Robustness to spontaneous emission of a variational quantum algorithm

We study theoretically the effects of dissipation on the performances of a variational quantum algorithm used to approximately solve a combinatorial optimization problem, the maximum independent set, on a platform of neutral atoms. We take a realistic model of dissipation with spontaneous emission, and show numerically that the detrimental effects of these incoherent processes are spontaneously attenuated by the variational nature of the procedure. We furthermore implement an alternative optimization scheme recently proposed, where one uses the expected shortfall of the mean energy as the objective function.

Figure 1

Principle of hybrid quantum-classical variational algorithms. These algorithms are composed of both a quantum and a classical processor, that exchange information within a feedback loop. The quantum processor is used to prepare and measure a $N$-qubit parametrized wave function $\left|\psi \right(\mathit{t},\mathit{\tau })\rangle$, starting from the initial state $|q_0,q_1,...,q_N\rangle$. The outcome of the measurement $\langle C\rangle$ is then used as the objective function in a standard classical optimization procedure, that updates the parameters $\mathit{t}$ and $\mathit{\tau }$ for the next iteration. In this paper, the parametrized quantum circuit representing the action of the quantum processor is a succession of time evolutions with Hamiltonians $H_M$ and $H_C$.

Figure 2

Upper panel shows an ensemble of neutral atoms in 2D with random initial positions, together with their Rydberg blockade region characterized by the Rydberg blockade radius $r_b$ (see Appendix pp1). It defines a graph in 2D where vertices are connected if they are closer than $r_b$, shown on the bottom panel. We display on that graph an example of an independent set in red. As there are no edges connecting two red vertices, this subset constitutes an independent set. The strong Rydberg-Rydberg interaction naturally prevents the excitation of connected atoms, so that the dynamics generated by $H$ in Eq. (1) is naturally restricted to independent sets.

Figure 3

Evolution of $r_1^{\mathrm{diss}}/r_0$ (blue dots and full curve) and $r_2^{\mathrm{diss}}/r_0$ (red squares and dashed curve) with respect to the spontaneous emission rate $\mathrm{\Gamma }/{\mathrm{\Omega }}_0$. All the results are averaged over 100 random graphs of $N=14$ atoms with density $\nu =2.6$. The optimization procedure is done with a small number of variational parameters for simplicity, and involves five randomly generated initial conditions, where each individual parameter is sampled uniformly in $[0,\pi /\left(\hslash {\mathrm{\Omega }}_0\right)[$. We use a direct time evolution of the density matrix with Eq. (3).

Figure 4

Histogram $d/\left(\hslash {\mathrm{\Omega }}_0\right)$ for 154 randomly generated graph instances with $N=18$ atoms and a density $\nu =2.6$, starting from the same 15 initial parameters for both the noisy and the noise-free optimization procedures. The histogram displays a large tail [region (ii)], for which the optimal parameters in the presence of spontaneous emission differ quantitatively from the ones obtained with an ideal time evolution. We use an average of over 2000 wave-function trajectories for the evaluation of the dynamics of the system with Eq. (3).

Figure 5

We show on panel (a) one particular UD graph (the positions of the vertices have been modified for more visibility, but the links remain unchanged). Panel (b) shows the evolution objective function with respect to the parameter $t_2$ for $\mathrm{\Gamma }=0$ (blue squares) and $\mathrm{\Gamma }/{\mathrm{\Omega }}_0=0.1$ (red dots). We have taken $t_1/\left(\hslash {\mathrm{\Omega }}_0\right)=1.5$ and ${\tau }_1/\left(\hslash {\mathrm{\Omega }}_0\right)=1$. The noisy time evolution is averaged over ${10}^3$ quantum trajectories.

Figure 6

Sketches of the probability density associated with the measurement of two distinct states ${\rho }_a$ (a) and ${\rho }_b$ (b) in the computational basis. The preparation of state ${\rho }_b$ would be favored with respect to the preparation of ${\rho }_a$ when using the cVAR objective function.

Figure 7

We show the evolution of the approximation ratio $r_0$ (and its standard deviation) as a function of the density $\nu$. We have three variational parameters and $N=18$ atoms, and each data point is averaged over 50 graph realizations, starting from the initial state with parameters initialized at $\pi /\left(\hslash {\mathrm{\Omega }}_0\right)$.

Figure 8

Left panel: illustration of the Rydberg blockade effect. The presence of the first atom in the Rydberg state induces a strong energy shift for the second atom at short distance smaller than $r_b$, thereby putting the Rydberg level strongly off resonance with the laser system.