13.2 Expressive Power
We saw in previous chapters how PQCs can be applied to solving optimisation problems (QAOA and VQE) as well as to various machine learning tasks covering both discriminative (QNN classifier) and generative (QCBM market generator) use cases. In general, the PQCs we used for quantum machine learning tasks can be divided into two types [88]: tensor network PQC (similar to the QNN circuit in Figure 8.4) and multilayer PQC (similar to the QCBM circuit in Figure 9.1). What is their expressive power and how can we rank them? Before trying to answer this question, let us have a look at a simple illustrative example: quantum circuits specified on a single quantum register.
![||||HHHHRRRRRR||||𝜃0000ZZZXXYψψψψx⟩⟩⟩⟩(𝜃(𝜃(𝜃(𝜃(𝜃(𝜃ABCDzzzxxy∼))))))⟩⟩⟩⟩U[− π,π], 𝜃y ∼ U[− π, π], &](https://static.packt-cdn.com/products/9781836209614/graphics/media/file1403.svg)
Figure 13.1: PQCs with different expressive powers.
Figure 13.1 displays four one-qubit circuits with dramatically different expressive powers, where U[−π,π] denotes the Uniform distribution over the closed interval [−π,π]. Let us go...
