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Data Complexity Measures for Quantum Circuits Architecture Recommendation

Fernando M de Paula Neto

TL;DR

The paper tackles the challenge of selecting compact, effective quantum circuit architectures for parametric quantum circuits under NISQ constraints by leveraging data-derived complexity measures. It introduces the Quantum Architecture Recommendation Processor (QARP), which uses 22 dataset metrics to predict the best PQC layer and the required number of layer repetitions, validated across 14 datasets and six circuit layouts. Key findings show that single metrics (notably T2 for layer count and N2 for layer decisions) can yield near-optimal or optimal performance, while combining all metrics does not always improve results. The work demonstrates a data-driven meta-learning approach to quantum circuit design with potential practical impact on automating architecture search for quantum classifiers.

Abstract

Quantum Parametric Circuits are constructed as an alternative to reduce the size of quantum circuits, meaning to decrease the number of quantum gates and, consequently, the depth of these circuits. However, determining the optimal circuit for a given problem remains an open question. Testing various combinations is challenging due to the infinite possibilities. In this work, a quantum circuit recommendation architecture for classification problems is proposed using database complexity measures. A quantum circuit is defined based on a circuit layer and the number of times this layer is iterated. Fourteen databases of varying dimensions and different numbers of classes were used to evaluate six quantum circuits, each with 1, 2, 3, 4, 8, and 16-layer repetitions. Using data complexity measures from the databases, it was possible to identify the optimal circuit capable of solving all problems with up to 100$\%$ accuracy. Furthermore, with a mean absolute error of 0.80 $\pm$ 2.17, one determined the appropriate number of layer repetitions, allowing for an error margin of up to three additional layers. Sixteen distinct machine learning models were employed for the selection of quantum circuits, alongside twelve classical regressor models to dynamically define the number of layers.

Data Complexity Measures for Quantum Circuits Architecture Recommendation

TL;DR

The paper tackles the challenge of selecting compact, effective quantum circuit architectures for parametric quantum circuits under NISQ constraints by leveraging data-derived complexity measures. It introduces the Quantum Architecture Recommendation Processor (QARP), which uses 22 dataset metrics to predict the best PQC layer and the required number of layer repetitions, validated across 14 datasets and six circuit layouts. Key findings show that single metrics (notably T2 for layer count and N2 for layer decisions) can yield near-optimal or optimal performance, while combining all metrics does not always improve results. The work demonstrates a data-driven meta-learning approach to quantum circuit design with potential practical impact on automating architecture search for quantum classifiers.

Abstract

Quantum Parametric Circuits are constructed as an alternative to reduce the size of quantum circuits, meaning to decrease the number of quantum gates and, consequently, the depth of these circuits. However, determining the optimal circuit for a given problem remains an open question. Testing various combinations is challenging due to the infinite possibilities. In this work, a quantum circuit recommendation architecture for classification problems is proposed using database complexity measures. A quantum circuit is defined based on a circuit layer and the number of times this layer is iterated. Fourteen databases of varying dimensions and different numbers of classes were used to evaluate six quantum circuits, each with 1, 2, 3, 4, 8, and 16-layer repetitions. Using data complexity measures from the databases, it was possible to identify the optimal circuit capable of solving all problems with up to 100 accuracy. Furthermore, with a mean absolute error of 0.80 2.17, one determined the appropriate number of layer repetitions, allowing for an error margin of up to three additional layers. Sixteen distinct machine learning models were employed for the selection of quantum circuits, alongside twelve classical regressor models to dynamically define the number of layers.

Paper Structure

This paper contains 44 sections, 19 equations, 12 figures, 10 tables.

Table of Contents

  1. Introduction
  2. Complexity database metrics
  3. Feature-based measures
  4. Maximum Fisher’s Discriminant Ratio (F1)
  5. The Directional-vector Maximum Fisher’s Discriminant Ratio (F1v)
  6. Volume of Overlapping Region (F2)
  7. Maximum Individual Feature Efficiency (F3)
  8. Collective Feature Efficiency (F4)
  9. Linearity Measures
  10. Sum of the Error Distance by Linear Programming (L1)
  11. Error Rate of Linear Classifier (L2)
  12. Non-Linearity of a Linear Classifier (L3)
  13. Neighborhood measures
  14. Fraction of Borderline Points (N1)
  15. Ratio of Intra/Extra Class Nearest Neighbor Distance (N2)
  16. ...and 29 more sections

Figures (12)

  • Figure 1: Measurement results in $0$ or $1$ with equal probability $\frac{1}{2}$.
  • Figure 2: An example of quantum circuit with one $\mathbf{CNOT}$ operator and two rotation gates, $\mathbf{Rx}$ and $\mathbf{Rz}$, one of them being controlled (controlled rotation).
  • Figure 3: Layer C-1.
  • Figure 4: Layer C-2.
  • Figure 5: Layer C-3.
  • ...and 7 more figures