Circuit-centric quantum classifiers

TL;DR

This work introduces a circuit-centric quantum classifier designed for near-term quantum devices, leveraging amplitude-encoded inputs and low-depth, variational circuits to achieve poly-logarithmic parameter growth in input size. The model employs a hybrid quantum-classical training loop where gradients are estimated via classical linear combinations of unitaries, enabling efficient optimization despite limited quantum coherence. Through simulations on standard benchmarks, the QC demonstrates competitive performance with far fewer trainable parameters than classical counterparts, and the authors show resilience to state-preparation and parameter noise, along with dropout-based regularization. The paper also provides a neural-network–style graphical interpretation of quantum gates and discusses architectural optimisations and challenges, offering a roadmap for practical quantum-assisted learning on near-term hardware.

Abstract

The current generation of quantum computing technologies call for quantum algorithms that require a limited number of qubits and quantum gates, and which are robust against errors. A suitable design approach are variational circuits where the parameters of gates are learnt, an approach that is particularly fruitful for applications in machine learning. In this paper, we propose a low-depth variational quantum algorithm for supervised learning. The input feature vectors are encoded into the amplitudes of a quantum system, and a quantum circuit of parametrised single and two-qubit gates together with a single-qubit measurement is used to classify the inputs. This circuit architecture ensures that the number of learnable parameters is poly-logarithmic in the input dimension. We propose a quantum-classical training scheme where the analytical gradients of the model can be estimated by running several slightly adapted versions of the variational circuit. We show with simulations that the circuit-centric quantum classifier performs well on standard classical benchmark datasets while requiring dramatically fewer parameters than other methods. We also evaluate sensitivity of the classification to state preparation and parameter noise, introduce a quantum version of dropout regularisation and provide a graphical representation of quantum gates as highly symmetric linear layers of a neural network.

Figures (9)

• Figure 1: Idea of the circuit-centric quantum classifier. Inference with the model $f(x,\theta) = y$ is executed by a quantum device (the quantum processing unit or QPU) which consists of a state preparation circuit$S_x$ encoding the input $x$ into the amplitudes of a quantum system, a model circuit$U_{\theta}$, and a single qubit measurement. The measurement retrieves the probability of the model predicting $0$ or $1$, from which in turn the binary prediction can be inferred. The classification circuit parameters $\theta$ are learnable and can be trained by a variational scheme.
• Figure 2: Supervised binary classification for $2$-dimensional inputs. Given the red circle and blue triangle data points belonging to two different classes, guess the class of the new input (pink square).
• Figure 3: Inference with the circuit-centric quantum classifier consists of four steps, here displayed in four colours, and can be viewed from three different perspectives, i.e. from a formal mathematical framework, a quantum circuit framework and a graphical neural network framework. In the first step, the feature map from the input space to the feature space $\mathbb{R}^N \rightarrow \mathbb{R}^K$ is executed for an input by a state preparation scheme. The quantum circuit applies a unitary transformation to the feature vector which can be understood as one linear layer (or, when decomposed into gates, several linear layers) of a neural network. The measurement statistics of the first qubit are interpreted as the continuous output of the classifier and effectively implement a weightless nonlinear layer in which every component of the last half of all units is mapped by an absolute square and summed up. The postprocessing stage binarises the result with a thresholding function via classical computing.
• Figure 4: Generic model circuit architecture for $8$ qubits. The circuit consists of two 'code blocks' $B_1$ and $B_3$ with a range of controls of $r = 1$ and $r=3$ respectively. The circuit consists of $17$ trainable single-qubit gates $G = G(\alpha, \beta, \gamma)$, as well as $16$ trainable controlled single qubit gates $C(G)$, which have in turn to be decomposed into the elementary constant gate set used by the quantum computer on which to implement it. If the optimisation methods are used to reduce the controlled gates to a single parameter, we have $3\cdot 33 + 1 = 100$ parameters to learn in total for this model circuit. These $100$ parameters are used to classify inputs of $2^8 = 256$ dimensions, which shows that the circuit-centric classifier is a much more compact model than a conventional feed-forward neural network.
• Figure 5: Illustration of first step of the proof from Observation \ref{['Obs:optimisation']} for an example of the first $5$ gates of a codeblock of $3$ qubits with range $r=1$. Decomposing the controlled rotations and merging single qubit gates reduces the number of parameters needed to represent the model circuit architecture. For simplification the gates are displayed without indices or parameters.

Theorems & Definitions (2)

• proof
• proof