Classical versus quantum: Comparing tensor-network-based quantum circuits on Large Hadron Collider data

Tensor networks (TN) are approximations of high-dimensional tensors designed to represent locally entangled quantum many-body systems efficiently. This paper provides a comprehensive comparison between classical TNs and TN-inspired quantum circuits in the context of machine learning on highly complex, simulated Large Hadron Collider data. We show that classical TNs require exponentially large bond dimensions and higher Hilbert-space mapping to perform comparably to their quantum counterparts. While such an expansion in the dimensionality allows better performance, we observe that, with increased dimensionality, classical TNs lead to a highly flat loss landscape, rendering the usage of gradient-based optimization methods highly challenging. Furthermore, by employing quantitative metrics, such as the Fisher information and effective dimensions, we show that classical TNs require a more extensive training sample to represent the data as efficiently as TN-inspired quantum circuits. We also engage with the idea of hybrid classical-quantum TNs and show possible architectures to employ a larger phase space from the data. We offer our results using three main TN Ansätze: Tree tensor networks, matrix product states, and multiscale entanglement renormalization Ansätze.

Figure 1

The upper panel shows the decomposition of the wave function into various classical tensor network realizations. Here the green edges represent Hilbert-space dimensions, and red lines represent auxiliary dimensions between tensors. The bottom panel shows their projection on quantum circuits and each tensor block has been represented as a certain transformation matrix between two qubits. Finally the arrow indicates a measurement. Images are only representative; varied number of inputs are given to simplify the interpretation of the network.

Figure 2

Representation of a tensor block in a quantum circuit. Each tensor block involves a unitary transformation, $U(\theta ,\phi ,\lambda )$, followed by a cnot gate.

Figure 3

The left panel shows an average of 5000 top events, and similarly the middle panel shows the same for the QCD background. The right panel shows the top signal for a single event. All images are reduced by cropping 12 pixels from each side and downsampling by averaging four adjacent pixels into one. The color bar shows the intensity of the modified $p_T$ deposited in each reduced pixel which has been fitted within $[0,\pi ]$.

Figure 4

Representation of the four-qubit classical MPS on the left and Q-MPS on the right.

Figure 5

Representation of the four-qubit classical TTN on the left and Q-TTN on the right.

Figure 6

Representation of the four-qubit classical MERA on the left and Q-MERA on the right.

Figure 7

Representation of the six-qubit classical TTN on the left and Q-TTN on the right.

Figure 8

Representation of the six-qubit classical MERA on the left and Q-MERA on the right.

Figure 9

ROC curve for four-qubit analysis where TTN, MERA, and MPS are shown with red, dark red, and light blue, respectively. Both plots compare classical TNs with quantum versions where QTNs are shown with a solid, dash-dotted, and dotted line for TTN, MERA, and MPS, respectively, where TNs are shown with dashed lines. The top panel shows the comparison where the QTNs are executed in IBM Quantum hardware (ibmq_quito), where the bottom panel shows the same for AER simulation. Multiple shaded lines for AER simulation represent different noise model assumptions. Each sample includes the same randomly chosen 10 000 events from the test sample, and the black dashed line shows a random guess.

Figure 10

Same as Fig. 9, showing this time six-qubit arrangement. The top panel shows the samples executed in IBM Quantum hardware (ibmq_perth).

Figure 11

Background rejection ratio between quantum and classical ROC curves has been presented for 16 feature analyses. From top to bottom, panels show TTN, MPS, and MERA architectures and each panel is presented with a respective AUC value. $(D,\chi )=(10,20)$ has been used for classical networks.

Figure 12

The upper panel shows the normalized effective dimension with respect to the number of samples for each TN realization, namely, TTN, MPS, and MERA, respectively. QTNs are shown with the solid blue line, and TNs are shown with dot-dashed, dashed, and dotted red lines with respect to corresponding Hilbert-space and bond dimensions shown with $(D,\chi )$. The bottom panel shows the eigenvalue distributions of the normalized Fisher matrix for each TN realization with the same color scheme. Again, the plot flow follows from left to right TTN, MPS, and MERA, respectively.

Figure 13

The left panel shows the representation of the classical TTN designed to capture 2D mapping. The right panel shows the representation of the hybrid classical-quantum TN design where the classical portion is TTN, and the quantum circuit can be any four-qubit variational circuit. Small black circles represent the pixel values, and green lines correspond to Hilbert-space dimensions. Blue circles represent tensor nodes, and the red connections are auxiliary dimensions. Finally, $l$ stands for the network decision.

Figure 14

Representation of hybrid classical-quantum TNs with MPS as a classical layer. The image has been divided into four blocks of nine nodes of MPS, where each MPS output goes into the quantum circuit. As in Fig. 13, green shows Hilbert dimensions and red shows the auxiliary dimensions.

Figure 15

Same as Fig. 9, showing this time the hybrid classical-quantum TN Ansatz with four-qubit arrangement. All quantum circuits have been executed in IBM Quantum hardware (ibmq_quito).