Dicke Simulators with Emergent Collective Quantum Computational Abilities

Using an approach inspired from spin glasses, we show that the multimode disordered Dicke model is equivalent to a quantum Hopfield network. We propose variational ground states for the system at zero temperature, which we conjecture to be exact in the thermodynamic limit. These ground states contain the information on the disordered qubit-photon couplings. These results lead to two intriguing physical implications. First, once the qubit-photon couplings can be engineered, it should be possible to build scalable pattern-storing systems whose dynamics is governed by quantum laws. Second, we argue with an example of how such Dicke quantum simulators might be used as a solver of “hard” combinatorial optimization problems.

Figure 1

In the Dicke model, photons (yellow lines) mediate a long range interaction between qubits (green circles). The drawing sketches schematically a six qubits system within its fully connected graph and its internal level structure. In the standard single-mode Dicke model, the exchange coupling is fixed at the same value for every pair of qubits. In systems where both many modes and disorder are present, the exchange couplings are qubit dependent and take the form given by Eq. (3).

Figure 2

Order parameter $m^{(1)}$ in Eq. (9) as a function of $\mathrm{\Delta }$ in the ultrastrong coupling regime ($\mathrm{\Omega }>{\mathrm{\Omega }}_c$) at $T=0$. On the theoretical curve (dashed line) are superposed the results (green dots) of a naive numerical optimization algorithm minimizing the ground state energy ansatz in Eq. (11). The parameters of the simulation are $\mathrm{\Omega }=4$, $N=100$, $M=10$. Green dots are the result of a single realization of the disorder. This plot suggests that at $N=100$ the system is already close to the thermodynamic limit behavior. The classical Hopfield model is recovered in the limit $\mathrm{\Delta }\to 0$.