Quantum reservoir computing with classical and nonclassical states in an integrated optical circuit

Abstract

Quantum reservoir computing (QRC) is a hardware-implementation-friendly quantum neural network scheme with minimal physical system requirements and a proven advantage over classical counterparts. We use an extension of the positive-P phase space method to efficiently simulate a bosonic, linear silicon-chip based QRC system excited with a single nonclassical state, a "kitten" state. In combination with input-encoding coherent states, our method allows to obtain exact results for all correlation functions without Hilbert space cutoff. Surprisingly, we find that such a setting - where the only "quantumness'' derives from a single input mode, is sufficient to obtain significant (over 9-fold) reduction of classification error over the classical counterpart. Our work provides a promising direction toward efficient quantum computation with accessible optical hardware.

Figures (6)

• Figure 1: Schematic representation of QRC using coherent input states interacting with a non-classical state in a linear waveguide interferometer. The data to be classified, which in this case consists of two spirals parametrized by the coordinates x and y, is encoded in phases or amplitudes of coherent inputs. Additionally, a non-classical state (in our case the optical "cat" or "kitten" state) and a coherent field with a fixed amplitude and phase are injected. The interferometer produces multi-mode entanglement even in the case of linear coupling. At the output of the network, average occupations $\langle \hat{n}_{i}\rangle$ and multi-mode correlations $g^{(2)}_{ij}$ are measured, effectively performing a nonlinear feature extraction. These features are used in the output layer for software logistic regression. During training, only the output weights are modified and the physical system remains intact.
• Figure 2: Panels a - d present the photon-number probability distribution calculated from the weighted complex-$\mathcal{P}$ distribution for different superpositions of the $\ket{\alpha}$ and $\ket{-\alpha}$ coherent states with coherent amplitude $\alpha = 2$. Panel a presents the photon number distribution for an "even" cat state built from even Fock states only and panel b for an "odd" cat state built from odd Fock states. The photon number distributions in panels c and d are identical, although the distribution on panel c corresponds to a cat state and the one on panel d to a coherent state. Panel e depicts the Wigner function for a cat state, together with the partial Wigner functions (W$_{\gamma,\delta}$) calculated from the compact complex-$\mathcal{P}$ distribution by choosing coherent states $\gamma$ and $\delta$. Panel f depicts a higher-order cat state being a superposition of four coherent states located in the vertices of a square centered at zero in the phase space.
• Figure 3: Panel a depicts a schematic representation of a two-mode waveguide interferometer together with the linear coupling strength between the two modes as a function of the waveguide length along the propagation direction. The sub-panels represent the partial single-mode Wigner functions of the states in each mode before entering (on the left) and after exiting (on the right) the coupling region, as calculated using Eq. (\ref{['WfromPP']}). One can see, that the kitten state has initially been in mode 1, after it is coupled to the second mode it is transferred to mode 2. Panel b shows the average photon number in both modes and the cross-mode $g_{12}^{(2)}=\langle\hat{a}^{\dagger}_1\hat{a}^{\dagger}_2\hat{a}_1\hat{a}_2\rangle / \langle \hat{n}_1\rangle \langle \hat{n}_2\rangle$ correlation function.
• Figure 4: Panel a depicts the average occupation on each mode of a two-mode waveguide interferometer, where initially mode 1 was in a kitten state and mode 2 in a coherent state. The continuous line represents the result of the exact PP calculation, and the dashed lines present the result of solving the Von Neumann equation with a density matrix represented in the Fock space basis and Fock space cutoff 4 (light colored) and 8 (dark colored) lines. Panel b presents the comparison between the multi-mode correlation function calculated in the PP framework and using the von Neumann equation. Panel c presents the mean-squared-error ($MSE_{g^{(2)}}$) of calculating the $g^{(2)}_{12}$ function using the master equation (ME) integrated along the propagation direction for different levels of Fock space truncation. The right axis depicts the density matrix dimension of a 4-mode network (as studied in Fig \ref{['fig:fig5']}) assuming a given Fock space cutoff. This highlights the importance of finding a optimal cutoff to minimize the error but keeping the density matrix dimension within computational possibilities. The interferometer setup and input states have been set to the same as in the calculations presented in Fig. \ref{['fig:fig3']}
• Figure 5: The figure presents a 4-mode waveguide interferometer with the network scheme depicted in panel a and the corresponding coupling strengths shown in panel c. Panels b and d show, respectively, the average photon number in different waveguides and the different $g^{(2)}$ cross-correlation functions. In both panels, two functions are emphasized to demonstrate the dynamics while ensuring readability. Panel e shows the single-mode reduced Wigner functions of all of the modes at the output of the interferometer. The input consists of coherent states at modes 1, 3 and 4 with amplitudes $\beta_{1} = e^{\mathrm{i} \pi/4}$, $\beta_{3} = 1.2\mathrm{i}$, and $\beta_{4} = 0.8e^{\mathrm{i} \pi5/4}$, respectively. Mode 2 input is a kitten state $(1/\mathcal{N})(\ket{\beta} + \ket{-\beta})$ with amplitude $\beta = 1$.