Adaptive random quantum eigensolver

We propose an adaptive random quantum algorithm to obtain an optimized eigensolver. Specifically, we introduce a general method to parametrize and optimize the probability density function of a random number generator, which is the core of stochastic algorithms. We follow a bioinspired evolutionary mutation method to introduce changes in the involved matrices. Our optimization is based on two figures of merit: learning speed and learning accuracy. This method provides high fidelities for the searched eigenvectors and faster convergence on the way to quantum advantage with current noisy intermediate-scaled quantum computers.

Figure 1

Scheme of a bioinspired adaptive random quantum algorithm. The individual is mapped on the encoding gate $G\left({\vec{\theta }}_t\right)$, the abiotic environment is represented by gate $E$, and the dead-alive probability is given by the decoding gate $G^{†}\left({\vec{\theta }}_t\right)$ and a measurement in the computational basis $\left\{\right|j\rangle \}$. We introduce changes in the encoding and decoding matrix (mutations) using a classical feedback loop (green lines) which depends on the classical communication of the measurement outcome (purple line).

Figure 2

Points $(x_j,y_j)$ (orange circles), and the monotonic and symmetric interpolation defined by Eq. (9). The green squares represent the points $p_0=(-1,0)$, $p_{n+1}=(0,0.5)$, and (1,1), which are fixed to ensure that the interpolation corresponds to a valid symmetric CDF. $f_j$ represents the function by parts that interpolates the points $p_j$ and $p_{j+1}$.

Figure 3

Histogram for mean number of interactions to converge without (left panel) and with optimization (right panel) for 100 different $U(\theta ,\varphi ,\lambda )$. Both panels are for optimization of learning speed.

Figure 4

Histogram for mean fidelity after convergence without (left panel) and with optimization (right panel) for 100 different $U(\theta ,\varphi ,\lambda )$, for optimization of learning accuracy.

Figure 5

Optimal PDF (solid blue line) and PCF (dashed orange line) for learning speed for $\tau \widehat{\mathcal{O}}={\sigma }_x$. Red dots are the points related to the parametrization; green dots are the fixed points of our PCF.

Figure 6

Optimal PDF (blue) and PCF (orange) for learning accuracy for $\tau \widehat{\mathcal{O}}={\sigma }_x$. Red dots are the points related to the parametrization; green dots are the fixed points of our PCF.

Figure 7

Optimal PDF (blue) and PCF (orange) for learning speed for $\tau \mathcal{O}$ given by Eq (22). Red dots are the points related to the parametrization; green dots are the fixed points of our PCF.

Figure 8

The histogram comparison of the mean number of iterations to converge without (left panel) and with optimization (right panel) for Eq (22).

Figure 9

Histogram for mean fidelity without (left panel) and with optimization (right panel) for 100 different $U(\theta ,\varphi ,\lambda )$, for the optimization of learning speed.