Variational simulation of higher-spin systems on qubit-based quantum simulators

Qubit-based quantum simulators naturally target two-level systems, whereas many quantum many-body problems are intrinsically $d$ level. Encodings from qudits to qubits then enlarge the Hilbert space and can introduce unphysical states that interfere with variational optimization. We formulate a variational framework for encoded $d$-level models that suppresses these illegitimate states with penalty terms and benchmark it for spin-1 and spin-3/2 bilinear-biquadratic Heisenberg chains. We compare binary encoding, which minimizes the qubit overhead, with symmetry encoding, which preserves the relevant spin symmetries and enables symmetry-conserving ans\"atze. Although binary encoding is more qubit efficient, its hardware-efficient ansatz is harder to train and less effective at exploiting conserved quantities. In contrast, symmetry encoding requires more qubits but reaches substantially higher fidelities, converges faster, and exhibits better trainability than the binary hardware-efficient ansatz. These results identify symmetry-preserving encodings as a practical route to simulating higher-spin models on existing qubit platforms.