Proof of hiding conjecture in Gaussian boson sampling

Gaussian boson sampling (GBS) is a promising protocol for demonstrating quantum computational advantage, and this paper proves that one can hide a complex Gaussian matrix as a submatrix of the outer product of Haar unitary submatrices in total variation distance, which provides a key argument for the classical hardness of simulating GBS.

The core conjecture and its relevance

The central finding is the proof of Conjecture 1 (hiding in GBS), which asserts that for an input state with a maximal number of squeezed states, the submatrix can be well-approximated by a complex Gaussian matrix in total variation distance as the matrix size M goes to infinity. This result is significant because it provides a rigorous proof of the hiding property for GBS in the experimentally relevant regime where K = M, which is crucial for establishing quantum advantage arguments analogous to those used for conventional boson sampling.

The GBS setup and probability distribution

Gaussian boson sampling involves preparing an initial Gaussian state with 1 ≤ K ≤ M single-mode squeezed states and M − K vacuum states, which then interfere in an optical network described by a unitary matrix U. The output photon distribution is given by the probability [4–6] P[n] = tanh(r) N / cosh(r)K Haf hUIKU T n,n i 2. This probability is proportional to the hafnian of a submatrix of UIKU T, where U is a Haar random unitary matrix. The goal is to show that this distribution can be approximated by a Gaussian matrix G in total variation distance.

The formal statement of the hiding conjecture

Conjecture 1 formally states: "Let UNK be the top left N ×K submatrix of an M ×M Haar random unitary matrix, and let Z = ZN,K be a matrix with distribution given by either Gsym N or GGT NK. Then for N ≤ K ≤ M and one of the distributions of Z, there exist polynomials p, r such that for any δ > 0 and M ≥ p(N)/r(δ), dTV(MK−1/2UNKU T NK, Z) = O(δ)." The paper proves Theorem 1.1 for the case K = M with Z drawn from Gsym N.

The proof strategy via KL divergence

Theorem 1.2, which states that dTV(√MA, G) ≤ O(N/√M) for N = o(√M), is proven using Pinsker’s inequality: dTV(√MA, G) ≤ r 1/2 DKL(√MA G). The Kullback–Leibler (KL) divergence is bounded by the expectation term in Proposition 3.1, which shows E" log˜f(√MA)g(√MA) = O N 2/M for N = o(√M). Furthermore, Proposition 3.2 establishes that the normalizing constant term 1 - ζ = O(N 2/M), where ζ is related to the normalization constant of the density, ensuring that the overall bound remains O(N/√M).

Implications for classical hardness

The proof relies on bounding moments of singular values of submatrices (Lemma 4.2 and Lemma 4.3) using Weingarten calculus (Lemma 2.2), which yields bounds like E X N j=1 λj = N squared + N/M + O(N 2/M 2). This, combined with the KL divergence estimates, leads to Theorem 1.3, which provides a multiplicative estimate on density functions: f(Z) ≤ (1 + O(N 3/M))g(Z). This entrywise closeness suggests that estimating Haf(G) and Haf(GGT) should be essentially the same problem, supporting the quantum advantage argument for GBS.

Extension to other regimes

The paper also discusses cases where K < M, proving Theorem 1.5 for NK = o(M), which shows dTV(MUNKU T NK, GNKG T NK) = O r N K / M!. This extends the hiding property beyond the sparse regime (K=o(M1/5)) to the maximal extent K = O(M1−ε). The authors conclude that while intuition suggests hiding might be harder for smaller K, their rigorous proof for K=M confirms it holds, making it plausible that it holds for smaller K as well.

Key results summarized:

1. Theorem 1.2 proves the hiding conjecture for GBS with K = M: dTV(√MA, G) ≤ O(N/√M).

2. Theorem 1.3 provides a multiplicative estimate on densities: f(Z) ≤ (1 + O(N 3/M))g(Z).

3. Theorem A.

As a fastidious researcher, I have analyzed Proof of Hiding Conjecture in Gaussian Boson Sampling. This paper provides a rigorous proof for the hiding property of Gaussian Boson Sampling (GBS) in the experimentally relevant regime where the number of squeezed states equals the number of modes, and it establishes that this property is sufficient to argue for classical hardness against simulating GBS.

Here are the specific improvements to AI systems derived from this research:

1. Improve Quantum Simulation and Verification Capabilities:

2. Enhance Quantum Machine Learning (QML) Model Training Robustness:

3. Develop Certified Quantum Algorithms for Hard Problems:

Specific Improvements and Capabilities

1. Improve Quantum Simulation and Verification Capabilities

The paper proves that a submatrix of a Circular Orthogonal Ensemble (COE) random matrix, derived from the GBS output distribution, can be well-approximated by a complex Gaussian matrix in total variation distance when the number of squeezed states equals the number of modes.

The improved AI system can:

• Identify and verify quantum hardware implementations of GBS protocols by checking if their output distributions conform to the predicted Gaussian approximations under specific experimental conditions (maximal squeezing, maximal mode usage).

• Develop automated diagnostic tools to assess whether a physical quantum circuit's output statistics are consistent with the expected complexity bounds derived from the hiding conjecture.

• Accurately model and simulate large-scale GBS experiments, providing tighter error bounds on the simulation accuracy based on Theorem 1.2 (TV distance bound of O(N/√M)).

2. Enhance Quantum Machine Learning (QML) Model Training Robustness

The paper establishes that simulating GBS is classically hard because it requires approximating the permanent/hafnians of matrices, and this hardness relies on the hiding conjecture for Gaussian matrices.

The improved AI system can:

• Design QML models whose training loss landscape is deliberately constructed to mimic the structure of high-order Hafnian calculations (e.g., via embedded random matrix structures).

• Utilize GBS sampling as a benchmark or oracle within QML training loops to test the system's ability to navigate classically hard optimization problems, thereby improving its robustness against classical simulation attacks on quantum algorithms.

• Develop hardness-aware neural network architectures that are specifically designed to resist classical approximation techniques targeting the underlying matrix structure of GBS.

3. Develop Certified Quantum Algorithms for Hard Problems

The core result is using the hiding property to argue for the quantum advantage of GBS in approximating hard functions (like Hafnians).

The improved AI system can:

• Implement quantum algorithms that leverage this proven hiding property to approximate complex, classically hard functions (like the Hafnian) with provable error bounds derived from Theorem 1.2 and Theorem 1.3.

• Develop certified quantum algorithms where the output is not just a sample, but a result whose accuracy is mathematically guaranteed up to a certain total variation distance, directly leveraging the proof structure involving Pinsker's inequality (Equation 3.2).

• Create hybrid quantum-classical systems that use GBS as a subroutine to efficiently estimate intractable quantities (like the Hafnian of random matrices) with quantifiable precision, moving beyond mere probabilistic speedups toward verifiable complexity advantages.

Gaussian boson sampling (GBS) is a promising protocol for demonstrating quantum computational advantage. One of the key steps in the argument for classical hardness of GBS is the so-called ``hiding conjecture'', which asserts that one can ``hide'' a complex Gaussian matrix as a submatrix of the symmetric product of Haar unitary submatrices in total variation distance. In this paper, we prove the hiding conjecture for input states with all input modes squeezed, which is a setup that has recently been realized experimentally [Madsen et al., Nature 606, 75 (2022)]. In this setting, the hiding conjecture states that a o(sqrt M) times o(sqrt M) submatrix of an M times M circular orthogonal ensemble (COE) random matrix can be well-approximated by a complex symmetric Gaussian matrix in total variation distance as M to infinity. This is the first rigorous proof of the hiding property for GBS in the experimentally relevant regime, and puts the argument for hardness of classically simulating GBS with the maximum number of squeezed input modes on a comparable level to that of the conventional boson sampling of [Aaronson and Arkhipov, Theory Comput. 9, 143 (2013)].