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I am refreshing my functional analysis knowledge to learn quantum machine learning and I am getting confused on Hilbert spaces. What does it mean for a "circuit to cover the Hilbert Space" I believe this means the can generate the entire space? I am looking for an example of this.

In the lecture I am watching, they say this is equivalent to "the extent to which states generated from the circuit deviates from the uniform distribution." Again, would it be possible to see an example of this?

Thank you. Here is the video: https://www.youtube.com/watch?v=20ftuhSV4sk&t=595s

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  • $\begingroup$ Yes, this means it can generate any pure state in the Hilbert space (it can generate any basis vector). It cannot produce a mixed state for you, even if it lives in the same Hilbert space $\endgroup$ Apr 26, 2023 at 15:03

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From the context and wording its not entirely clear what they are talking about. I suspect it is something similar to this: Say $U(\theta)$ is a $d$-dimensional unitary describing a parameterized circuit with $m$ parameters $\theta \in \Theta \subset \mathbb{R}^m$ and define $|\psi(\theta)\rangle := U(\theta)|0\rangle$. Then you can evaluate average-case behavior of the circuit (means, variances) with respect to to these parameters using a operator like this: $$ T_t = \int_{\Theta} |\psi(\theta)\rangle \langle \psi(\theta)|^{\otimes t} \mu_{\theta}(d\theta), \tag{1} $$ where $\mu_{\theta}(\theta) = p(\theta)d\theta$ is a measure assigning probabilities to each value of $\theta$. For example the expected value of an observable $O$ averaged over values of $\theta$ would be $$ \mathbb{E}_{\theta \sim \Theta}[\text{Tr}(O|\psi(\theta)\rangle \langle \psi(\theta)|] = \text{Tr}(T_1O). \tag{2} $$ Then, one way to define the expressivity of your circuit is to compare $T_t$ to an integral over the Haar measure (Eq. 3, Sim, 2019): $$ A_t = \int_{\text{U}(d)} (U|0\rangle \langle 0|U^\dagger)^{\otimes t} \mu(dU) - T_t, \tag{3} $$ where $\mu(dU)$ is the Haar measure on the set of $d$-dimensional unitaries $\text{U}(d)$. This difference gets smaller as the set of states generated by randomly sampling $\theta$ gets closer to a "uniform distribution" over $d$-dimensional states. The paper I linked has some nice figures and discussion on this point.

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