Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

From Reachability to Learnability: Geometric Design Principles for Quantum Neural Networks

Vishal S. Ngairangbam, Michael Spannowsky

TL;DR

This work reframes QNN design from state reachability to controllable geometry of hidden quantum representations, and introduces Classical-to-Lie-algebra maps and the criterion of almost Complete Local Selectivity (aCLS), which combines directional completeness with data-dependent local selectivity.

Abstract

Classical deep networks are effective because depth enables adaptive geometric deformation of data representations. In quantum neural networks (QNNs), however, depth or state reachability alone does not guarantee this feature-learning capability. We study this question in the pure-state setting by viewing encoded data as an embedded manifold in $\mathbb{C}P^{2^n-1}$ and analysing infinitesimal unitary actions through Lie-algebra directions. We introduce Classical-to-Lie-algebra (CLA) maps and the criterion of almost Complete Local Selectivity (aCLS), which combines directional completeness with data-dependent local selectivity. Within this framework, we show that data-independent trainable unitaries are complete but non-selective, i.e. learnable rigid reorientations, whereas pure data encodings are selective but non-tunable, i.e. fixed deformations. Hence, geometric flexibility requires a non-trivial joint dependence on data and trainable weights. We further show that accessing high-dimensional deformations of many-qubit state manifolds requires parametrised entangling directions; fixed entanglers such as CNOT alone do not provide adaptive geometric control. Numerical examples validate that CLS-satisfying data re-uploading models outperform non-tunable schemes while requiring only a quarter of the gate operations. Thus, the resulting picture reframes QNN design from state reachability to controllable geometry of hidden quantum representations.

From Reachability to Learnability: Geometric Design Principles for Quantum Neural Networks

TL;DR

This work reframes QNN design from state reachability to controllable geometry of hidden quantum representations, and introduces Classical-to-Lie-algebra maps and the criterion of almost Complete Local Selectivity (aCLS), which combines directional completeness with data-dependent local selectivity.

Abstract

Classical deep networks are effective because depth enables adaptive geometric deformation of data representations. In quantum neural networks (QNNs), however, depth or state reachability alone does not guarantee this feature-learning capability. We study this question in the pure-state setting by viewing encoded data as an embedded manifold in and analysing infinitesimal unitary actions through Lie-algebra directions. We introduce Classical-to-Lie-algebra (CLA) maps and the criterion of almost Complete Local Selectivity (aCLS), which combines directional completeness with data-dependent local selectivity. Within this framework, we show that data-independent trainable unitaries are complete but non-selective, i.e. learnable rigid reorientations, whereas pure data encodings are selective but non-tunable, i.e. fixed deformations. Hence, geometric flexibility requires a non-trivial joint dependence on data and trainable weights. We further show that accessing high-dimensional deformations of many-qubit state manifolds requires parametrised entangling directions; fixed entanglers such as CNOT alone do not provide adaptive geometric control. Numerical examples validate that CLS-satisfying data re-uploading models outperform non-tunable schemes while requiring only a quarter of the gate operations. Thus, the resulting picture reframes QNN design from state reachability to controllable geometry of hidden quantum representations.
Paper Structure (27 sections, 1 theorem, 57 equations, 5 figures, 3 tables)

This paper contains 27 sections, 1 theorem, 57 equations, 5 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Infinitesimal geometry and quantum feature learning
  3. Building a geometric classical-to-quantum bridge
  4. Necessity of parametrically controllable directions
  5. Necessity of tunable data dependence
  6. Towards a geometric definition of quantum layers
  7. Classical-to-Lie-algebra maps
  8. General structure of quantum layers
  9. Almost Complete Local Selectivity
  10. Geometry of hidden quantum feature spaces
  11. Global non-selectivity of data-independent unitaries
  12. Non-adaptive selectivity in pure-data re-uploading
  13. Geometrically flexible CLA maps
  14. Accessing exponentially scaling manifold dimensions
  15. Benchmarking against pure data re-upload baselines
  16. ...and 12 more sections

Key Result

Proposition 1

A parametrised quantum gate $e^{i\;\alpha(\mathbf{w},\mathbf{x})\;G}$ acting on $\mathcal{Q}$ has tunable unitary action under the weight $w_j$ i.e. $\frac{\partial{D_{FS}(|\psi_\mathbf{w}(\mathbf{x}_1)\rangle,|\psi_\mathbf{w}(\mathbf{x}_2)\rangle)}}{\partial w_j}\neq 0$ if $\frac{\partial^2\alpha}{

Figures (5)

  • Figure 1: Fidelity (top) between two states $|\psi_w(x_1)\rangle$ and $|\psi_w(x_2)\rangle$ where $|\psi_w(x)\rangle=e^{\frac{i}{2}wx\sigma_z}\;e^{\frac{i}{2}x\sigma_x}|0\rangle$. The partial derivative $\frac{\partial F}{\partial w}$ with respect to the tunable parameter $w$ is shown in the bottom row. We see that when $x_2=0$, since $|\psi_w(x_2)\rangle=|0\rangle$, fidelity is independent of $w$ with a zero partial derivation for any value of $x_1$.
  • Figure 2: The parametrised circuit block corresponding to the unitary operator $U_{SE}(\mathbf{a}_{SE})$ for $n=3$ qubits.
  • Figure 3: Output histograms for the $S^5_A$ vs $S^5_B$ binary classification experiment comparing the best performing aCLS and PDR models for the $|\bar{r}_A-\bar{r}_B|=3\sigma$ case (top-right) and $|\bar{r}_A-\bar{r}_B|=1\sigma$ (top-left), along with their corresponding ROC curves (bottom).
  • Figure 4: Output histogram (left) of the best-performing aCLS and PDR models for the top-decay classification and the corresponding ROC curves (right).
  • Figure 5: Averaged confusion matrices normalised by true labels of the aCLS (right) and PDR (left) models for the multi-class jet classification scenario.

Theorems & Definitions (3)

  • Proposition 1
  • Definition 1
  • Definition 2