Brief Reviews on Supervised Learning with Quantum-enhenced Feature Spaces

This review is written by non-expert and used for personal study only. This post content is continuously improving.

The complexity associated with quantum computing can be effectively managed by deploying short-depth circuits. Notably, these circuits have demonstrated compatibility with noisy devices while effectively reducing decoherence through error-mitigation techniques. However, the challenge lies in simulating these simple circuits using classical computational methods.

To overcome this, the focus has shifted towards supervised learning and developing classifiers, specifically designed for quantum computing.

Classically, problems in this space are tackled using support vector machines (SVMs). However, proposals have now emerged for a quantum version of SVMs, designed to handle data in a coherent superposition.

By embracing a quantum perspective, two binary classifiers have been put forward that can process data supplied in a classical format. Uniquely, these classifiers utilize the quantum state space as a feature space, achieving remarkable results. A proof of concept implementation on a superconducting quantum processor was able to classify data with a remarkable 100% success ratio.

For conventional SVMs, efficiency in training and classification is attained when the inner products between feature vectors can be effectively evaluated. However, quantum circuits do not hold a quantum advantage over conventional SVMs if the feature vector kernel can be efficiently computed on a classical computer.

To gain an edge over classical approaches, there is a pressing need for a feature map grounded in circuits that pose significant challenges to classical simulation.

A circuit exhibiting excellent performance in trials and not overly complex is employed to define a feature map on n-qubits. The circuit is activated on the initial state, utilizing the coefficients to encode the data.

While it is #P-hard to evaluate the inner product between two states originating from a similar circuit, simulation of a single-layer preparation circuit can be efficiently achieved classically.

The quantum variational classification protocol consists of four primary steps: first, mapping data to a quantum state; second, applying a short-depth quantum circuit to the feature state; third, performing a binary measurement for classification; and finally, making a decision based on repeated measurements.

The optimization is executed by defining a cost function that is predicated on the error probability of incorrectly assigning a label from the empirical distribution. This empirical risk is calculated by averaging the error probability across the full training set.

The entire process entails two phases: training the classifier and optimizing the parameters, followed by employing the classifier to assign labels to unlabelled data.

The successful implementation of the quantum variational classifier on a superconducting quantum processor was achieved for different depths, with the anticipation of higher classification success correlating with increased depth. The binary measurement was derived from the parity function, $f = Z_1Z_2$, and the data was generated from three different random unitaries for each depth.

Furthermore, the optimization of the empirical risk was displayed for various training sets and depths, applying an error mitigation technique based on zero-noise extrapolation.

The quantum state space plays a vital role as it can be interpreted as a feature space with vectors and inner products. The trace signifies the Hilbert-Schmidt inner product between the normal vector and the mapped datum.

This protocol is remarkable as it employs a quantum computer twice: first to estimate the kernel for all pairs of training samples, and then to estimate the kernel for a new datum in the classification phase. The convergence of this method and the classification results have been successfully demonstrated, leading to the successful classification and convergence of the cost function.

The study underlines the significance of feature maps that are hard to estimate using classical methods. This difficulty is a key component in establishing a quantum advantage and hints at the possibility of extending the technique beyond binary classification.

Reference

1. Havlíček, V. et al. Supervised learning with quantum-enhanced feature spaces. Nature 567, 209–212 (2019).