Noise-reduction of multimode Gaussian Boson Sampling circuits via Unitary Averaging

We improve Gaussian Boson Sampling (GBS) circuits by integrating the unitary averaging (UA) protocol, previously demonstrated to protect unknown Gaussian states from phase errors [Phys. Rev. A 110, 032622]. Our work extends the applicability of UA to mitigate arbitrary interferometric noise, including beam-splitter and phase-shifter imperfections. Through comprehensive numerical analysis, we demonstrate that UA consistently achieves higher fidelity and success probability compared to unprotected circuits, establishing its robustness in noisy conditions. Remarkably, enhancement is maintained across varying numbers of modes with respect to the noise. We further derive a power-law formula predicting performance gains in large-scale systems, including 100-mode and 216-mode configurations. A detailed step-by-step algorithm for implementing the UA protocol is also provided, offering a practical roadmap for advancing near-term quantum technologies.

💡 Research Summary

The paper addresses a critical obstacle in scaling Gaussian Boson Sampling (GBS) – the accumulation of interferometric noise in large linear‑optical networks. Building on earlier work that showed unitary averaging (UA) could protect unknown Gaussian states from pure phase errors, the authors extend the protocol to mitigate arbitrary linear‑optical imperfections, including fluctuations in beam‑splitter transmissivities and phase‑shifter angles.

The core idea of UA is to duplicate the unknown input state across n identical “replica” channels using a passive 50:50 beam‑splitter network (the encoding stage). Each replica then experiences independent stochastic unitary noise, modeled as Gaussian‑distributed variations of the beam‑splitter angles θ and phase shifts ϕ with zero mean and variance σ². After the noisy unitaries, a decoding network (the inverse of the encoding) recombines the replicas. By heralding vacuum on the n – 1 “error modes”, the remaining mode is post‑selected and exhibits a reduced effective noise. Mathematically, the output covariance matrix is derived, and in the low‑noise limit analytical approximations for ⟨tanh r′⟩ and ⟨cos ϕ_β⟩ are obtained, leading to explicit formulas for fidelity F and success probability P as functions of n and σ.

The authors first analyze a two‑mode squeezed‑vacuum input. Numerical simulations show that with a single replica (n = 1) the fidelity drops to ~92 % at σ = 0.08, whereas with n = 2 it rises to ~98 % and the success probability exceeds 97.5 %. The improvement is robust against variations in the individual squeezing parameters as long as the average photon number is kept constant.

Scaling up, the paper presents results for three, four, five, and ten‑mode GBS circuits. In each case, applying UA with n = 2 already yields a noticeable fidelity boost; increasing n to 4 or 8 further improves performance until a saturation point is reached. The authors fit the success probability to a power‑law form

 P_success ≈ 1 – C · σ^α · n^–β,

where C, α, β are empirically determined constants. This expression accurately predicts the behavior of much larger systems. Using it, the authors extrapolate to 100‑mode and 216‑mode GBS devices (the current experimental record) and predict that UA will still provide a several‑percent fidelity gain and maintain high success probabilities even under realistic noise levels.

A practical, step‑by‑step algorithm for experimental implementation is provided: (1) prepare the multimode squeezed Gaussian state; (2) encode it across n replicas with a balanced beam‑splitter mesh; (3) let each replica traverse an interferometer subject to independent stochastic noise; (4) decode with the inverse mesh; (5) perform vacuum heralding on the n – 1 error modes; (6) carry out the standard GBS measurement on the surviving mode. The protocol is compatible with existing loss‑tolerant techniques and does not require active feed‑forward, making it attractive for near‑term photonic platforms.

In conclusion, the work demonstrates that unitary averaging is a versatile, hardware‑efficient error‑mitigation strategy that goes far beyond phase‑only correction. It can suppress generic interferometric imperfections in multimode GBS circuits, scales favorably with system size, and can be integrated into current large‑scale photonic experiments. This positions UA as a promising tool for advancing quantum advantage demonstrations and for future fault‑tolerant photonic quantum computing architectures.