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Classification with Quantum Neural Networks on Near Term Processors

Edward Farhi, Hartmut Neven

TL;DR

The paper introduces a quantum neural network (QNN) framework that uses parameterized quantum circuits and a readout qubit to perform supervised binary classification on both classical and quantum data. It provides constructive representation results based on Reed–Muller expansions and demonstrates learning algorithms for subset parity, subset majority, and real-world digit data, including a MNIST-derived task, as well as quantum-state properties. Through simulations, the authors explore gradient-based and batch-learning strategies, analyze fundamental limitations (e.g., parity learning becoming vanishingly hard for large n), and illustrate early practical feasibility on near-term devices. The work lays out a principled, hardware-conscious approach to quantum-assisted learning, emphasizes the potential of quantum data processing, and identifies directions for future hardware-enabled exploration and possible hybrid architectures.

Abstract

We introduce a quantum neural network, QNN, that can represent labeled data, classical or quantum, and be trained by supervised learning. The quantum circuit consists of a sequence of parameter dependent unitary transformations which acts on an input quantum state. For binary classification a single Pauli operator is measured on a designated readout qubit. The measured output is the quantum neural network's predictor of the binary label of the input state. First we look at classifying classical data sets which consist of n-bit strings with binary labels. The input quantum state is an n-bit computational basis state corresponding to a sample string. We show how to design a circuit made from two qubit unitaries that can correctly represent the label of any Boolean function of n bits. For certain label functions the circuit is exponentially long. We introduce parameter dependent unitaries that can be adapted by supervised learning of labeled data. We study an example of real world data consisting of downsampled images of handwritten digits each of which has been labeled as one of two distinct digits. We show through classical simulation that parameters can be found that allow the QNN to learn to correctly distinguish the two data sets. We then discuss presenting the data as quantum superpositions of computational basis states corresponding to different label values. Here we show through simulation that learning is possible. We consider using our QNN to learn the label of a general quantum state. By example we show that this can be done. Our work is exploratory and relies on the classical simulation of small quantum systems. The QNN proposed here was designed with near-term quantum processors in mind. Therefore it will be possible to run this QNN on a near term gate model quantum computer where its power can be explored beyond what can be explored with simulation.

Classification with Quantum Neural Networks on Near Term Processors

TL;DR

The paper introduces a quantum neural network (QNN) framework that uses parameterized quantum circuits and a readout qubit to perform supervised binary classification on both classical and quantum data. It provides constructive representation results based on Reed–Muller expansions and demonstrates learning algorithms for subset parity, subset majority, and real-world digit data, including a MNIST-derived task, as well as quantum-state properties. Through simulations, the authors explore gradient-based and batch-learning strategies, analyze fundamental limitations (e.g., parity learning becoming vanishingly hard for large n), and illustrate early practical feasibility on near-term devices. The work lays out a principled, hardware-conscious approach to quantum-assisted learning, emphasizes the potential of quantum data processing, and identifies directions for future hardware-enabled exploration and possible hybrid architectures.

Abstract

We introduce a quantum neural network, QNN, that can represent labeled data, classical or quantum, and be trained by supervised learning. The quantum circuit consists of a sequence of parameter dependent unitary transformations which acts on an input quantum state. For binary classification a single Pauli operator is measured on a designated readout qubit. The measured output is the quantum neural network's predictor of the binary label of the input state. First we look at classifying classical data sets which consist of n-bit strings with binary labels. The input quantum state is an n-bit computational basis state corresponding to a sample string. We show how to design a circuit made from two qubit unitaries that can correctly represent the label of any Boolean function of n bits. For certain label functions the circuit is exponentially long. We introduce parameter dependent unitaries that can be adapted by supervised learning of labeled data. We study an example of real world data consisting of downsampled images of handwritten digits each of which has been labeled as one of two distinct digits. We show through classical simulation that parameters can be found that allow the QNN to learn to correctly distinguish the two data sets. We then discuss presenting the data as quantum superpositions of computational basis states corresponding to different label values. Here we show through simulation that learning is possible. We consider using our QNN to learn the label of a general quantum state. By example we show that this can be done. Our work is exploratory and relies on the classical simulation of small quantum systems. The QNN proposed here was designed with near-term quantum processors in mind. Therefore it will be possible to run this QNN on a near term gate model quantum computer where its power can be explored beyond what can be explored with simulation.

Paper Structure

This paper contains 10 sections, 47 equations, 1 figure.

Table of Contents

  1. Introduction and Setup
  2. Representation
  3. Representing Subset Parity and Subset Majority
  4. Learning
  5. Learning Subset Parity
  6. Learning Subset Majority
  7. Learning to Distinguish Digits
  8. Classical Data Presented in Superposition
  9. Learning a Property of Quantum States
  10. Conclusions and Outlook

Figures (1)

  • Figure 1: Schematic of the proposed quantum neural network on a quantum processor. The input state $\left|\psi, 1\right\rangle$ is prepared and then transformed via a sequence of few qubit unitaries $U_i(\theta_i)$ that depend on parameters $\theta_i$. These get adjusted during learning such that the measurement of $Y_{n+1}$ on the readout qubit tends to produce the desired label for $\left|\psi\right\rangle$.