Regimes of Classical Simulability for Noisy Gaussian Boson Sampling

As a promising candidate for exhibiting quantum computational supremacy, Gaussian boson sampling (GBS) is designed to exploit the ease of experimental preparation of Gaussian states. However, sufficiently large and inevitable experimental noise might render GBS classically simulable. In this work, we formalize this intuition by establishing a sufficient condition for approximate polynomial-time classical simulation of noisy GBS—in the form of an inequality between the input squeezing parameter, the overall transmission rate, and the quality of photon detectors. Our result serves as a nonclassicality test that must be passed by any quantum computational supremacy demonstration based on GBS. We show that, for most linear-optical architectures, where photon loss increases exponentially with the circuit depth, noisy GBS loses its quantum advantage in the asymptotic limit. Our results thus delineate intermediate-sized regimes where GBS devices might considerably outperform classical computers for modest noise levels. Finally, we find that increasing the amount of input squeezing is helpful to evade our classical simulation algorithm, which suggests a potential route to mitigate photon loss.

Figure 1

Simplification of noise model. (a) A realistic GBS experiment suffers from imperfect squeezing, modeled by an ideal squeezer followed by a loss channel with transmission ${\eta }_S$, a lossy interferometer, described by a subunitary matrix $\mathit{A}$, and inefficient noisy threshold detectors, characterized by $q_D≔p_D/{\eta }_D$, where $p_D$ and ${\eta }_D$ are the dark count rate and its quantum efficiency, respectively. (b) Under the assumption of uniform loss within the interferometer, we can absorb all losses into the thermalized squeezed state $\sigma$. We also assume that $r$ and $q_D$ do not change as the device size increases, which is reasonable for the regime of near-term experiments.

Figure 2

(a),(b) Sufficient condition for efficient simulation of GBS by using the quantum fidelity. In both cases $\epsilon =0.01$. Solid lines are the conditions given by Eq. (1). Dashed lines correspond to the conditions given by $\eta ={\eta }_{\infty }$. Above such lines GBS can be simulated approximately and exactly in polynomial time, respectively. In (a) we fix input squeezing to 1 dB, and from top to bottom we choose $q_D={10}^{-2},{10}^{-3},{10}^{-4}$. In (b) we fix $q_D={10}^{-2}$, and from top to bottom we choose 1, 5, and 10 dB squeezing. (c),(d) Sufficient condition for efficient simulation of GBS by using the $\alpha$-order sandwiched Rényi relative entropy, as given in Eq. (7). From top to bottom $\alpha =0.5$, 0.6, 0.7, 0.8, 0.9, 0.99, 0.999. Each line is obtained by numerically optimizing $D_{\alpha }(\sigma \parallel \tau )$ over $\tau \in {\mathcal{C}}_G^{(t)}$ for $t\in [1-2q_D,1]$. In both cases we fix 1 dB of squeezing. In (c) we choose $q_D={10}^{-2}$ while in (d) we consider perfect detectors.