Scalable Quantum Machine Learning: Trainability, Expressivity and Efficiency
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 u(n) and is surjective onto U(n) via Givens rotations, enabling Haar initialization. Two-body correlator readouts achieve gradient variance Θ(k3/n5), polynomial in n throughout n−2k=Ω(n). Expressive. Hardness is controlled by k: best-known classical sampling runs in time 2Θ(k), worst-case #P-hardness holds from k=nε, and the average-case machinery of Fermion Sampling applies at k=Θ(n). The sampler is input-dependent, so costs are paid per query: at our operating point k=100 one query needs ∼ 1015 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(n2) gradients from 4kn circuit evaluations per step, a factor n/k reduction over the standard rule's 4n2, growing linearly with n at fixed k. The unitary butterfly targets all-to-all hardware, with 2log n layers of parallel gates and (3/2)nlog n parameters, similar hardness guarantees, and 4klog 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 Θ(k3/n5) rate needs a conjectured scalar comparison with the two-particle Haar second moment.
Related
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- Random layers for quantum optimal control with exponential expressivity
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BibTeX
@misc{kerenidis2026scalable,
title = {{Scalable Quantum Machine Learning: Trainability, Expressivity and Efficiency}},
author = {Iordanis Kerenidis},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2607.24014},
eprint = {2607.24014},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.24014},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Scalable Quantum Machine Learning: Trainability, Expressivity and Efficiency
AU - Iordanis Kerenidis
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2607.24014
UR - https://arxiv.org/abs/2607.24014
AB - 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 u(n) and is surjective onto U(n) via Givens rotations, enabling Haar initialization. Two-body correlator readouts achieve gradient variance Θ(k^3/n^5), polynomial in n throughout n−2k=Ω(n). Expressive. Hardness is controlled by k: best-known classical sampling runs in time 2^Θ(k), worst-case #P-hardness holds from k=n^ε, and the average-case machinery of Fermion Sampling applies at k=Θ(n). The sampler is input-dependent, so costs are paid per query: at our operating point k=100 one query needs ∼ 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 2log n layers of parallel gates and (3/2)nlog n parameters, similar hardness guarantees, and 4klog 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 Θ(k^3/n^5) rate needs a conjectured scalar comparison with the two-particle Haar second moment.
ER -