Symmetry-enhanced Counterdiabatic Quantum Algorithm for Qudits

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

💡 Research Summary

The manuscript introduces a novel framework for designing shallow variational quantum circuits that simultaneously compresses three distinct resources: circuit depth, information encoding, and the number of variational parameters. The authors achieve this by (i) incorporating counter‑diabatic (CD) driving into a digitized quantum‑approximate‑optimization‑algorithm (DC‑QAOA) to suppress diabatic transitions and thus reduce the number of Trotter steps required; (ii) replacing binary qubits with higher‑dimensional qudits (specifically qutrits) so that a single physical system encodes log₂ d times more Hilbert‑space information, thereby lowering the total number of quantum registers needed; and (iii) exploiting the spatial permutation symmetries of the underlying problem Hamiltonian to group together CD parameters that are related by symmetry, cutting the total number of trainable parameters by roughly 40‑60 % for the tested instances.

The paper first reviews the necessary background on qudit gate sets (e.g., equatorial rotations R_{i,j}(θ,φ) and Mølmer‑Sørensen entangling gates) and on variational algorithms such as QAOA. It then derives the CD term A^{(ℓ)}_λ as a linear combination of nested commutators of the adiabatic Hamiltonian H_a(t) and shows that, for ℓ = 1, the resulting operators are at most two‑body, making them compatible with existing hardware. The CD coefficients α_k(t) are obtained via an action‑minimization principle, and the resulting CD Hamiltonian is discretized using a first‑order Trotter expansion. In the “impulse regime” where the CD contribution dominates, the algorithm reduces to a pure CD evolution U(θ)=exp(−i∑_k θ_k P_k).

The symmetry analysis proceeds by defining the permutation group S_P that leaves the problem Hamiltonian H_P invariant. The authors prove that applying any π∈S_P merely relabels the CD operators A_k, so parameters belonging to the same orbit under S_P can be set equal. Numerical enumeration of ZZ‑Ising models on up to eight qubits shows an average reduction to 59 ± 19 % of the original parameter count. This reduction directly translates into fewer circuit evaluations per optimization step, mitigating the measurement overhead that plagues NISQ‑scale VQAs.

Two benchmark applications validate the approach. First, the Max‑3‑Cut problem on graphs with up to 12 vertices is tackled. Compared with standard QAOA and a fully parameterized CD‑enhanced ansatz, the symmetry‑enhanced CD ansatz converges 2–3× faster and achieves 5–10 % higher cut values while using roughly 30–40 % fewer Trotter layers. Second, the preparation of a three‑level W state is demonstrated. Using qutrits, the CD‑driven circuit prepares the state with fidelity > 0.99 in a depth of ≈ 7 gates, versus ≈ 12 gates required by a conventional three‑qubit circuit.

Overall, the work provides a concrete recipe for constructing more resource‑efficient variational algorithms on near‑term qudit hardware. By unifying counter‑diabatic control, high‑dimensional encoding, and symmetry‑based parameter reduction, the authors show that shallow, low‑overhead quantum circuits can achieve performance comparable or superior to deeper, more parameter‑heavy counterparts. The methodology is hardware‑agnostic and can be extended to higher‑dimensional qudits, more complex symmetry groups, and other combinatorial optimization problems, offering a promising pathway toward practical quantum advantage on NISQ devices.