Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Properties of multiqubit variational quantum states representing weighted graphs and their computing with quantum programming

Kh. P. Gnatenko, A. Kaczmarek

Abstract

We study multiqubit variational quantum states that can be considered as weighted quantum graph states. These states are constructed as single-layer variational circuits with $RX$ rotations and $RZZ$ entangling gates, corresponding to graphs of arbitrary structure. In general case of quantum graph states of arbitrary structure we derive the geometric measure of entanglement and evaluate quantum correlators. It is shown that these quantities are directly related to the degrees of the corresponding vertices in graph. As an example, we analyze the state associated with the star graph $K_{1,4}$ using noisy quantum computing on the AerSimulator. The results are in good agreement with theoretical predictions. These findings demonstrate a connection between graph structure and quantum properties, enabling the study of classical graphs via quantum computing.

Properties of multiqubit variational quantum states representing weighted graphs and their computing with quantum programming

Abstract

We study multiqubit variational quantum states that can be considered as weighted quantum graph states. These states are constructed as single-layer variational circuits with rotations and entangling gates, corresponding to graphs of arbitrary structure. In general case of quantum graph states of arbitrary structure we derive the geometric measure of entanglement and evaluate quantum correlators. It is shown that these quantities are directly related to the degrees of the corresponding vertices in graph. As an example, we analyze the state associated with the star graph using noisy quantum computing on the AerSimulator. The results are in good agreement with theoretical predictions. These findings demonstrate a connection between graph structure and quantum properties, enabling the study of classical graphs via quantum computing.

Paper Structure

This paper contains 4 sections, 24 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Properties of one-layer variational quantum states corresponding to weighted graphs
  3. Quantum computing of the entanglement of quantum graph states and quantum corelators
  4. Conclusions

Figures (6)

  • Figure 1: Quantum circuit for preparation of quantum state $\ket{\psi_G}$ corresponding to $K_{1,4}$.
  • Figure 2: Surface plots of geometric measure of entanglement for quantum graph state $\ket{\psi_G}$ Fig. 1 with $\phi_i = \phi$ and $\theta_{ij} =\theta$. Yellow surface shows analytical result, blue dots were obtained from ideal simulation on AerSimulator and red dots from noisy simulation on AerSimulator with readout error probability of $10^{-2}$, X and SX gate error of order $10^{-4}$ and CNOT gate error of order $10^{-2}$.
  • Figure 3: Surface plots of $\langle \sigma_0^{x} \sigma_1^{x} \rangle$, $\langle \sigma_0^{y} \sigma_1^{y} \rangle$, $\langle \sigma_0^{z} \sigma_1^{z} \rangle$ in quantum state $\ket{\psi_G}$ corresponding to $K_{1,4}$ graph. Surface shows analytical result, and dots are obtained from noisy quantum computing on AerSimulator
  • Figure 4: Bar plots of absolute differences between the analytical results for $\langle \sigma_0^{x} \sigma_1^{x} \rangle$, $\langle \sigma_0^{y} \sigma_1^{y} \rangle$, $\langle \sigma_0^{z} \sigma_1^{z} \rangle$ in quantum state $\ket{\psi_G}$ corresponding to $K_{1,4}$ graph and results of noisy quantum computing on AerSimulator.
  • Figure 5: Surface plots of $\langle \sigma_0^{x} \sigma_1^{y} \rangle$, $\langle \sigma_0^{y} \sigma_1^{z} \rangle$, $\langle \sigma_0^{x} \sigma_1^{z} \rangle$ in quantum state $\ket{\psi_G}$ corresponding to $K_{1,4}$ graph. Surface shows analytical result, and dots are obtained from noisy quantum computing on AerSimulator
  • ...and 1 more figures