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quantum-kittens
GitHub Repository: quantum-kittens/platypus
 Path: blob/main/notebooks/summer-school/2021/lec6.1.ipynb
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Kernel: Python 3

From Variational Classifiers to Linear Classifiers

The lecture is divided into two broad parts:

  • Part-1: Linear classifiers

    • Introduces the classical concepts of linear classifier, the two formulations of its optimisation function, along with the widely used Kernel trick.

  • Part-2: Quantum Feature Maps

    • Extends the idea of the classical kernel trick to a quantum version, first explaining the concept of quantum feature maps and applying them in introducing a model called Variational Quantum Classifier (VQC). Thereafter, linking in spirit with part-1, Bryce presents a proof that the VQC is indeed a linear classifier, implying that the kernel trick is valid for use in a VQC; finally presenting the quantum kernel estimator (QKE) in the lecture’s conclusion.

  • Download the lecturer's notes here

FAQ

Is VQC a mere variation of the previously seen VQE (variational quantum estimator)? No. VQE seeks to search for the lowest energy state of a hamiltonian that encodes a problem, whereas, in VQC the data to be classified is first encoded via a quantum feature map in a state, and thereafter an iterative optimisation is carried out on a parameterized hamiltonian. The optimal hamiltonian is what we seek in VQC.
What are Lagrange Multipliers? Lagrange multipliers appear in the mathematical optimisation method of Lagrange multipliers, which is used to find the local maxima and minima of a function subject to certain constraint equations. A detailed explained is deferred to the notes by Andrew Ng referred to above.
What about the quantum advantage of a VQC? The quantum advantage of a VQC comes in play only when using a quantum feature map that is classically hard to compute, hence rendering a computation of the kernel function a classically tough task but an easy-to-do task on a quantum computer.
Examples of a classically hard to compute kernel that is easier to do on a quantum computer? The DLog i.e. the discrete log kernel, however it’s not of much “practical” use/applications.

Suggested reading

  • CS220 Lecture Notes Part V: Support Vector Machines (SVM) by Andrew Ng

  • Supervised Learning with Quantum Enhanced Feature Spaces by Vojtech Havlicek et al