# Techniques: Trainability and Expressivity of Hamming-Weight Preserving Quantum Circuits for Machine Learning

## What the paper contributes (one paragraph)

The paper analyses Hamming-weight (HW) preserving variational quantum circuits built from Reconfigurable Beam Splitter (RBS) and Fermionic Beam Splitter (FBS) gates, which keep the state inside the fixed-weight subspace of dimension binom(n,k). It contributes a trainable amplitude-encoding data loader that encodes any binom(n,k)-dimensional vector on n qubits, together with quantum-optimal-control tools (Dynamical Lie Algebra dimension and Quantum Fisher Information Matrix rank) and two constructive algorithms for deciding existence and building such a loader. It proves an exact analytic expression for the cost-gradient variance of these circuits, yielding conditions for the absence (and presence) of Barren Plateaus that depend on the subspace dimension rather than on a 2-design or controllability assumption, and shows this refutes a controllability-trainability conjecture in this setting. It also proves that the QFIM rank of any VQC is almost-everywhere constant, and notes when fixed-weight circuits are classically simulable.

## Techniques offered

- **Hamming-weight preserving amplitude-encoding quantum data loader** - a trainable n-qubit RBS/FBS circuit that prepares the amplitude encoding of an arbitrary binom(n,k)-dimensional real vector in the fixed Hamming-weight subspace.
  - guarantee: existence supported when the subspace Dynamical Lie Algebra dimension or the QFIM rank reaches the controllability threshold (d_k - 1); training is classically simulable so any quantum speed-up is at most polynomial.
  - quote: "In this work, we propose a method to achieve the amplitude encoding of any $\binom{n}{k}$-dimensional vector using $n$ qubits by training a VQC made of Hamming weight (HW) preserving gates."

- **DLA and QFIM-rank controllability characterisation for data-loader design** - uses the Dynamical Lie Algebra dimension and the maximal Quantum Fisher Information Matrix rank as controllability metrics, with two greedy algorithms (add gates until the QFIM rank saturates, or prune an overparametrised circuit) to certify and construct a loader from a given qubit connectivity.
  - guarantee: a static controllability certificate; maximal QFIM rank equal to d_k - 1 is taken as evidence the circuit can reach any state on the sphere S^{d_k-1}, and DLA dimension below d_k - 1 proves a loader is impossible.
  - quote: "We propose to use quantum optimal control tools to show that, according to the circuit connectivity, we can prove the existence of a quantum data loader."

- **Absence-of-Barren-Plateaus trainability guarantee for RBS/FBS circuits** - an exact analytic formula for the variance of the l2 cost-function gradient that gives conditions of existence or absence of Barren Plateaus for HW-preserving circuits without a 2-design or controllability assumption.
  - guarantee: statistical bound; the gradient variance scales as k(n-k)/(n(n-1)) * 8/d_k, i.e. as O(1/binom(n,k)), so trainability is preserved for small fixed k and Barren Plateaus appear for large k.
  - quote: "Lastly, we analyze the trainability of Hamming weight preserving circuits, and show that the variance of the $l_2$ cost function gradient is bounded according to the dimension $\binom{n}{k}$ of the subspace."

- **Almost-constant QFIM rank theorem** - proves that for any standard VQC the rank of the Quantum Fisher Information Matrix is constant almost everywhere on parameter space, so a single random parameter sample yields the maximal rank.
  - guarantee: static proof (via analyticity of VQC functions); the QFIM rank equals r_max with probability 1 for a uniformly drawn parameter point.
  - quote: "we provide a theoretical justification for the fact that the rank of the QFIM of any VQC state is almost-everywhere constant, which is of separate interest."

- **Classical simulation of fixed-Hamming-weight subspace circuits** - identifies that an HW-preserving circuit confined to a fixed-weight subspace can be simulated directly with polynomially sized d_k by d_k matrices, with matchgate and Lie-algebraic simulation as alternatives.
  - guarantee: classical reproduction with cost polynomial in n when k is fixed (d_k polynomial); exponential when k grows so that d_k = Omega(exp(n)).
  - quote: "If a HW-preserving circuit is employed entirely in a subspace of fixed HW $k$ (independent of the number of qubits $n$), then it may be classically simulated directly, i.e. by constructing all the polynomially-sized unitary matrices ($d_k \times d_k$) and state vectors ($d_k$) classically and performing matrix multiplications."

## Where it could apply

- Target primitive(s): VQE / QAOA and general variational quantum circuits, amplitude encoding (data loading), and quantum neural network / orthogonal neural network layers.
- Target application group(s): quantum_ml, platforms_tooling (trainability diagnostics and classical simulation), with chemistry_materials and optimization as downstream consumers of symmetry-preserving ansatze.
- Code family / hardware assumptions: gate-model hardware with native or easy RBS gates and a connected qubit connectivity graph; results also cover FBS gates; no error-correcting code assumed.

## Caveats

The encoding and trainability results are specific to RBS and FBS (matchgate-type) Hamming-weight preserving circuits and to the squared-Euclidean (l2) cost function; the general-case variance theorem assumes input and target states drawn from a distribution between a spherical 1-design and 2-design, so it does not cover all learning tasks. The CPSA absence-of-Barren-Plateaus theorem depends on an unproven spectral-gap conjecture for certain stochastic matrices (supported only by numerical evidence). Because the data loaders are classically simulable, any quantum advantage from them is at most polynomial.