Sufficient conditions for hardness of lossy Gaussian boson sampling

TL;DR

This work establishes complexity-theoretic foundations for the classical hardness of lossy Gaussian boson sampling under photon loss. By deriving a loss-dependent analytic form for the output probabilities and constructing a low-degree polynomial interpolation strategy, the authors show that lossy GBS remains as hard as ideal GBS when the average number of lost photons scales as $O(\\log N)$. They introduce a pair of average-case hardness problems and laborious reductions (via Stockmeyer’s framework) that connect lossy-output probabilities to the ideal case, alongside an information-theoretic total-variation-distance bound as an alternative route. The results provide a rigorous threshold and methodology for asserting quantum advantage with near-term, lossy GBS experiments, clarifying the interplay between photon loss, post-selection, and computational hardness in Gaussian-based photonic platforms.

Abstract

Gaussian boson sampling (GBS) is a prominent candidate for the experimental demonstration of quantum advantage. However, while the current implementations of GBS are unavoidably subject to noise, the robustness of the classical intractability of GBS against noise remains largely unexplored. In this work, we establish the complexity-theoretic foundations for the classical intractability of noisy GBS under photon loss, which is a dominant source of imperfection in current implementations. We identify the loss threshold below which lossy GBS maintains the same complexity-theoretic level as ideal GBS, and show that this holds when at most a logarithmic fraction of photons is lost. We additionally derive an intractability criterion for the loss rate through a direct quantification of the statistical distance between ideal and lossy GBS. This work presents the first rigorous characterization of classically intractable regimes of lossy GBS, thereby serving as a crucial step toward demonstrating quantum advantage with near-term implementations.

Figures (4)

• Figure 1: (a) Schematic of our lossy GBS setup, composed of input $M$ squeezed vacuum states, an $M$-mode random linear optical circuit, beam splitter loss described in (b), and measurement of $N$ output photons. (b) The beamsplitter loss model oszmaniec2018classicalgarcia2019simulatingqi2020regimesoh2024classicalbarnett1998quantumdemkowicz2015quantum. Each input mode interacts with an ancillary vacuum mode via a beamsplitter of transmittance $\sqrt{\eta}$, after which the ancillary mode is traced out, as indicated by the hatched square in the figure.
• Figure 2: Outline of our hardness analysis on lossy GBS. To summarize, we show that when the transmission rate $\eta^*$ of lossy GBS exceeds a threshold $\eta_{\rm{th}}$ given in Theorem \ref{['thm: main result informal']}, such lossy GBS is classically hard to simulate under certain complexity-theoretic assumptions.
• Figure 3: Schematics for the proof of Theorem \ref{['thm: main result informal']}. Given an oracle $\mathcal{O}$ that estimates $P(\eta,X)$ in Eq. \ref{['qqe']} for $\eta \leq \eta^*$ and $X \sim \mathcal{N}(0,1)_{\mathbb{C}}^{N \times M}$, one can estimate the $N$-degree polynomial $R_{X}(\eta)$ (black curve) through $\mathcal{O}(\eta, X)Q(\eta)$ (blue dots) by explicitly computing $Q(\eta)$. This also allows estimation of the low-degree polynomial $R_X^{(l)}(\eta)$ (red curve) for $\eta$ satisfying $R_{X}^{(l)}(\eta) \approx R_{X}(\eta)$ (grey area) and $\eta \leq \eta^*$. Using the estimated values of $R_{X}^{(l)}(\eta)$ for different $\eta$ values, one can construct the corresponding Lagrange interpolation polynomial (magenta curve), which in turn allows estimating the value $R_{X}^{(l)}(1) = P(X)$ in Eq. \ref{['eq: rescaled ideal probability']}.
• Figure S4: Graph representation of the $2N$ by $2N$ adjacency matrix $\mathcal{A}$ described in Eq. \ref{['adjacencymatrix']}. The corresponding graph $G = (V,E)$ can be represented as a bipartite graph, whose vertices $V$ can be divided into the vertex sets $V_1$ and $V_2$. Here, for each $i \in [N]$, the $i$th vertex in $V_1$ corresponds to the matrix index $i$ of $\mathcal{A}$, and the $i$th vertex in $V_2$ corresponds to the matrix index $N + i$ of $\mathcal{A}$. Also, the edge set $E$ can be divided into the edge sets $E_1$, $E_2$, and $E_I$. Here, the edges in $E_1$ and $E_2$, denoted as the blue edges in the figure, connect the vertices inside $V_1$ and $V_2$, respectively. Accordingly, the adjacency matrix of the subgraph $(V_1,E_1)$ and $(V_2, E_2)$ is given by $XX^T$ and $X^*X^{\dag}$, respectively. Finally, the edges in $E_I$, denoted as the red edges in the figure, connect the vertices between $V_1$ and $V_2$. Specifically, the $i$th edge in $E_I$ connects the $i$th vertex of $V_1$ with $i$th vertex of $V_2$ with its weight $(1-\eta)M\tanh r$, for all $i \in [N]$.

Theorems & Definitions (16)

• Conjecture 1: hamilton2017gaussiankruse2019detailed
• Theorem 1
• Conjecture 2: Loss reduces complexity of GBS
• Corollary 1: Hardness of lossy GBS
• proof : Proof Sketch of Theorem \ref{['thm: main result informal']}
• Lemma 1
• proof
• Theorem 2
• Theorem 3: Restatement of Theorem \ref{['thm: main result informal']}
• Lemma 2