  Current quantum simulators are primarily qubit-based, making them naturally
suitable for simulating 2-level quantum systems. However, many systems in
nature are inherently $d$-level, including higher spins, bosons, vibrational
modes, and itinerant electrons. To simulate $d$-level systems on qubit-based
quantum simulators, an encoding method is required to map the $d$-level system
onto a qubit basis. Such mapping may introduce illegitimate states in the
Hilbert space which makes the simulation more sophisticated. In this paper, we
develop a systematic method to address the illegitimate states. In addition, we
compare two different mappings, namely binary and symmetry encoding methods,
and compare their performance through variational simulation of the ground
state and time evolution of various many-body systems. While binary encoding is
very efficient with respect to the number of qubits it cannot easily
incorporate the symmetries of the original Hamiltonian in its circuit design.
On the other hand, the symmetry encoding facilitates the implementation of
symmetries in the circuit design, though it comes with an overhead for the
number of qubits. Our analysis shows that the symmetry encoding significantly
outperforms the binary encoding, despite requiring extra qubits. Their
advantage is indicated by requiring fewer two-qubit gates, converging faster,
and being far more resilient to Barren plateaus. We have performed variational
ground state simulations of spin-1, spin-3/2, and bosonic systems as well as
variational time evolution of spin-1 systems. Our proposal can be implemented
on existing quantum simulators and its potential is extendable to a broad class
of physical models.
