Symmetry-enhanced Counterdiabatic Quantum Algorithm for Qudits

TL;DR

This work introduces a symmetry-based enhancement to digitized counterdiabatic quantum algorithms, applicable for qudits of any dimension, which leads to a better design of shallow variational quantum circuits, improving the feasibility of their implementation on near-term qudit devices.

Abstract

Qubit-based variational quantum algorithms have undergone rapid development in recent years but still face several challenges. In this context, we propose a symmetry-enhanced digitized counterdiabatic quantum algorithm utilizing qudits instead of qubits. This approach offers three types of compression as compared to with respect to conventional variational circuits. First, compression in the circuit depth is achieved by counterdiabatic protocols. Second, information about the problem is compressed by replacing qubits with qudits, allowing for a more efficient representation of the problem. Lastly, the number of parameters is reduced by employing the symmetries of the system. We illustrate this approach by tackling a graph-based optimization problem Max-3-Cut and a highly-entangled state preparation, the qutrit W state. As our numerical results show, we achieve a better convergence with a lower circuit depth and less measurement overhead in all the cases considered. This work leads to a better design of shallow variational quantum circuits, improving the feasibility of their implementation on near-term qudit devices

Figures (6)

• Figure 1: Number of reduced parameters employing spatial permutation symmetries vs. total number of parameters without taking into account the symmetries. Here, we have checked all $ZZ$-Ising Hamiltonians as in Eq. \ref{['eq: ZZIsingQudit']} with a connected graph structure up to 8 nodes that can be found in the database from Coolsaet2023.
• Figure 2: Example of graphs and their symmetries. From top to bottom: $\mathcal{G}$, $\mathcal{G}$ with vertex and edge orbits, $\mathcal{H}$, and $\mathcal{H}$ with the grouped terms in colors, which indeed are the vertex and arc orbits.
• Figure 3: Expected value of the energy at the end of each iteration of the variational procedure for a Max-3-Cut instance. We show the results for the different ansätze for a single layer. We only show the results after 30 steps of the classical optimizer to discard the initial fluctuations. As shown, the CD ansatz with the grouped parameters shows a faster convergence compared to the fully parameterized CD ansatz. After a high number of iterations, the fully parameterized ansatz obtains better results due to its higher expressivity. These results are also true for the DCQAOA variants. QAOA fails to solve the problem due to its limited expressivity even though the depths of the ansätze are similar.
• Figure 4: Fidelity defined in Eq. \ref{['eq: fidlity']} (top) and the expectation value of the Hamiltonian of Eq. \ref{['eq: Hamiltonian W']} (bottom) for the ansätze considered, shown as a function of the optimizer iteration. The results refer to the case $N=3$ and the three figures show the results for different numbers of layers: (a) $p=2$, (b) $p=3$, and (c) $p=5$, as ordered from left to right. Also here, we omit the first $30$ iterations to avoid showing initial fluctuations of the optimizer. We see that the grouped CD ansatz outperforms the other ones for a low number of layers, as expected.
• Figure 5: Results for the different algorithms and instances for the Max-3-Cut. The solid lines correspond to mean values and the colored area corresponds to the interquartile range. The upper plots show the energy at each iteration step. The plots in the middle of each sub-figure show the probability of measuring any of the states that correspond to the correct solution. The inset graphs correspond to each of the graphs that define the instances. We have run 20, 12, 12, and 10 runs respectively for each problem using uniformly random initial parameters.