Scalable Quantum Machine Learning: Trainability, Expressivity and Efficiency

Abstract

Designing scalable parameterized quantum circuits for machine learning faces three obstacles: barren plateaus, the absence of 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 $k$ is a tunable dial trading classical simulation hardness against training cost. Trainable. The brick-wall has dynamical Lie algebra $\mathfrak{u}(n)$ and is surjective onto $U(n)$ via Givens rotations, enabling Haar initialization. Two-body correlator readouts achieve gradient variance $\Theta (k^3/n^5)$, polynomial in $n$ throughout $n-2k=\Omega (n)$. Expressive. Classical hardness is controlled by $k$: best-known classical sampling algorithms run in time $2^{\Theta (k)}\mathrm{poly}(n)$, worst-case #P-hardness holds from $k=n^{\epsilon }$, and the average-case machinery of Fermion Sampling applies at $k=\Theta (n)$. At our operating point $k=60$, best-known classical simulation exceeds $10^{24}$ operations at every $n$. Efficient. A multi-layer parallel parameter-shift rule computes all $O(n^2)$ gradients from $4kn$ circuit evaluations per gradient step, a factor $n/k$ reduction over the $4n^2$ evaluations of the standard rule, growing linearly with $n$ at fixed $k$. The unitary butterfly variant targets all-to-all hardware, with depth $2\log n$ and $(3/2)n\log n$ parameters, similar hardness guarantees, and $4k\log n$ evaluations per gradient step -- the same factor-$n/k$ reduction. Its trainability holds at two levels: absence of exponential barren plateaus is unconditional, while the sharp $\Theta (k^3/n^5)$ rate holds under a two-particle approximate-2-design conjecture.

45 pages. v2: Exact closed-form gradient variance with improved $\Theta (k^3/n^5)$ rate; improved parameter-shift count. The simulability proposition now covers arbitrary fixed body number via $r$-RDM propagation (we thank Erfan Amidi), making triplet-block two-body readouts polynomial-time classically; that encoding is removed. Sampling hardness unchanged. Extended ML pipeline presentation