# Is QST a inherently supervised or unsupervised problem in Machine Learning?

I am studying how to apply neural networks to the problem of Quantum State Tomography and I got confused when it comes to decide if this is a supervised or unsupervised learning problem.

At first, I came across the usage of Restricted Boltzmann Machines in the domain of generative models, which are used for unsupervised learning, which seems right since the state tomography is a problem where I do not have access to the state in question and the only information available are measurement outcomes: thus, I am feeding my learning algorithm with unlabeled data, that is, in an unsupervised fashion.

But then I found this paper and also this one describing QST in the PAC-learning framework, which as far as I know is a framework applied to supervised problems only, but the paper seems ok since they were dealing with a training set of $$(E, Tr(E\rho))$$ where $$E$$ are measurement operators and $$Tr(E\rho)$$ are the trace of the measurement operator times the density matrix. The elements of the training set were drawn i.i.d from a probability distribution.

My problem here is: in QST the probability distribution and the density matrix are unknown. So how can I formulate QST as pac-learning and how can it be suited into the framework of a supervised learning algorithm? Or it depends on what kind of algorithm I am using?

RBMs are not the only approach. Nowadays, researchers are using deep neural networks and a lot of generative modeling, the most promissing being the usage of Conditional GANs.

What am I missing?

• can you link the papers you're referring to?
– glS
Commented Mar 9 at 13:30
• @glS yes! Just eddited the question with them Commented Mar 11 at 13:43

To me, QST (at least the nonadaptive kind) is more like a parameter estimation problem - a more constrained type of learning problem than ordinary supervised ML.$$^1$$

The typical setup for parameter estimation starts with an unknown variable $$\theta$$ sampled from a prior distribution $$p(\theta)$$. This $$\theta$$ is fixed for the duration of the learning task, and (for simplicity) induces a distribution of observations $$p(\{(x_1, y_1), \dots, (x_n, y_n)\}|\theta):= p(B|\theta), \tag{1}$$ where $$B$$ represents the "training data", i.e. learner's observations. The learner's goal is, upon sampling observations from $$p(B|\theta)$$, to output an estimate $$\hat{\theta} = \hat{\theta}(B)$$ that is close to $$\theta$$. This might look like minimizing a distance $$d(\theta, \hat{\theta})$$ where the function $$d$$ does not depend on $$p(B|\theta)$$.

In basic QST, the observations look like $$B=\{(E_i,\langle \hat{E_i}\rangle_{\rho} )\}_{i=1}^n$$, where $$E_i$$ denotes a measurement setting and $$\langle \hat{E_i}\rangle_{\rho}$$ is an empirical estimate for the result of that measurement (subject to statistical noise, other experimental errors, etc). To connect this to parameter estimation, the variable $$\theta:=\rho$$ consists of all of the entries of $$\rho$$, and so $$\hat{\theta}:= \hat{\rho}$$ represents the learner's best guess for $$\rho$$. Typically we consider $$d(\rho, \hat{\rho}) = \lVert \rho - \hat{\rho} \rVert_1$$, and the learner wants to find $$\hat{\rho}$$ that minimizes this distance. If there's a prior distribution of states $$p(\rho)$$, the learner might try to minimize $$$$\mathbb{E}_{p(\rho)} \left(\mathbb{E}_{p(B|\rho)} \lVert \rho - \hat{\rho}(\{(E_1,\langle \hat{E_1}\rangle_{\rho}), \dots, (E_n,\langle \hat{E_n}\rangle_{\rho})\}) \rVert_1\right) \tag{2}$$$$

The reason this differs from supervised learning is that the term in the parentheses above depends explicitly on $$\rho$$. In most supervised learning problems, you care less about learning the exact value of an underlying parameter, and more about predicting each new label $$y_i$$ given $$x_i$$.

Given the above, phrasing QST as a supervised learning problem is kind of sketchy, but I'll try anyways. Here's one possible setup: Suppose $$\rho$$ is still sampled according to $$p(\rho)$$, but that instead of learning $$\rho$$, you want to be able to predict measurement outcomes for randomly sampled measurements $$E_i$$. Mathematically, you want to come up with a function $$f$$ that minimizes $$$$\mathbb{E}_{p(\rho)} \left(\mathbb{E}_{p(E)} |f(E) - \langle \hat{E}\rangle_{\rho}|\right) \tag{3}$$$$ This now resembles a supervised learning problem. In fact, its almost identical to linear regression: You randomly sample pairs $$(E_i, y_i)$$ where $$y_i:= \langle \hat{E}_i, \rho \rangle$$, and your goal is to predict $$y=\langle E, \rho \rangle$$ with respect to the distribution of $$E$$.

This problem should not be much harder than QST, since if we had a good estimate $$\hat{\rho}$$ for $$\rho$$, then we could compute $$\text{tr}(\hat{\rho}E_i) = y_i$$. On the other hand, it might be that our distribution over measurements only contains a handful of measurements $$E$$ so that full knowledge of $$\hat{\rho}$$ is simply unnecessary for the task (this is some of the intuition behind classical shadows techniques).

$$^1$$ The way I use terminology here (learning, prediction, parameter estimation) isn't necessarily standard - its always better to just explain the learning setting in detail.

• it was a great answer, very well argued. Thanks for the time Commented Mar 11 at 14:32