Data re-uploading for a universal quantum classifier

Adrián Pérez-Salinas; Alba Cervera-Lierta; Elies Gil-Fuster; José I. Latorre

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Data re-uploading for a universal quantum classifier

Adrián Pérez-Salinas, Alba Cervera-Lierta, Elies Gil-Fuster, José I. Latorre

arXiv:1907.02085v3Jul 3, 2019quant-ph

TL;DR

The paper addresses how to perform universal classification with very limited quantum resources, especially the constraints of a single qubit and data loading. It uses repeated data re-uploading with single-qubit processing and reports successful complex-pattern classification, while multi-qubit entanglement improves efficiency. The authors also identify limits in single-qubit quantum advantage and in the scope of their multi-qubit analysis.

Problem
A single qubit has limited representational and processing capacity, while loading classical information into quantum circuits is difficult for higher-dimensional data.
Method
The classifier repeatedly re-uploads classical data through a sequence of data-loading and single-qubit processing units, with multi-qubit variants using alternative measurement strategies.
Results
More than 90% success rate is achieved by the single-qubit classifier across tested pattern types, while additional qubits and entanglement increase success and reduce required layers.
Takeaways & Limitations
Data re-uploading enables a single-qubit classifier to represent multidimensional complex figures and supports a universal-classifier construction.
Takeaways & Limitations
The single-qubit classifier does not provide quantum advantage over classical classification techniques, and the extended multi-qubit performance study is beyond the paper’s scope.

Abstract

from arXiv · show

A single qubit provides sufficient computational capabilities to construct a universal quantum classifier when assisted with a classical subroutine. This fact may be surprising since a single qubit only offers a simple superposition of two states and single-qubit gates only make a rotation in the Bloch sphere. The key ingredient to circumvent these limitations is to allow for multiple data re-uploading. A quantum circuit can then be organized as a series of data re-uploading and single-qubit processing units. Furthermore, both data re-uploading and measurements can accommodate multiple dimensions in the input and several categories in the output, to conform to a universal quantum classifier. The extension of this idea to several qubits enhances the efficiency of the strategy as entanglement expands the superpositions carried along with the classification. Extensive benchmarking on different examples of the single- and multi-qubit quantum classifier validates its ability to describe and classify complex data.

1 Introduction

The paper investigates the minimum quantum resources needed for supervised classification and proposes combining data uploading with processing in small quantum circuits. It argues that repeated data re-uploading lets single-qubit classifiers represent complex functions while supporting broader classification tasks.

Motivation: The work emphasizes that few-qubit algorithms may remain useful as components of larger circuits even without pursuing quantum advantage.This motivates studying quantum classifiers with small numbers of quantum resources.
Motivation: The paper asks what minimum number of qubits, quantum operations, and classically optimized parameters can support general supervised classification.It frames this as a refined resource question rather than estimating quantum cost from classical analogies.
Approach: The proposed approach combines data uploading and information processing repeatedly instead of separating them into distinct circuit stages.Single-qubit rotations are used multiple times along the circuit to generate non-trivial functions of the data.
Approach: Data re-uploading addresses the inability of quantum systems to copy data by introducing classical inputs repeatedly during computation.The paper relates this strategy to the Universal Approximation Theorem and argues that one qubit can reproduce any continuous function in principle with enough re-uploading.
Contribution: The paper focuses on reducing quantum resources for classification, identifying a trade-off between the number of qubits and repeated data re-uploading.Fewer qubits can be used at the price of entering the data several times.
Evaluation: The classifiers are benchmarked on binary, multiclass, multidimensional, and non-convex patterns using parametrized circuits and performance analyses based on circuit architecture.The examples include planar regions, multidimensional patterns, and non-convex figures for single- and multi-qubit classifiers.

2 Structure of a single-qubit quantum classifier

The classifier alternates classical data re-uploading with parametrized single-qubit processing, then classifies patterns from measurements of the final state. Repeated layers increase representation capacity, while input dimensionality can be expanded with linear circuit-complexity growth.

2.1 Re-uploading classical information: A single qubit cannot universally classify higher-dimensional data using one upload and one Bloch-sphere rotation.The limitations are its two degrees of freedom and the restricted separation capability of a single rotation.
2.1 Re-uploading classical information: Repeated re-uploading lets successive processing units combine prior quantum information with the original classical input.This architecture is inspired by neural-network processing across hidden units while respecting the no-cloning constraint.
2.1 Re-uploading classical information: The circuit alternates data-uploading rotations U(x) with parametrized single-qubit processing gates across repeated layers.Each processing layer combines a data upload and a tunable unitary, and the full circuit is trained for classification.
2.2 Processing along re-uploading: The tested encoding introduces data linearly into rotation gates, while nonlinearities arise from the structure and composition of those gates.The authors present linear encoding as a proof of concept and note that it is particularly suited to rotationally symmetric data.
2.2 Processing along re-uploading: The input space can be enlarged by dividing data into three-dimensional vectors, with circuit complexity increasing linearly with input size.A compact layer can combine data and tunable parameters in one rotation, reducing circuit depth by half, although excessive layer combination can lose nonlinearity.
2.3 Measurement: The final classification uses measurements of the output state, with a fidelity-based cost comparing it against class-specific target states.For four classes represented by tetrahedron vertices, the expected fidelity is 1 for the correct class and 1/3 for the others.

3 Universality of the single-qubit classifier

The paper connects repeated single-qubit rotations with the Universal Approximation Theorem to explain how a single-qubit classifier can approximate arbitrary classification functions. Its construction uses trigonometric functions generated by SU(2) layers and classically optimized parameters.

3 Universality of the single-qubit classifier: The authors report evidence that the single-qubit classifier can approximate any classification function up to arbitrary precision.They motivate this claim by relating the classifier to the Universal Approximation Theorem for neural networks.
3.1 Universal Approximation Theorem: The Universal Approximation Theorem states that a single-hidden-layer neural network can approximate continuous functions using a nonconstant, bounded, continuous activation function.The theorem uses weights, biases, and neuron-output coefficients to construct the approximation.
3.2 Universal Quantum Circuit Approximation: Data and trainable parameters are encoded together in the unitary gate, with each layer contributing functions of the form θ_i + w_i ◦ x.The resulting trigonometric functions are nonconstant, bounded, and continuous, matching the relevant approximation-theorem conditions.
3.2 Universal Quantum Circuit Approximation: The classifier composes repeated SU(2) rotations into a single exponential whose coefficient functions depend trigonometrically on the input.Baker-Campbell-Hausdorff correction terms remain proportional to Pauli matrices and can be absorbed into the input-dependent functions.
3.2 Universal Quantum Circuit Approximation: The authors state that additional parameters are needed to map the quantum construction fully onto the Universal Approximation Theorem expression.The cost function can also include class-specific parameters when weighted fidelity is used.
3.2 Universal Quantum Circuit Approximation: In the quantum analogy, circuit weights correspond to neural-network weights, rotation parameters to biases, layers to hidden neurons, and trigonometric functions to activations.This correspondence links repeated parametrized quantum processing with the neural-network approximation construction.

4 From single- to multi-qubit quantum classifier

The paper generalizes the single-qubit classifier to multiple qubits and proposes measurement strategies, circuit architectures, and entanglement structures for multi-qubit classification.

The single-qubit classifier cannot provide quantum advantage over classical artificial neural networks, motivating extensions to multiple qubits and layers.
A multi-qubit classifier generalizes the single-qubit formalism, while entanglement is proposed to reduce the required number of layers.
Measurement strategy and cost function: Two measurement strategies compare the final state with computational-basis states or classify using thresholds on one selected qubit.
Measurement strategy and cost function: The computational-basis measurement strategy becomes unrealizable for many qubits because tomography requires exponentially many measurements.
Quantum circuits examples: Entangling layers use CZ gates between rotations, omit CZ gates from the final layer, and alternate the pairings in four-qubit circuits.
Quantum circuits examples: For fixed layers, two-qubit classifiers require twice and four-qubit classifiers four times the single-qubit parameters; entangling circuits have depth 2N versus N without entanglement.

5 Minimization methods

Training the parametrized quantum classifier is formulated as minimizing a cost function over circuit parameters, whose landscape may contain local minima. The study compares SGD with L-BFGS-B and uses L-BFGS-B for the reported results because it was more accurate and relatively fast, particularly for small training sets.

Parameter minimization: Training minimizes a cost function over the circuit’s angles, weights, and, when applicable, class-specific parameters.For a single-qubit classifier, the parameter count is (3 + d)N, with C additional parameters for a weighted fidelity cost function.
Parameter minimization: The cost-function landscape is generally unknown, so gradient descent is not guaranteed to find a suitable minimum; SGD is the commonly used machine-learning variant.The text notes that local minima are unavoidable in sufficiently large parameter landscapes.
Optimization methods: L-BFGS-B is presented as an alternative minimization method previously used successfully in classical machine learning.
Optimization methods: L-BFGS-B was used for the reported results because it was found to be accurate and relatively fast, with an open-source minimization package serving as the core.The minimizer was treated as a black box with default parameter settings.
Optimization methods: For small training sets, L-BFGS-B was generally better than SGD because it is less sensitive to local minima; SGD can be more useful with many training points because it is faster.The authors used small training sets because of computational constraints.

6 Benchmark of a single- and multi-qubit classifier

The benchmarks test single- and multi-qubit data-reuploading classifiers across binary, multi-class, multidimensional, and non-convex problems. Single-qubit models exceed 90% success, while additional qubits and entanglement can reduce the layers needed.

The benchmarks cover binary, multiple-pattern, multidimensional, and non-convex classification problems using circuits with varied layers, qubits, and entanglement.Training uses random data, independently generated test sets, and several circuit architectures.
6.1 Simple example: classification of a circle: More than 90% success is achieved for the circle problem with a single-qubit classifier using two layers and the weighted fidelity cost function.The two-layer circuit uses 12 parameters; two- and four-qubit classifiers reach 96% with two layers.
6.1 Simple example: classification of a circle: One layer divides the circle-classification plane in half, whereas two layers capture the circular shape and additional layers adjust its radius.The result follows from the nonlinear behavior of the rotational gates.
6.2 Classification of multiple patterns: The four-class 3-circles problem reaches 92% success with a single qubit and 10 layers, while two entangled qubits achieve the same result with four layers.The single-qubit result uses 54 parameters, compared with 34 for the two-qubit entangled classifier.
6.3 Multidimensional classification: Multidimensional data can be uploaded in subsets when its dimension exceeds three, so the input dimension need not be limited by a single qubit’s degrees of freedom.The classifier can upload up to three values per rotation and split higher-dimensional vectors across sublayers.
Across the tested problems, single-qubit performance is at least comparable to classical methods and exceeds them on the 3-circles and binary-annulus problems.The comparison includes a neural network and a support vector classifier.

7 Conclusions

The paper concludes that repeated data re-uploading enables a universal-looking single-qubit classifier, while extending the architecture to multiple qubits introduces entanglement and can improve efficiency. Benchmarks support strong classification performance, with optimization and ansatz scope boundaries remaining.

Data re-uploading is the core mechanism enabling a single-qubit classifier to represent multidimensional complex figures.The paper connects this construction with the Universal Approximation Theorem.
The classifier repeatedly uploads data and processing parameters through one-qubit rotations, whose parameters are optimized with classical minimization.Two cost functions are defined for training.
Extending the classifier to multiple qubits permits entanglement between qubits through two-qubit gates inserted between rotation layers.The paper uses one entangling ansatz as a proof of concept.
Across the benchmark suite, single-qubit classifiers exceed 90% success, while more qubits and entanglement increase success and reduce the required layers.The weighted fidelity cost function generally performs better than the fidelity cost function.
The probability of becoming trapped in local minima increases with the number of layers, and alternative ansatzes are outside the study’s scope.These are stated boundaries of the benchmarked approach.

A The Stochastic Gradient Descent method (SGD)

This section describes stochastic-gradient-based parameter optimization and reports that it was less effective and more computationally demanding than L-BFGS-B for the authors’ experiments.

The proposed derivative algorithm is inspired by stochastic gradient descent and can be interpreted as neural-network back-propagation.It uses intermediate circuit states to compute the gradient exactly.
Full access to the wave function at intermediate computation steps makes this gradient procedure costly to implement experimentally.

A.1 To compute the gradient

The gradient procedure factors each circuit derivative into forward and backward state components, then updates parameters through batch-based optimization rather than point-by-point updates.

A.1 To compute the gradient: Intermediate states are defined forward from the circuit input and backward from the target state to express layer-specific gradients.The layer index identifies where the gradient is computed.
A.1 To compute the gradient: Each layer can contain three rotation angles and three data weights, with additional sublayers for data dimension d > 3.Only the gate associated with the differentiated parameter is affected.
A.1 To compute the gradient: The SU(2) parametrization simplifies derivatives because derivatives of the layer matrix are nearly the matrix itself.Parameter derivatives with respect to weights include the corresponding data component.
A.1 To compute the gradient: Derivative calculations can use a parameter shift by π to measure the relevant gradient component.
A.2 To update parameters: Batched optimization averages gradients over shuffled subsets, improving efficiency over point-by-point updates while remaining statistically equivalent.The described stochastic-gradient setup uses batches of training examples.
A.2 To update parameters: Updating from all examples at once can cancel opposing gradients and stabilize before the optimization should end.
A.2 To update parameters: The stochastic-gradient implementation was less accurate and more computationally expensive than L-BFGS-B, so it was discarded.

B Classification results in other problems

The classifier was evaluated on varied, unbiased classification tasks using datasets of 200 random entries, or 500 for the sphere, and 4000 random test points. Performance generally improves with additional layers before reaching a stationary regime, while more qubits and entanglement can reach that regime with fewer layers.

The extra tasks cover different kinds of training data and test whether the classifier adapts to varied problems.
The datasets contain 200 random entries, except for the sphere with 500, and use 4000 random test points.
Unbiased tasks give blind-classifier success rates of ∼50% for binary, ∼33% for ternary, and ∼25% for quaternary classification.
Performance generally improves as the number of layers increases until reaching a stationary regime.
More qubits and entanglement usually allow the stationary regime to be reached with fewer layers.

Non-convex problem

The non-convex problem tests classification across two mutually non-convex zones whose small regions cannot be left unclassified. The quantum classifier achieves high success rates, including 98% with one qubit and 97% with two entangled qubits, while outperforming SVC in the cited comparison.

The non-convex problem divides a 2D area into two zones separated by a sinusoidal border, with no sufficiently small unclassified area permitted.
98% is achieved by weighted fidelity with 1 qubit, 6 layers, and 32 parameters.
The quantum classifier classifies the non-convex data without major difficulty, whereas SVC performs worse than neural networks.
The binary annulus problem requires connecting disconnected regions, involving a to-and-fro path for the parameters.
97% is achieved by weighted fidelity with 2 non-entangled qubits, 4 layers, and 40 parameters, while fidelity cost reaches 94% with 2 entangled qubits.

Sphere

The classifier was also tested on multidimensional sphere data. For the three-dimensional sphere, weighted fidelity reaches 96% with two or four qubits in two layers, while fidelity cost reaches 93%.

The experiments include multidimensional data, including a four-dimensional hypersphere and a three-dimensional sphere.
93% is achieved with fidelity cost using a single qubit and 10 layers, or two entangled qubits and 6 layers.
96% is achieved with weighted fidelity using two or four qubits in 2 layers, with or without entanglement, requiring 24 or 48 parameters.
The four-quadrant squares problem is constructed to compare quantum classification with neural networks on straight-line separations.
99% is achieved by fidelity cost with two non-entangled qubits, 6 layers, and 60 parameters, exceeding the cited classical success rates of up to 98% and 96%.

Wavy lines

The four-class wavy-lines problem contains small non-convex regions that account for most failures. The classifier reaches 94% with both fidelity cost and weighted fidelity, approximately matching neural networks and outperforming SVC in the reported comparison.

Wavy lines: The four-class problem divides the area into four regions using the borders x2 = sin(πx1) ± x1.
Wavy lines: Some regions are too small for the classifier to detect reliably.
Wavy lines: Most failure points occur in the small non-convex regions shown in Figure 9(d).
Wavy lines: 94% is achieved with fidelity cost using two entangled qubits, 10 layers, and 200 parameters.
Wavy lines: 94% is also achieved with weighted fidelity using four entangled qubits, 4 layers, and 80 parameters.
Wavy lines: The quantum classifier approximately equals neural networks at 94% and outperforms SVC at 82%.