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Description
Training a parametrized quantum circuit $U(θ)$ to reproduce an n-qubit target state |ψ⟩ is usually posed as maximizing the fidelity $|⟨ψ|ψ_θ⟩|²$. This faces two obstacles: 1) fidelity requires coherent access to |ψ⟩, and 2) being a global observable, fidelity suffers barren plateaus, with gradients vanishing as $2^{−Ω(n)}$. Huang, Preskill and Soleimanifar recently proposed the shadow overlap, a fidelity surrogate estimated from single-qubit measurements plus classical query access to the amplitudes of |ψ⟩, and conjectured it could avoid barren plateaus. That claim was left unproven and untested on parametrized circuits.
This work generalizes, proves, and simulates the conjecture. We derive an exact Pauli decomposition of the shadow-overlap's observable L, showing its locality is governed by a single classical quantity: the Walsh–Hadamard (WHT) degree $d_\star$ of the target state. We prove a gradient-variance formula and the resulting trainability criterion the loss is free of barren plateaus when $d_\star = O(\log(n))$, and exhibits a plateau when $d_\star = \Theta(n)$. This gives a computable, a-priori test of which states are efficiently learnable. We then classify concrete families: product states, bounded-range local phase states, graph/cluster states, and gapped (disordered-phase) TFIM ground states are all trainable ($d_\star = O(1)$ or $O(\log(n))$, whereas global-phase states such as the parity state are not. We further show that trainability is provably distinct from certifiability. Numerical simulations on TFIM ground states and range-R phase states confirm the predicted variance scaling.
| Topic | Quantum state reconstruction from data |
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| Additional Topic | Quantum variational circuits and their applications |