Scalable Quantum Machine Learning: Trainability, Expressivity and Efficiency

Designing scalable parameterized quantum circuits for machine learning faces three fundamental obstacles: barren plateaus that prevent gradient-based training, the absence of provable guarantees that the learned function class is classically hard, and prohibitive circuit evaluations per gradient step. We propose the unitary brick-wall: a $k$-particle fermionic architecture for nearest-neighbor hardware, combining Reconfigurable Beam Splitter gates with interleaved single-qubit phase gates and a non-Gaussian magic-state encoding, where the particle number $k$ is a tunable dial trading classical simulation hardness against training cost. Trainable. The brick-wall has dynamical Lie algebra $\mathfrak{u}(n)$ and directly parametrizes $U(n)$ via Givens rotations, enabling Haar initialization. Two-body correlator readouts achieve gradient variance $\Theta(k^2/n^5)$. Expressive. Classical hardness is controlled by the particle number $k$: best-known classical algorithms for sampling and for two-body expectation values run in time $2^{\Theta(k)}\mathrm{poly}(n)$, worst-case #P-hardness holds at $k = n^{\epsilon}$, and average-case hardness applies at $k = \Theta(n)$. Efficient. A multi-layer parallel parameter-shift rule computes all $O(n^2)$ gradients from $k(8n+4)$ circuit evaluations per gradient step, a factor $3n/(8k)$ reduction over the $3n^2$ evaluations required by the standard parameter-shift rule. The unitary butterfly variant targets all-to-all hardware, with depth $2\log n$ and $n\log n$ parameters. It achieves similar hardness guarantees at $8k\log n$ evaluations per gradient step, the same factor $3n/(8k)$ reduction. Its trainability is established at two levels: the absence of exponential barren plateaus is unconditional, whereas the sharp $\Theta(k^2/n^5)$ rate holds under a two-particle approximate-2-design conjecture.