Quantum photonic neural networks in time

Abstract

We introduce the architecture and timing algorithm to realize a time-bin-encoded quantum photonic neural network (QPNN): a reconfigurable nonlinear photonic circuit inspired by the brain and trained to process quantum information. Unlike the typical spatially-encoded QPNN, time-encoded networks require the same number of photonic elements (e.g. phase shifters or switches) regardless of their size or depth. Here, we present a model of such a network and show how to include imperfections such as losses, routing errors and most notably distinguishable photons. As an example, we train the QPNN to realize a controlled-NOT gate, based on a hypothetical ideal Kerr nonlinearity. We then extend our model to a realistic two-photon nonlinearity due to scattering from a single, semiconductor quantum dot coupled to a photonic waveguide. We show that, using this realistic nonlinearity, the QPNN can be trained to act as a Bell-state analyzer which operates with a fidelity of 0.96 and at a rate only limited by losses. We further show that time gating can raise this fidelity to over 0.99, while still maintaining an efficiency exceeding 0.9. Overall, this work lays a framework for the first QPNN encoded in time, and provides a clear path to the scaling of these networks.

Figures (5)

• Figure 1: Time- and space-encoded QPNNs. (a) The architecture of a spatially encoded $N \times N$ QPNN comprised of meshes of linear interferometers (light blue) separated by single-site nonlinearities (dark blue, $\Sigma$). Each MZI contains two phase shifters ($\phi$, $2\theta$) and two $50:50$ directional couplers, as shown in the inset, performing a unitary transformation $T^{(\ell,c)}_{(m_{i},m_{j})}$ between modes $m_i$ and $m_j$. (b) The equivalent time-encoded QPNN architecture: two inner loops connected by a reconfigurable MZI realize the linear mesh, while the nonlinearity is part of the external loop. Different modes enter the QPNN successively through the bottom bus fiber, separated by a time-bin $\tau_\mathrm{B}$. Each photon pulse has a width $\sigma_\mathrm{p}$, and the time jitter of the pulses corresponds to a normal distribution with width $\sigma_\mathrm{j}$. Switches S1, S2, S3 direct modes in and out of the linear and nonlinear loops, and PS is the output phase shifter. The required number of (c) nonlinear components and (d) phase shifters for the spatial (orange) and temporal (blue) QPNNs as a function of their size. (e) The requisite fiber length of the temporal QPNN (red, left axis), as well as the repetition rate at which the network operates normalized to $\tau_\mathrm{B}$, as a function of the network size, for the temporal (blue) and spatial (orange) networks (right axis).
• Figure 2: A linear CNOT gate, in time. (a) CNOT gate timing algorithm, showing the reflectivity of the MZI, the state of switch S2 and output phase shifter PS, at each timestep. For the timesteps where photons enter or leave the circuit, the modes ($m_n$) are labeled in correspondence with the equivalent spatial circuit (shown in Figs. S1 and S2 of the supplemental material). (b) The corresponding fidelity (colored dots, left axis) and efficiency (gray squares, right axis) as a function of visibility (or time jitter, top axis). (c) Hinton diagrams of the resulting CNOT operation for the visibilities corresponding to the solid markers in b ($V = 1, 0.5, 0$). Here, $|0\rangle$ and $|1\rangle$ denote the computational basis state for each input and output, and the output states are re-normalized to this basis.
• Figure 3: Time-encoded 2-layer, 4-mode QPNN trained to perform a CNOT gate. (a) Training cost for 100 trials of 250 epochs each, for different visibilities. Successful training attempts clump together, resulting is overlapping curves that together appear darker. (b) Fidelity $F$ (colored, left axis) and efficiency $\eta$ (gray squares, right axis) as a function of visibility and time jitter. Circular markers denote online training while the dashed line shows offline training, which is done by taking the optimal solutions for $V=1$ and applying them to the system at different visibilities. (c) Hinton diagrams showing CNOT implementations at $V=1$, $0.5$ and $0$ (solid circles in b).
• Figure 4: Time-encoded QPNN trained to act as a BSA, using the nonlinearity of a waveguide-coupled QD. (a) Schematic of a QD in a waveguide, showing how an input one- (two-) photon wavefunction (in time) is altered after scattering. (b) Fidelity distributions for QPNNs of different sizes, each trained to act as BSAs. The width of each box corresponds to the number of QPNNs that achieved the fidelity at which it is found. The arrow denotes the optimal QPNN with $N=6$ and $L=4$, which reached a fidelity of 0.96, and is studied further in Fig. \ref{['fig:fig5']}.
• Figure 5: Approaching unity fidelity for a QD-based QPNN trained as a BSA. (a) The two-photon temporal wavefunction in each logical output defined for the BSA while training the QPNN shown with an arrow in Fig. \ref{['fig:fig4']}b. The fraction of the probability distribution found in each output is annotated, with those that signal a correct result outlined in green. In each bin, we mark exemplary time filters calculated as contours at 0.2, 0.5, and 0.9 of the maximum amplitude for the distribution in the correct output. For clarity, we expand these in the inset, showing states $\left|0\right\rangle_\mathrm{out}$ (correct state) and $\left|1\right\rangle_\mathrm{out}$ (false positive; amplified by a factor of 30) with filters. (b) Infidelity (purple) and efficiency (red) of the BSA as a function of the filter drawn at a percentage of the maximum value. The points on each curve correspond to the filters shown in a.