Optimizing photon-number distributions of Gaussian states in the presence of loss: Towards minimizing the impact of loss in Gaussian boson sampling

Abstract

We analyze the impact of photon loss on the photon-number statistics of Gaussian states. Specifically, we propose and carefully evaluate several methods to mitigate deviations in the photon-number distributions of lossy (displaced) squeezed vacuum states from those of their lossless counterparts. These methods rely on appropriately redefining the parameters of Gaussian states when the loss budget is known in order to recover, as closely as possible, the desired photon-number distribution associated with each target state. While it is intrinsically hard to directly optimize the photon-number distribution of high-dimensional, correlated multimode Gaussian states, the proposed methods are instead based on optimizing specific key properties such as fidelity, phase-space functions, low-order moments of the underlying photon-number statistics, or overlap with the vacuum state. In particular, our results show that optimizing the fidelity between a pure Gaussian target state and a modified Gaussian state that has passed through a loss channel does typically not result in closeness of the corresponding photon-number distributions. Furthermore, we show that correcting for the vacuum overlap minimizes the deviation in the photon-number distribution for large parameter ranges which we explicitly prove for single-mode squeezed vacuum and provide numerical evidence for general (displaced) squeezed vacuum states. As photon loss is a key limitation for Gaussian boson sampling, our results provide insights into the feasibility and limitations of such photonic quantum simulations in lossy environments and offer guidelines for mitigating these imperfections.

Figures (12)

• Figure 1: Two-mode GBS system with additional beam splitters modeling loss at relevant locations. Each $\eta_i^{(l)}$ gives the transmissivity for an incoming photon, while $1-\eta_i^{(l)}$ is the corresponding loss probability. The index $i$ denotes the mode and the superscript $l$ the layer in the system.
• Figure 2: Effect of loss on photon-number distribution. For a displaced squeezed vacuum state with $\alpha = 0.25$ and $\xi = 0.9$, we plot, the ratio $\mathcal{P}'_{\xi,\alpha;\eta}(m)/\mathcal{P}_{\xi,\alpha}(m)$ as a function of loss ($1-\eta$) for selected photon numbers $m=0,\dots,5$. $\mathcal{P}_{\xi,\alpha}(m)$ denotes the probabilities of the target distribution given by the displaced squeezed vacuum state while $\mathcal{P}'_{\xi,\alpha;\eta}(m)$ is the corresponding probability after a pure-loss channel with transmissivity $\eta$. Ratios larger than unity arise because loss redistributes probability weight from higher to lower photon numbers, thereby potentially increasing the latter.
• Figure 3: The upper left plot shows $\delta$ (black solid line) and $F$ (blue dashed line) for an target squeezing pa-ra-me-ter $\tilde{\xi}=1.5$ under variation of $\xi$ at a loss parameter of $1-\eta=0.5$. Both $\xi_\mathrm{min}$ and $\xi_\mathrm{F}$ are depicted with a black dot and labeled. The vertical dotted black line indicates $\xi=\tilde{\xi}$. The lower left plot shows $\xi_\mathrm{min}$ (solid black line) and $\xi_\mathrm{F}$ (dashed blue line) for different loss parameters at a fixed $\tilde{\xi}=1.5$. The values for $\xi_\mathrm{min}$ and $\xi_\mathrm{F}$ from the upper plot are also depicted in the lower plot by black dots. The right plot shows the photon-number distribution of the target state (green), its lossy version (red), the fidelity-optimized state (blue), and the $\delta$-minimized state (black) from left to right for a target squeezing parameter of $\tilde{\xi}=1.5$ and $\eta = 0.5$. The typical even-odd signature of a perfect squeezed vacuum state can be seen.
• Figure 4: Squeezing parameters obtained from different phase-space distribution optimization methods as a function of loss for a target state with $\tilde{\xi} = 1.5$. Among the phase-space approaches, the Wasserstein metric (solid yellow) increases more gradually at small losses, whereas the other methods [$\mathrm{PS}$ (dashed dark green), $\mathrm{BHA}$ (dotted orange), and $\mathrm{KLDsym}$ (dash-dotted turquoise)] show a steeper initial increase. For larger losses, all phase-space methods converge to similar squeezing values.
• Figure 5: Parameter regions for the location of the minimizer $\xi_{\mathrm{min}}$. In the green region, $\xi_{\min} = \xi_{\mathrm{vac}}$ as proven analytically. In particular, for all $\xi_\mathrm{vac}<2.290047\ldots$ (dashed black line) we have $\xi_{\min} = \xi_{\mathrm{vac}}$ for all loss parameters $\eta$. For $\xi_\mathrm{vac}>2.290047\ldots$, the minimizer coincides with vacuum-overlap correction for $\eta < 14/15$. For $\eta > 14/15$, $\xi_\mathrm{vac}$ is no longer optimal in the red region where we find $\xi_{\min} < \xi_{\mathrm{vac}}$.