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A Unified Theory of Quantum Neural Network Loss Landscapes

ICLR 2025 · Probabilistic and scientific modeling › Uncertainty and Gaussian processes · approach: Infinite-width networks as Gaussian processes · 0 citations (OpenAlex)

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Proves that randomly initialized quantum neural network loss landscapes converge to Wishart processes determined by the Jordan algebra of the network, yielding general barren plateau, Gaussian-process-limit, and local-minima results.

Problem

Quantum neural networks generally do not behave as Gaussian processes at random initialization, and no unified asymptotic description of their loss landscapes (barren plateaus, local minima, NTK-type limits) exists.

Approach

Shows that every QNN loss and its first two derivatives, under approximately uniform random initialization, form Wishart processes whose degrees of freedom are set by the Jordan algebra generated by the objective observable and ansatz (the JAWS framework), unifying prior barren plateau, QNTK and local minima results.

Key results

• Loss variance under random initialization is the sum over Jordan components of Tr((O^a)^2)Tr((rho^a)^2)/dim Aut(A_a), generalizing prior barren plateau results.
• Local minima density is a convolution of gamma distributions; minima concentrate near the global minimum only when the network is overparameterized (p at least roughly max_a Tr(O^a)^2/Tr((O^a)^2)).
• Gives necessary and sufficient conditions for a QNN to converge to a Gaussian process limit via vanishing higher cumulants.

Surprising findings

• Rank-1 objective with complex Hermitian algebra gives exponentially distributed loss, so such QNNs do not converge to a Gaussian process.
• Poor local minima in polynomially sized shallow QNNs can only be avoided when effective Hilbert space dimension grows at most polynomially, narrowing the space for superpolynomial training advantage.

Limitations

• Results are asymptotic in dim(A) and rely on approximately uniform (approximate t-design) initialization.
• Assumes approximate Gaussianity of ansatz matrix elements; exceptional octonionic Jordan algebra excluded.

Experimental setup

Builds on: Lie algebra-supported ansatz quantum neural tangent kernel Kac-Rice formula Wishart distribution

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