Variational quantum algorithm for Gaussian discrete solitons and their boson sampling

In the context of quantum information, highly nonlinear regimes, such as those supporting solitons, are marginally investigated. We miss general methods for quantum solitons, although they can act as entanglement generators or as self-organized quantum processors. We develop a computational approach that uses a neural network as a variational ansatz for quantum solitons in an array of waveguides. By training the resulting phase space quantum machine-slearning model, we find different soliton solutions, varying the number of particles and interaction strength. We consider Gaussian states that enable measuring the degree of entanglement and sampling the probability distribution of many-particle events. We also determine the probability of generating particle pairs and unveil that soliton bound states emit correlated pairs. These results may have a role in boson sampling with nonlinear systems and in quantum processors for entangled nonlinear waves.

Figure 1

Double-head model for a layer representing a Gaussian.

Figure 2

Double-head model representing a Gaussian state with a bias.

Figure 3

Graphical representation of the model for the characteristic function.

Figure 4

Graphical representation of the generalized model for the characteristic function with bias $\mathit{a}$, which is adopted in the linear transformations.

Figure 5

Graphical representation of the transformed characteristic function in terms of the original characteristic function with modified input and bias.

Figure 6

Linear layer to transform the input variables to the $\chi$ layer.

Figure 7

Graphical representation of a linear transformation as a pullback of a linear layer from the original characteristic function. The resulting model corresponds the transformed characteristic function in Fig. 5. Note that here $\mathit{a}=0$.

Figure 8

Simplified model of Fig. 7, omitting the internal variables $\mathbf{y}$ and $\mathit{b}$ and the zero input bias $\mathit{a}$.

Figure 9

Single-layer multiheaded model for a linear transformation of a Gaussian state.

Figure 10

Two-layer multiheaded model equivalent to Fig. 9.

Figure 11

Linear transformations and pullbacks. A single transformation with unitary operator ${\widehat{U}}_1$ corresponds to a single pullback. A double transformation with unitary operators ${\widehat{U}}_1$ and ${\widehat{U}}_2$ corresponds to a double pullback. Note that the flow of data from $\mathbf{x}$ to the output through the network is in reverse order with respect to the two transformations.

Figure 12

Equivalence of two cascaded linear layers with a single linear layer.

Figure 13

Example of cascading three transformations. Note the reverse order of layers from right to left.

Figure 14

Model for a coherent state obtained by pullback of a displacement operator (Glauber layer) from the vacuum.

Figure 15

A squeezed coherent state obtained by pullback with a single mode squeezing layer from the vacuum.

Figure 16

Pullback model for a squeezed coherent state by displacing a squeezed vacuum (top panel). Pullback model for squeezed coherent state by squeezing a displaced vacuum (bottom panel). Here all the layers act on the same mode.

Figure 17

(a) Quantum processor for solitons in coupled optical waveguides. $n$ vacuum spatial modes are squeezed, displaced, and mixed by a trainable interferometer $\widehat{U}$ ($n=5$ in the figure). (b) Multiple pullbacks as a quantum soliton variational ansatz.

Figure 18

(a), (b) Ground state for $N_T=10$ and $\gamma =-0.01$: (a) displacement $|\langle {\widehat{\psi }}_j\rangle {|}^2$; (b) mean boson number $\langle {\widehat{n}}_j\rangle$; (c), (d) as in (a), (b) with $N_T=10$ and $\gamma =-1$ ($n=17$).

Figure 19

Training history for Fig. 18: (a) $\langle H\rangle$ rapidly reaches the minimum for $\gamma =-0.01$ (dashed line is the coherent state with $\langle \widehat{H}\rangle \simeq -2N_T$); (b) as in (a) for $\gamma =-1$. Dashed line is Eq. (143). The model explores plateaus corresponding to solitons with different degrees of entanglement; (c) logarithmic negativity for $\gamma =-0.01$, no entanglement (dashed line); (d) as in (c) for $\gamma =-1$, the ground state is entangled.

Figure 20

(a) Boson distribution and mean displacement (right axis) for two representative bound-soliton solutions with $N_T=40$ and $\gamma =-1.0$; (b) logarithmic negativity versus the boson number $N_T$ for different $\mathrm{\Delta }$; (c) probabilities for events containing ${\overline{n}}_T=2$ particles in two varying sites $A$ and $B$ and $N_T=10$; (d) as in (c) with two particles per site (${\overline{n}}_T=4$). The probability is not vanishing only when the particles are at the soliton sites, denoting the generation of entangled pairs ($n=10$).

Figure 21

Graph of the multihead model for the quantum solitons. One can notice the central computational core represented by the cascade of multihead layers, including the interferometers LinearRandomMultiHead, displacements LinearConstantMultiHead, and squeezing layers SingleModeSqueezer. Then we have the activation layer, computing the Gaussian characteristic function ResidualGaussianMultiHead, and the layer to determine the covariance matrix CovarianceLayer. After that, a series of lambda layers, i.e., simple computing layers for the cost functions and the various metrics during the training.