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Simulation of quantum systems is expected to be one of the most importantapplications of quantum computing, with much of the theoretical work so farhaving focused on fermionic and spin-$\frac{1}{2}$ systems. Here, we insteadconsider encodings of $d$-level (i.e. qudit) quantum operators into multi-qubitoperators, studying resource requirements for approximating operatorexponentials by Trotterization. We primarily focus on spin-$s$ and truncatedbosonic operators in second quantization, observing desirable properties forapproaches based on the Gray code, which to our knowledge has not been used inthis context previously. After outlining a methodology for implementing anarbitrary encoding, we investigate the interplay between Hamming distances,sparsity patterns, bosonic truncation, and other properties of local operators.Finally, we obtain resource counts for five common Hamiltonian classes used inphysics and chemistry, while modeling the possibility of converting betweenencodings within a Trotter step. The most efficient encoding choice is heavilydependent on the application and highly sensitive to $d$, although clear trendsare present. These operation count reductions are relevant for runningalgorithms on near-term quantum hardware because the savings effectivelydecrease the required circuit depth. Results and procedures outlined in thiswork may be useful for simulating a broad class of Hamiltonians on qubit-baseddigital quantum computers.

Resource-efficient digital quantum simulation of $d$-level systems for photonic, vibrational, and spin-$s$ Hamiltonians