2026-10-28 · arXiv Quantum Physics
arXiv:2607.24014v3 Announce Type: replace
Abstract: Scalable parameterized quantum circuits for machine learning face three obstacles: barren plateaus, no guarantee 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 Splitters with interleaved single-qubit phases and a non-Gaussian magic-state encoding, where $k$ trades classical 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. Hardness is controlled by $k$: best-known classical sampling runs in time $2^{\Theta(k)}$, worst-case #P-hardness holds from $k=n^{\epsilon}$, and the average-case machinery of Fermion Sampling applies at $k=\Theta(n)$. The sampler is input-dependent, so costs are paid per query: at our operating point $k=100$ one query needs $\sim 10^{15}$ operations, beyond commercial query rates, and at $k=120$ an exascale machine sustains under one query per second.
Efficient. A multi-layer parallel parameter-shift rule computes all $O(n^2)$ gradients from $4kn$ circuit evaluations per step, a factor $n/k$ reduction over the standard rule's $4n^2$, growing linearly with $n$ at fixed $k$.
The unitary butterfly targets all-to-all hardware, with $2\log n$ layers of parallel gates and $(3/2)n\log n$ parameters, similar hardness guarantees, and $4k\log n$ evaluations per 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 needs a conjectured scalar comparison with the two-particle Haar second moment.
原文链接：https://arxiv.org/abs/2607.24014
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