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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?

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

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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 \begin{equation} \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} \end{equation}

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 \begin{equation} \mathbb{E}_{p(\rho)} \left(\mathbb{E}_{p(E)} |f(E) - \langle \hat{E}\rangle_{\rho}|\right) \tag{3} \end{equation} 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.

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  • $\begingroup$ it was a great answer, very well argued. Thanks for the time $\endgroup$
    – Dimitri
    Mar 11 at 14:32

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