Expressive power of parametrized quantum circuits

Parametrized quantum circuits (PQCs) have been broadly used as a hybrid quantum-classical machine learning scheme to accomplish generative tasks. However, whether PQCs have better expressive power than classical generative neural networks, such as restricted or deep Boltzmann machines, remains an open issue. In this paper, we prove that PQCs with a simple structure already outperform any classical neural network for generative tasks, unless the polynomial hierarchy collapses. Our proof builds on known results from tensor networks and quantum circuits (in particular, instantaneous quantum polynomial circuits). In addition, PQCs equipped with ancillary qubits for postselection may possess expressive power stronger than that of those without postselection. We employ them as an application for Bayesian learning, since it is possible to learn prior probabilities rather than assuming they are known. We expect that it will find many more applications in semisupervised learning where prior distributions are normally assumed to be unknown. Lastly, we conduct several numerical experiments using the Rigetti Forest platform to demonstrate the performance of the proposed Bayesian quantum circuit.

Figure 1

Simulation of controlled unitary gates.

Figure 2

A general framework of IQP circuits.

Figure 3

Illustration of MPQCs.

Figure 4

An example of a TPQC that inherits the layout of a tree tensor network, where the cnot gates in different layers have different local constraints.

Figure 5

An example of TPQCs that inherit the layout of MPSs.

Figure 6

A general framework of an AD-MPQC. The arrangement of quantum gates in each block is identical.

Figure 7

The general scheme of the proposed BQC.

Figure 8

An example of conditionally applying $U\left({\mathit{\theta }}_{{\lambda }_k}^1\right)$ to data qubits iff $|{\lambda }_k\rangle =|01\rangle$.

Figure 9

The generative results obtained from DDQCL, the QCBM, and our model. Since the BAS dataset can be regard as a set of binary images, it can be mapped into different integers, as the $x$ axis of figures. Panels (b), (c), and (e) are the generated result of $2\times 2$ BAS images using the QCBM, DDQCL, and the BQC respectively. Figures (d) and (f) are the generated results of $3\times 3$ BAS images using the QCBM and the BQC, respectively.

Figure 10

The figure illustrates the MMD loss functions for $p\left({\lambda }_1\right)=0.70$ with different parameter settings.

Figure 11

The figure illustrates the MMD loss functions for $p\left({\lambda }_1\right)=0.85$ with different parameter settings.

Figure 12

The mapping between a MPQC and a MPS.

Figure 13

The first preprocessing step to transform the RBM to MPQCs. The upper left panel shows the original RBM. The upper right panel illustrates that the hidden neuron $h_k$, as highlighted by purple shadow, is selected and split. The lower right panel demonstrates the splitting result, where each two new hidden neurons $h_o$ and $h_l$ only connect to the visible neurons $v_i$ and $v_j$, respectively. Following the same routine, the lower left panel exhibits the result of splitting all hidden neurons of the original RBM.

Figure 14

The second preprocessing step to transform the RBM to MPQCs. The left (right) panel shows the RBM after the first (second) preprocessing step. In particular, all hidden neurons that connect to the same visible neuron are merging into the new hidden neuron. For example, hidden neurons $h_j$ and $h_k$ that connect to $v_i$ are substituted by $h_l$.

Figure 15

The mapping rule from the RBM to MPQCs. The left panel shows the RBM and the right panel illustrates the equivalent MPQC representation. An iterative updating rule is employed to map RBM to MPQCs, where each hidden neuron corresponds to a parametrized rotation single-qubit gate along the $Y$ axis.

Figure 16

The reformulation from a DBM to an RBM by introducing virtual visible neurons. The blue and orange nodes refer to the visible and hidden neurons, respectively. The hidden layers with odd numbers, as highlighted by green dash-dotted boxes, form the hidden layer of the reformulated RBM. The hidden layers with even numbers, as highlighted by red dash-dotted boxes, form the virtual visible neurons of the reformulated RBM.

Figure 17

The arrangement of quantum gates in a general IQP circuit.

Figure 18

The arrangement of quantum gates in each block.

Figure 19

The left panel illustrates an equivalent circuit described by swap operation. The right panel shows the implementation of swap by two cnot gates and one reversed cnot gate.

Figure 20

Simulating a single cnot gate by using 4 blocks.