Input phase noise in Gaussian Boson sampling

TL;DR

The paper investigates how relative phase noise between input modes affects Gaussian Boson Sampling (GBS) and the associated classical simulability. Using Matrix Product Operators (MPOs) to model dephasing via a Wrapped Gaussian distribution, it shows that the entanglement entropy between bipartitions grows linearly with the number of input states, even under substantial phase noise. An analytical argument confirms that local phase noise yields per-mode entropies that sum to give linear scaling, and this behavior persists even when phase noise is intensified or combined with photon loss. The findings imply that phase stabilization may not be essential to preserve the classical hardness of GBS, with potential implications for practical experimental designs seeking quantum advantage without perfect phase control.

Abstract

Gaussian boson sampling is an important protocol for testing the performance of photonic quantum simulators. As such, various noise sources have been investigated that degrade the operation of such devices. In this paper, we examine a situation with phase noise between different modes of the input state leading to dephasing of the system. This models the phase fluctuations which remain even when the mean phase is controlled. We aim to determine whether these phase-noisy input states still form a computationally difficult problem. To do this, we use Matrix Product Operators to model the system, a technique recently used to model boson sampling scenarios. Our investigation finds that the Entanglement entropy grows linearly with the number of input states even for noisy input states. This implies that, unlike boson loss, this form of experimentally relevant noise remains difficult to simulate with tensor networks and may allow for the demonstration of quantum advantage without the need for implementing the challenging task of input-state phase stabilisation.

Figures (7)

• Figure 1: Standard Gaussian Boson Sampling set-up, where the first $N$ input modes of an $M\times M$ interferometer are filled with Gaussian states and the other $M-N$ modes are left empty. All of the output modes are measured by photon number resolving detectors.
• Figure 2: An example of obtaining matrix $A_{\overline{n}}$ for $M=4$ and output pattern $\overline{n} = (0,0,1,1)$. $A_{\overline{n}}$ is constructed from the dark violet elements of the original matrix and is in this case of size $4\times 4$.
• Figure 3: Haar random interferometer of size $M\times M$ corresponding to unitary operation $U$ decomposed into $M$ layers of beamsplitters which in this case represent local operations on 2 modes.
• Figure 4: Simulation of MPO Maximal Entanglement Entropy for phase-noisy input squeezed states with fixed squeezing parameter $r=0.4$ and different amounts of phase noise. The model of phase noise is a Wrapped Gaussian distribution with different $\sigma$. The lines with corresponding colors are linear fits of the data.
• Figure 5: Simulations of MPO Maximal Entanglement Entropy for completely phase-noisy input squeezed states with different squeezing parameters $r$ (uniform distribution). The lines with corresponding colors are linear fits of the data.