0,\, \forall\, g\in\textit{SU}(d), \, \mathrm{and}\, \forall\, \tau\in{S}_N$, implies that the decomposition of $U_g^{\otimes N}$ in eq. (9) is block-diagonal with respect to the irrep decomposition of $V_\tau$ and vice versa. Schur-Weyl duality tells us that one may use the labels y to decompose both representations, where only products of irreps with the same labels appear in the decomposition. For $\textit{SU}(2)$, λ labels the values of the total angular momentum J of N ${\text{spin-}}\frac{1}{2}$ particles and can be obtained from the corresponding integer partitions as $\lambda=\frac{y_1-y_2}{2}$.

The block decomposition of eq. (9) induces a similar decomposition on the state space $\mathcal{H}_d^{\otimes N}$ as

$$\begin{equation} \mathcal{H}_d^{\otimes N}\cong \sum_y \mathcal{H}^{(y)}:=\sum_y \mathcal{U}^{(y)}\otimes \mathcal{V}^{(y)}, \end{equation} \tag{ 11 }$$

where we have used the irrep labels of $\mathrm{S}_N$ to label the various invariant subspaces $\mathcal{H}^{(y)}$. The latter can be further decomposed into a tensor product of subspaces $\mathcal{U}^{(y)}$, $\mathcal{V}^{(y)}$ which support the irreps $U_g^{(y)}$ and $V_\tau^{(y)}$ respectively. The congruence sign in eq. (11) indicates that there exists an orthonormal basis relative to which the total state space $\mathcal{H}_d^{\otimes N}$ assumes the specific block decomposition. This change of basis is accomplished by the Schur transform for which efficient circuits that implement it exist [9,10].

It is worthwhile pausing briefly to understand the nature and role of the block-diagonal decomposition in eq. (11). The irrep labels $\lambda,\, y$ encode globally invariant properties of the dataset, an example of which is the spectrum of eigenvalues of any ρ when $\mathcal{D}_Q=\rho^{\otimes N}$. Indeed, the vector y/N concentrates on the spectrum of ρ [11–14]. In general, the subspace $\mathcal{U}^{(y)}$ encodes information pertaining to properties associated with the symmetry group $\textit{SU}(d)$, such as the expectation value of some generator of $\textit{SU}(d)$ (see footnote  ⁴ ), which are invariant under all permutations. Conversely, the subspace $\mathcal{V}^{(y)}$ encodes information associated with the permutation group S_N —such as the labels of the constituent particles— which are in turn invariant under arbitrary rotations.

A case of particular relevance is when every admissible quantum dataset is invariant with respect to a group of transformations, i.e., $\mathrm{Inv}:=\{\mathcal{D}_Q^{(i)}\, \vert W_g\,\mathcal{D}^{(i)}_Q\, W_g^\dagger=\mathcal{D}^{(i)}_Q, \, \forall \, g\in G, \mathrm{and}\, \forall\, i\}$. In this case it is sufficient to consider invariant measurements [7], i.e.,

$$\begin{equation} M^{\mathrm{inv}}:=\{M_i\, \vert\, W_g\, M_i\, W^\dagger_g=M_i, \forall\, g\in G\}. \end{equation} \tag{ 12 }$$

Using (11) every element, M_i , of such a POVM can be written in block-diagonal form. For instance, if $G=\textit{SU}(d)$ then

$$\begin{equation} M_i\cong \sum_y \mathbb{1}^{(y)}\otimes M_i^{(y)}. \end{equation} \tag{ 13 }$$

A second case of particular importance is that of learning covariant properties of admissible quantum datasets, i.e., $\mathrm{Cov}:=\{\mathcal{D}_Q^{(g)}\,\vert\, \mathcal{D}_Q^{(g)}=W_g \mathcal{D}_Q W_g^\dagger, \, g\in G\}$ for some fiducial dataset $\mathcal{D}_Q$. Notice that learning a covariant property reduces to learning the group element $g\in G$. In this case it is sufficient to consider covariant measurements [7],

$$\begin{equation} M^{\mathrm{cov}}:=\{M_g \,\vert\, W_g\, M_0\, W^\dagger_g = M_g, \forall\, g\in G\}. \end{equation} \tag{ 14 }$$

Since $\sum\!\!\!\!\!\!{\int} M_g\, \rm{d}g=\mathbb{1}$, where $\mathrm{d}g$ denotes the invariant (Haar) measure of G, Schur-Weyl duality imposes a block-diagonal structure on the fiducial POVM element M₀. If $G=\textit{SU}(d)$ then

$$\begin{equation} M_0\cong \sum_y M_0^{(y)}, \end{equation} \tag{ 15 }$$

where $M_0^{(y)}\in\mathcal{L}(\mathcal{H}^{(y)})$ and $tr_{\mathcal{U}^{(y)}}M_0^{(y)}=\mathbb{1}^{(y)}\in\mathcal{L}(\mathcal{V}^{(y)})$. Similarly, if $G=S_N$, $tr_{\mathcal{V}^{(y)}}M_0^{(y)}=\mathbb{1}^{(y)}\in\mathcal{L}(\mathcal{U}^{(y)})$.

We note that both covariant and invariant measurement strategies are known to be optimal both when the figure of merit is chosen to be the average and the worst case [7]. We also note that in hybrid cases where the admissible quantum datasets are covariant but the property of interest, f, is invariant, i.e., $f(\mathcal{D}_Q)=f(W_g\mathcal{D}_Q W_g^\dagger), \forall\, g\in G$, then the optimal measurement is invariant.

We are now ready to review a selection of more complex structured quantum learning problems that can be addressed with the methods we have presented. They have a common structure in that the relevant information is extracted from the distribution of the irrep label y in the global Schur-Weyl decomposition and, possibly, from a measurement which is non-trivial only on the factor $\mathcal V^{(y)}$. Specifically, for a quantum dataset $\mathcal{D}_Q=\sigma=\bigotimes_{i=1}^l\rho_i^{\otimes n_i}$ we can apply Schur-Weyl decomposition to each iid part $\rho_i^{\otimes n_i}$, $\sum_{i=1}^l n_i=n$ and obtain

$$\begin{equation} \sigma=\bigotimes_{i=1}^l\sum_{y_i\vdash n_i}\pi^{(y_i)}(\rho_i)_i\otimes \mathbb{1}_i^{(y_i)}, \end{equation} \tag{ 16 }$$

where $y_i\vdash n_i$ denotes the various integer partitions of n_i , and $\pi^{(y_i)}(\rho_i)_i$ is a positive semi-definite operator that depends on $\rho_i$ (see footnote  ⁵ ). According to the same decomposition, we can write

$$\begin{align} \mathcal H_d^{\otimes n}&=\bigotimes_{i=1}^l\bigoplus_{y_i\vdash n_i}\mathcal U^{(y_i)}\otimes \mathcal V^{(y_i)}=\bigoplus_{\vec{y}\in\mathfrak{S}}\bigotimes_{i=1}^l\mathcal U^{(y_i)}\otimes \mathcal V^{(y_i)}\nonumber\\ &=\bigoplus_{\vec{y}\in\mathfrak{S}}\bigoplus_{y\vdash n}\mathcal{U}^{(y)}\otimes \mathcal W_{\vec{y};y}\otimes_{i=1}^l \mathcal V^{(y_i)}\nonumber\\ &=\bigoplus_{y\vdash n} \mathcal U^{(y)}\otimes \left(\bigoplus_{\vec{y}\in\mathfrak{S}} \mathcal W_{\vec{y};y}\otimes_{i=1}^l \mathcal V^{(y_i)}\right), \end{align} \tag{ 17 }$$

where $\mathfrak{S}:=\lbrace \vec{y}=(y_1,\ldots, y_l)\, \vert y_1\vdash n_1,\ldots, y_l\vdash n_l\rbrace$, $\mathcal{U}^{(y)}$ arises from the decomposition of $\otimes_{i=1}^l \mathcal{U}^{(y_i)}$ into irreps with $\mathcal{W}_{\vec{y};y}$ the corresponding multiplicity space of this decomposition (possibly zero-dimensional). Note that the second factor in eq. (17) is isomorphic to the space $\mathcal{V}^{(y)}$ in eq. (11). Under an $\textit{SU}(d){\text{-invariant}}$ measurement, the state σ of eq. (16) produces the same statistics over measurement outcomes as an $\textit{SU}(d){\text{-invariant}}$ state of the form

$$\begin{equation} \tilde{\sigma}=\bigoplus_{y\vdash n} \frac{\mathbb{1}^{(y)}}{\mu_y}\otimes \left(\bigoplus_{\overrightarrow{y}\in\mathscr{S}} \xi_{\overrightarrow{y} ;y}(\rho_{1},\ldots,\rho_l)\bigotimes_{i=1}^l \frac{\mathbb{1}^{(y_i)}}{\nu_{y_i}}\right), \end{equation} \tag{ 18 }$$

where all the information is contained within the positive semi-definite operators $\xi_{\vec{y};y}(\rho_{1},\ldots,\rho_l)$.

Programmable state discrimination

Supervised quantum learning

Unsupervised classification of quantum data

Quantum change point detection

Learning of distance measures

Learning properties of quantum datasets will be an indispensable step of upcoming quantum technological applications, be it as part of their normal operation or as a tool to certify that they work as intended. Indeed, the efficient certification of quantum states and processes has already been identified as a pressing need in near-term quantum applications [40], and protocols have been devised to test elementary properties assuming iid runs of quantum experiments. In this Perspective we have focused on quantum learning algorithms that go beyond the iid assumption and aim at more general types of properties of highly structured quantum datasets, leveraging symmetries to handle the increased complexity of the problems. Conceivably, this kind of problems will become more and more relevant in the context of future quantum networks, where quantum data are the natural carriers of information [41,42].

While the works reviewed here are concerned with optimal algorithms, it is worth noting that these may require applying collective measurements over a (typically) large quantum dataset, which are generally hard to implement in practice. Still, analyzing optimal performance provides an ultimate benchmark of the learning task at hand against which we can test more feasible strategies, e.g., based on LOCC or sequential measurements [43].

Finally, let us mention that the approach to quantum learning covered here can also be formally extended to quantum objects beyond states, such as quantum channels or quantum processes. In these, the quantum "dataset" would consist in, e.g., a number of independent uses of an unknown device that we regard as a black box, and over which we may make certain structural assumptions. One example of such a problem would be detecting the presence and position of an anomaly in a sequence of allegedly identical unitary operations [44], in a setting where the unitaries are not perfectly characterized.

We acknowledge financial support from the Spanish Agencia Estatal de Investigación, Grant No. PID2019-107609GB-I00 and Catalan Government for the project QuantumCAT 001-P-001644, co-financed by the European Regional Development Fund (FEDER). JC also acknowledges support from ICREA Academia award.

Data availability statement: No new data were created or analysed in this study.