Tavis-Cummings Model: Controllability, Multi-qubit gates, and an Accidental Symmetry

Widely used in atomic and superconducting qubit systems, the Jaynes-Cummings (JC) Hamiltonian is a simple, yet powerful model for a two-level system interacting with a quantum harmonic oscillator (bosonic mode). In this paper, we focus on a system of N qubits, identically coupled to a single oscillator via the JC interaction, which is also known as the Tavis-Cummings (TC) Hamiltonian. We show that all permutation-invariant (PI) N-qubit unitaries can be realized using this PI Hamiltonian, which couples the qubits to an oscillator initialized in its vacuum state, together with global uniform z and x fields on all qubits. This includes useful many-body gates, such as multi-controlled-Z gate, with an arbitrary number of control qubits. As a corollary, we find that all permutationally invariant states -- including useful entangled states such as GHZ and Dicke states -- can be prepared using this interaction and global fields.

We present various examples of explicit quantum circuits for the case of N=2 qubits. In particular, we develop new methods for implementing controlled-Z, SWAP, iSWAP,and sqrt(iSWAP) gates using only the TC interaction and a global z field. Our work also reveals an unexpected, or ``accidental'' symmetry in the TC Hamiltonian and shows that it can be explained using Schwinger's oscillator model of angular momentum.