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Preprint Proves a Conditional GBS Hiding Result Across Squeezer Counts
An arXiv preprint proves a Gaussian boson sampling hiding result for arbitrary squeezed-input counts, with vanishing error when the photon count is o(√K). Its hardness reduction has additional qualifications, including a failure of density-
Quantum.world Team · Written with AI · fact-checked ·
An arXiv preprint reports a proof of the hiding conjecture for Gaussian boson sampling (GBS) with arbitrary numbers of squeezed input modes. The distributional result is conditional: for an interferometer with M modes, K equally squeezed inputs and N detected photons, the stated error is O(N/√K), which tends to zero when N=o(√K). The preprint was submitted on August 19, 2026; it is not the Quantum paper previously associated with this result.
“We complete the proof of the hiding conjecture for Gaussian boson sampling with an arbitrary number of squeezers K, which is a part of the argument for classical hardness of GBS.” Laura Shou, Alexey V. Gorshkov, Victor Galitski and Sarah H. Miller, the paper’s authors, write.
What the proof hides inside a random interferometer
In Gaussian boson sampling, squeezed light passes through a network of optical modes and detectors record photon counts. For collision-free outcomes, the probability of an output depends on the squared magnitude of a matrix’s hafnian, a function that sums over pairings of matrix elements. Complexity arguments connect that calculation to the difficulty of reproducing the sampler’s output distribution on a classical computer.
The hiding step asks whether a random complex Gaussian matrix can be embedded, in distribution, as a submatrix of the matrix describing the optical network. If the distributions are close in total variation distance, a hypothetical sampler can be used in a reduction to estimate hafnians of Gaussian matrices. Earlier proofs covered the sparse-squeezer regime and the fully squeezed case; the preprint establishes the hiding result across the remaining squeezer counts, subject to its parameter conditions.
The bound requires N=o(√K)
For an interferometer with M modes, K squeezed inputs and N detected photons, the authors show that the appropriately scaled submatrix is within total variation distance O(N/√K) of a symmetric complex Gaussian matrix with independent entries. This error tends to zero only in the stated asymptotic regime N=o(√K), with N≤K≤M. They also show that the corresponding scaled product of an independent Gaussian matrix and its transpose approaches the same Gaussian distribution in this regime. The range N≤K≤M alone is not the condition that makes the error vanish.
That distinction matters because the hiding argument needs the photon-count submatrix to behave like the random matrices used in the hardness analysis. The result is mathematical, not an experimental demonstration: it does not report a new sampler, photon count or device benchmark. The preprint’s distributional bound addresses arbitrary K, but does not erase the restrictions on N.
Why the earlier rejection-sampling route breaks
Proving distributional closeness does not by itself provide an efficient way to construct the hidden instances needed for a hardness reduction. The familiar density-based rejection-sampling method relies on a pointwise comparison between the submatrix distribution and a Gaussian distribution. The authors show that this comparison fails when K=cM for c<1/2: the density ratio can grow exponentially with M, or the relevant density may not exist.
Instead, they use approximate instance generation, adapting a method that produces nearly Haar-random optical unitaries and nearly uniform output locations. The hardness reduction relies on an assumption that the required random unitaries and matrices can be represented and computed at sufficient finite precision. This is a qualification of the argument, not an experimental engineering result.
A hardness argument, not a proof of quantum advantage
The preprint’s result supports one component of the classical-hardness case for Gaussian boson sampling; it does not establish that every such experiment is classically intractable. The broader conclusion also depends on an average-case conjecture about approximating squared hafnians of random complex Gaussian matrices. The authors state that, if this conjecture holds, a classical approximate sampler under the reduction’s stated conditions would imply a collapse of the polynomial hierarchy.
That conjecture marks a boundary of what the work establishes. The preprint proves a hiding result across squeezer counts with the stated asymptotic bound, while its hardness argument also depends on the hafnian conjecture and the finite-precision implementation assumption.