Variational simulation of $d$-level systems on qubit-based quantum simulators

TL;DR

This paper presents a systematic method for simulating $d$-level systems on qubit-based quantum computers, comparing binary and symmetry encoding methods, and demonstrating the superiority of symmetry encoding in variational simulations.

Contribution

The authors develop a new approach to address illegitimate states in $d$-level system simulations and compare encoding strategies, highlighting the advantages of symmetry encoding over binary encoding.

Findings

Symmetry encoding outperforms binary encoding in efficiency and resilience.

Symmetry encoding requires fewer two-qubit gates and converges faster.

The method is applicable to current quantum simulators and various physical models.

Abstract

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.