Machine learning of high dimensional data on a noisy quantum processor

Evan Peters; João Caldeira; Alan Ho; Stefan Leichenauer; Masoud Mohseni; Hartmut Neven; Panagiotis Spentzouris; Doug Strain; Gabriel N. Perdue

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Machine learning of high dimensional data on a noisy quantum processor

Evan Peters, João Caldeira, Alan Ho, Stefan Leichenauer, Masoud Mohseni, Hartmut Neven, Panagiotis Spentzouris, Doug Strain, Gabriel N. Perdue

arXiv:2101.09581v1Jan 23, 2021quant-ph

TL;DR

The paper addresses whether quantum kernel methods can process real, high-dimensional classical data without dimensionality reduction. It develops hardware-compatible quantum circuits and demonstrates competitive supernova classification on noisy Sycamore hardware, while showing that shot statistics and kernel magnitudes require careful control.

Problem
Quantum machine learning's contribution to real-world classical datasets remains uncertain, particularly for high-dimensional data that may produce statistically unreliable, vanishing kernel elements.
Method
The authors encode 67-dimensional supernova data with a quantum kernel circuit using up to 17 nearest-neighbor-connected hardware qubits, then train a classical support vector machine with error mitigation and readout correction.
Results
Classification accuracy was competitive with noiseless simulation despite hardware noise and no quantum error correction, while remaining reasonable when observed bitstring probabilities were suppressed by 50%-70%.
Takeaways & Limitations
Quantum kernel machine learning can operate on natural, high-dimensional data without dimensionality reduction or significant classical preprocessing, motivating further study of noisy kernels.
Takeaways & Limitations
Performance evaluation requires careful attention to finite shot statistics and kernel-element magnitudes, and the experiments used orders of magnitude fewer repetitions than needed for robust classification.

Abstract

from arXiv · show

We present a quantum kernel method for high-dimensional data analysis using Google's universal quantum processor, Sycamore. This method is successfully applied to the cosmological benchmark of supernova classification using real spectral features with no dimensionality reduction and without vanishing kernel elements. Instead of using a synthetic dataset of low dimension or pre-processing the data with a classical machine learning algorithm to reduce the data dimension, this experiment demonstrates that machine learning with real, high dimensional data is possible using a quantum processor; but it requires careful attention to shot statistics and mean kernel element size when constructing a circuit ansatz. Our experiment utilizes 17 qubits to classify 67 dimensional data - significantly higher dimensionality than the largest prior quantum kernel experiments - resulting in classification accuracy that is competitive with noiseless simulation and comparable classical techniques.

I. INTRODUCTION

Quantum kernel methods offer a route to use quantum processors for machine learning, but prior experiments largely used artificial or heavily preprocessed data. This work extends the approach to high-dimensional supernova data on up to 17 hardware qubits.

Prior quantum-kernel experiments focused on artificial or heavily pre-processed data and hardware implementations involving few qubits.
The method prepares a kernel matrix on a quantum processor for a classical support vector machine to learn patterns in 67-dimensional supernova data.
The experiment extends quantum-kernel machine learning to 17 hardware qubits using only nearest-neighbor connectivity.
The implementation uses Google’s Sycamore processor, with 23 active qubits in the accessed machine.
Results are competitive on a nontrivial classical dataset, with classifier robustness under moderate circuit-fidelity loss.

II. QUANTUM KERNEL SUPPORT VECTOR MACHINES

The quantum-kernel SVM replaces classical input-space inner products with quantum feature-space similarities, then uses the resulting kernel matrix in a classical SVM pipeline. The processor estimates kernel entries by sampling circuit outputs, while noise and readout error can alter those estimates.

An SVM uses a symmetric positive definite kernel to replace input-space inner products and enable nonlinear decision boundaries in the original input space.
Quantum kernel methods map classical inputs into a high-dimensional complex Hilbert space and use the resulting kernel for classification.
The quantum processor estimates each kernel entry by sampling U†(xi)U(xj) and computing the empirical probability of the all-zeros bitstring.
After estimating the kernel matrix, a classical SVM is trained on the quantum outputs and evaluated on test data using additional kernel evaluations.
Device noise and readout error can make the sampled kernel matrix differ substantially from the ideal kernel matrix.

A. Data and preprocessing

The experiment uses simulated PLAsTiCC astronomical time series transformed into fixed spectral features for binary classification. Inputs are scaled for circuit preparation without dimensionality reduction, using preprocessing consistent with competitive classical methods.

The dataset comes from the PLAsTiCC challenge and contains simulated astronomical time series across six wavelength bands.
Variable-length time series are converted into fixed derived quantities, including statistics of flux, flux error, and their ratios.
Lognormal-distributed spectral inputs are converted to log scale and normalized before entering the quantum circuit.
No dimensionality reduction is performed, and the data-processing pipeline follows treatment used by state-of-the-art classical methods.

B. Circuit design

The circuit encodes individual input features into local rotations and is designed to avoid small kernel values that would amplify sampling uncertainty. Its performance is evaluated against an RBF-kernel SVM using noiseless 17-qubit simulations.

Circuit design: Kernel estimation requires repeated circuit executions, and relative statistical error grows as kernel magnitude or repetition count decreases.
Circuit design: The design balances the need for large kernel elements against the need for a mapping that is difficult to compute classically.
Circuit design: The encoding uses a Hadamard followed by three data-parameterized single-qubit rotations and hardware-native iSWAP entanglers.
Circuit design: The circuit maps input features into quantum states using local rotations parameterized by individual data elements and entangling operations.
Circuit design: The noiseless 17-qubit classifier is compared with an RBF-kernel SVM through learning curves based on stratified 10-fold splits and repeated downsampling.
Circuit design: The circuit consistently produces large-magnitude inner products, with median K ≥ 10^-1 and little statistical error in the estimates.

A. Dataset selection

The study balances SVM training cost against generalization by selecting representative train/test subsets for 67-dimensional supernova data. Simulations show the 17-qubit classifier is competitive with a classical RBF-kernel SVM on identical subsets.

O(m^2) training complexity motivates minimizing the SVM training set while preserving generalization on the reserved test set.
Learning curves simulated varying train-set sizes to explore the tradeoff between computational cost and test-set prediction quality.
Hardware train/test sets were selected from simulated 1000 × 1000 kernels using repeated 4-fold cross-validation on 280-point subsets.

B. Hardware classification and Postprocessing

The experiment evaluates quantum-kernel SVMs on Sycamore hardware using 10, 14, and 17 qubits for 67-dimensional supernova data. Across qubit counts, performance remains reasonable despite 50%-70% suppression of observed kernel values from limited circuit fidelity.

The Sycamore processor used 23 superconducting qubits with nearest-neighbor connectivity and native iSWAP entangling gates.
10-, 14-, and 17-qubit circuits classified 67-dimensional supernova data using 210 training and 70 test examples with 5000 repetitions per circuit.
The SVM penalty hyperparameter C was selected by leave-one-out cross-validation before evaluating reserved-test accuracy.
Hardware QKM test accuracy was competitive with noiseless simulations across qubit counts, while hardware noise reduced overfitting.
50%-70% suppression of observed bitstring probabilities and kernel elements still yielded reasonably good QKM classification performance.

IV. CONCLUSION AND OUTLOOK

The work demonstrates quantum-kernel classification of real, high-dimensional data on noisy intermediate-scale hardware without dimensionality reduction, while emphasizing that shot statistics and kernel magnitudes constrain practical evaluation. It establishes competitive classification performance but leaves quantum advantage and suitable beyond-classical datasets as open questions.

Conclusion and outlook: 10–17 qubits support quantum machine learning on natural datasets using a superconducting quantum computer, including high-dimensional data without dimensionality reduction.The demonstrated circuits process real scientific data despite hardware noise and the absence of quantum error correction.
Conclusion and outlook: Competitive classification results were achieved despite hardware noise and lack of quantum error correction, matching noiseless simulation within the reported scope.The circuits are not presented as candidates for demonstrating quantum advantage.
Conclusion and outlook: Careful control of shot statistics and kernel-element magnitudes is necessary when evaluating quantum-kernel performance.Finite-shot uncertainty becomes problematic as sampled kernel magnitudes shrink, motivating further work on noisy kernels.
Conclusion and outlook: Identifying a natural dataset that could yield beyond-classical quantum-machine-learning performance remains the main open problem.The proposed candidates involve correlations difficult or inefficient to represent, store, learn, or infer classically.
Statistical uncertainty and vanishing kernels: Vanishing kernel elements make realistic hardware experiments infeasible in parts of the Type 1 parameter space, especially as qubit count or encoding parameters increase.Type 1 circuits require PCA compression to match input dimension to qubit count, whereas the Type 2 experiments showed no such trend for 67-dimensional inputs.
Circuit design: Type 2 circuits were designed to encode high-dimensional inputs without dimensionality reduction while maintaining large-magnitude kernels through an approximately orthogonal parameterization.The encoding uses hardware-efficient layers with local entanglers and maps data into a subspace of C2^n.

Appendix C: Dataset selection and preprocessing

The preprocessing pipeline transformed the 67-feature PLAsTiCC data, robustly scaled inputs to reduce outlier effects, and tuned rotation parameters. Automated qubit selection was also constrained by Sycamore connectivity and diminishing performance gains near 19 qubits.

The dataset comprised 67 engineered floating-point features with binary labels from the PLAsTiCC challenge.
Logscale transformation addressed lognormal feature distributions, while absolute-value processing caused approximately 4% information loss for median flux entries.
Robust scaling reduced the influence of large outliers using percentile-based bounds equivalent to a (0.01, 0.99) quantile range.
Qubit selection used Sycamore-grid entangler patterns and T1 annotations, with no more than 19 assignable qubits and diminishing effects as n →19.
A multiplicative factor c1 was tuned after preprocessing to adjust the circuit’s rotation parameters.

Appendix D: Error mitigation

Error mitigation combined calibration-based readout correction with automated selection of qubits and entangling edges. The selection objective normalized heterogeneous calibration data and weighted single-qubit and two-qubit performance metrics.

Readout error probabilities p00 and p11 supported post-processing error correction, while other calibration metrics informed automated qubit selection.
A graph traversal algorithm selected qubits using calibration diagnostics to optimize an objective over vertices and edges forming an Eulerian graph.
The optimization required k selected vertices and k −1 selected edges, after calibration values were normalized to [0, 1] and error metrics were inverted.
The scoring function applied logarithmic transforms to T1, T2, and fXEB,2q metrics, but linear transforms to p00 and p11.

3. Readout error correction

The paper corrects readout errors in quantum kernel methods by modeling bit flips, constructing a response matrix, and applying a truncated pseudoinverse to measured frequencies. Truncation makes correction practical, but introduces bias and depends on assumptions about error independence and calibration stability.

Response-matrix construction: Readout error is modeled as stochastic bit flips, with an n-qubit response matrix built from single-qubit transition probabilities.The matrix element gives the probability of transitioning between bitstrings as a product of per-qubit flip probabilities.
Response-matrix construction: Independent bit-flip channels are assumed, although the technique can also use experimentally measured likelihoods for dependent errors.
Truncated correction: Correction uses only the low-Hamming-weight subspace, applying a truncated response-matrix pseudoinverse to measured readout frequencies.Because the kernel depends on the all-zero bitstring frequency, the full 2^n-dimensional response matrix is unnecessary.
Truncated correction: The probability of many simultaneous readout errors is exponentially suppressed, supporting truncation of the response matrix.A Chernoff-bound analysis motivates measuring the probability outside the retained Hamming-weight subspace.
Empirical behavior: For the 10-qubit circuit, kmax = 1 provides significant correction, while larger truncation weights yield diminishing returns.
Implications: Readout error can appear approximately linear in the kernel, so its impact on the classifier depends on deviations from a linear transformation rather than error magnitude alone.

Appendix E: Hardware error and performance

Hardware noise does not necessarily prevent useful quantum-kernel classification because SVM performance can remain comparable to noiseless results despite low circuit fidelity. However, reliable evaluation requires sufficient sampling, careful hyperparameter validation, and attention to readout-correction and calibration effects.

Hardware noise and classifier performance: Hardware kernel elements are strongly biased toward zero by decoherence, while SVM scale invariance can preserve performance under approximately uniform rescaling.
Hardware noise and classifier performance: Comparable test accuracy to noiseless simulation was achieved for n = 14 qubits despite circuit fidelity near 30%.SVM performance is affected by how sampled kernel elements differ from a linear transformation of exact elements, not by fidelity alone.
Hardware noise and classifier performance: The mean all-zero kernel element acts as a proxy for circuit fidelity and tends toward 30% for the investigated qubit counts.
Readout correction: Readout-correction bounds become increasingly loose as the measured kernel approaches typical readout-error probabilities, motivating kernels with larger elements.
Readout correction: Readout correction did not consistently improve classifier performance because validation-selected hyperparameters did not reliably predict improved test accuracy.
Shot statistics: Beyond R = 5000 circuit repetitions, simulated validation accuracy shows diminishing returns, although about 50,000 repetitions may be needed to approach noiseless accuracy.
Model selection: Hardware SVM models can overfit less but exhibit pathological hyperparameter behavior, requiring validation of the penalty parameter C.