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Hierarchical quantum classifiers
Edward Grant, Marcello Benedetti, Shuxiang Cao, Andrew Hallam, Joshua Lockhart, Vid Stojevic, Andrew G. Green, Simone Severini
TL;DR
The paper addresses open questions about the advantages, noise robustness, and suitable architectures of quantum machine-learning classifiers. It evaluates hierarchical TTN and MERA circuits across classical and quantum data, finding that MERA outperforms TTN and that trained models tolerate depolarizing noise and run on ibmqx4.
Problem
The advantages of quantum computation for machine learning, including speedup, noise robustness, and suitable network architectures, remain open questions.
Method
The paper evaluates hierarchical TTN and MERA quantum circuits with multiple unitary parameterizations on classical and quantum classification tasks, using stochastic-gradient-based training.
Results
MERA classifiers outperform TTN classifiers, while the trained models show depolarizing-noise resistance and are successfully deployed on ibmqx4.
Takeaways & Limitations
Hierarchical quantum circuits can classify both classical and quantum data using gate sets implementable on existing quantum computers.
Takeaways & Limitations
The study leaves weight sharing and the role of entanglement in classification circuits for future investigation.
Abstract
from arXiv · showhide
Quantum circuits with hierarchical structure have been used to perform binary classification of classical data encoded in a quantum state. We demonstrate that more expressive circuits in the same family achieve better accuracy and can be used to classify highly entangled quantum states, for which there is no known efficient classical method. We compare performance for several different parameterizations on two classical machine learning datasets, Iris and MNIST, and on a synthetic dataset of quantum states. Finally, we demonstrate that performance is robust to noise and deploy an Iris dataset classifier on the ibmqx4 quantum computer.
1 Introduction
The paper investigates hierarchical quantum circuits for classifying classical and quantum data, asking how architecture, parameterization, noise, and hardware affect performance. It compares TTN and MERA classifiers across classical datasets and synthetic quantum states.
- Hierarchical quantum circuits are motivated by tensor networks, which represent quantum states and circuits while sharing structural similarities with neural networks.
- TTN and MERA circuits are compared with single-qubit, two-qubit, and three-qubit parameterizations, including real and complex forms.
- MERA adds unitary transformations to TTN-like structures to capture a broader range of quantum correlations.
- Experiments are restricted to one-dimensional circuits because large two-dimensional MERA circuits cannot be classically simulated efficiently.
- The study tests binary classification on Iris, MNIST, and synthetic quantum datasets, including both separable qubit encoding and entangled amplitude encoding.
- The experiments examine depolarizing-noise robustness and deploy an Iris classifier on the ibmqx4 quantum computer.
2 Results
The paper defines the classical and quantum datasets used for binary classification and represents each input as a quantum state before circuit processing. Quantum inputs may already be entangled amplitude-encoded states.
- Classical data: Classical binary-classification data consists of input vectors paired with labels in {0, 1}.
- Classical data: Classical vectors must be encoded into quantum states, with qubit encoding using single-qubit rotations for the experiments.
- Quantum data: Quantum datasets consist of unit-length complex state vectors paired with binary labels.
- Quantum data: Quantum data is used as-is when it arrives already in superposition, avoiding a relevant state-preparation cost.
- Classifier setup: Figure 1 depicts 8-qubit TTN and MERA classifiers built from input states, unitary blocks, and a measurement operation.
2.2 Circuit architecture
The classifiers use hierarchical tree-like circuits that progressively reduce the number of qubits before measuring one remaining qubit. MERA extends TTN with additional unitaries that capture correlations across length scales.
- TTN: TTN applies nearest-neighbour two-qubit unitaries, discards one output from each pair, and halves the qubit count at every layer.
- Measurement: The classifier evaluates a single-qubit expectation value after the parameterized quantum circuit acts on the input.
- MERA: MERA retains the TTN unitaries and adds two-qubit unitaries before each TTN layer.
- MERA: MERA’s additional unitaries allow correlations at a particular length scale to be captured within the corresponding network layer.
2.3 Unitary parameterization
The paper compares real and complex unitary parameterizations ranging from hardware-friendly rotation-and-CNOT blocks to general two- and three-qubit gates. Ancillas enable richer nonlinear operations but increase compilation demands.
- Real and complex forms: Real unitaries are restricted to SO(·), whereas complex unitaries are drawn from SU(·).
- Gate parameterizations: The simplest block uses two arbitrary single-qubit rotations followed by a CNOT and is readily implementable on available quantum computers.
- Gate parameterizations: An arbitrary two-qubit gate provides a more general parameterization but requires compilation into hardware-dependent low-level gates.
- Gate parameterizations: A three-qubit gate with an ancilla can implement a rich class of nonlinear functions after the ancilla is traced out.
- Measurement: Binary classification thresholds the expectation value of a chosen Pauli measurement performed on a specific qubit.
2.4 Learning process and complexity
The classifiers minimize mean square error using Adam-based stochastic gradient descent, while circuit evaluation and gradient estimation determine computational cost and introduce sampling or device noise.
- Learning process: The training objective is mean square error between circuit predictions and class labels rather than direct accuracy optimization.The objective is treated as a tractable bound on classification error.
- Learning process: Adam updates circuit parameters using stochastic gradients estimated from small batches, which can help escape local minima.The update uses a learning rate and noisy minibatch gradients.
- Complexity: Circuit evaluation is required to obtain measurement outcomes, and its cost is summarized by the number of scalar multiplications.Table 1 reports this complexity for different hierarchical circuits and encodings.
- Learning process: Exact gradients are available for efficiently contractable networks, whereas non-contractable circuits require finite differences or approximations with additional sampling and device noise.The latter strategies are used when classical contraction is infeasible.
2.5 Experimental results: Iris dataset
The experiments evaluate hierarchical classifiers on Iris and MNIST, using binary tasks and reduced quantum encodings. More expressive MERA and unitary parameterizations generally improve accuracy, while PCA reduction limits MNIST performance.
- Iris dataset: The Iris study converts the three-class dataset into three balanced binary tasks using four-qubit encodings of four flower attributes.Each binary task uses one-third of examples for testing, with results averaged over five initializations.
- Iris dataset: TTN classifiers performed extremely well on all Iris binary classification tasks.Table 2 reports mean test accuracy and one standard deviation across five random initializations.
- MNIST dataset: MNIST evaluation uses four binary digit tasks after reducing each 28 × 28 image to eight principal components for eight-qubit simulation.The tasks distinguish 0 versus 1, 2 versus 7, even versus odd, and greater than 4 versus not.
- MNIST dataset: General unitaries, complex rotations, and MERA classifiers improved accuracy relative to simpler alternatives across the MNIST tasks.Complex rotations helped except on the already above 99.5% ‘0 or 1’ task, and MERA exceeded TTN in all cases.
- MNIST dataset: Hybrid classifiers matched standard MERA accuracy, but required 2.45 times more total training steps and only 0.825 times as many post-training steps.This pattern indicates classical pre-training may reduce training performed on a quantum computer.
- MNIST dataset: 87.09% test accuracy without PCA exceeded 70.7% with PCA for logistic regression on the ‘Is > 4’ task.The result supports the authors’ conclusion that dimensionality reduction can discard information useful for classification.
2.6 Experimental results: Quantum data
The study evaluates hierarchical TTN classifiers on synthetic quantum states generated by random circuit layouts. More expressive TTN parameterizations classify these states better, while classical logistic regression performs no better than random.
- Dataset construction: Maximum bipartite entanglement entropy provided an intractable feature that separated the synthetic classes sufficiently to make the classification tasks meaningful.The authors use this feature only to show that a class-correlated property exists, not as an expected tractable classifier strategy.
- Classifier results: TTNs with general two-qubit gates performed no better than random class assignment across all synthetic tasks.The reported results were averaged over five random initializations.
- Classifier results: Adding an ancilla qubit and using arbitrary three-qubit gates enabled TTNs to classify the quantum states with some accuracy.Accuracy was higher for the 1 or 10 task, whose entropy distributions had less overlap than the other tasks.
- Classical comparison: A classical logistic regression model trained on the amplitude vector achieved close to 50% test accuracy, no better than random.The authors also state that providing the quantum input in classical form is not feasible in practice.
2.7 Experimental results: Characterizing the effect of noise on classification performance
The Iris TTN classifier retains high mean accuracy under moderate depolarizing and finite-sampling noise. Stronger depolarizing noise lowers accuracy and increases variability because class probabilities approach an even split.
- Noise experiment: 96.5% mean accuracy was obtained at λ = 0 with 401 measurements per example, compared with 100% under exact computation.Finite sampling therefore introduced error even without depolarizing noise.
- Noise experiment: Mean accuracy remained above 95% for depolarizing noise up to λ = 0.07.The depolarizing channel was applied after every single-qubit rotation and CNOT gate in the simulated Iris TTN circuit.
- Noise experiment: As depolarizing noise increased beyond λ = 0.07, mean accuracy decreased and the standard deviation increased.The experiment varied λ from 0 to 0.2 in increments of 0.01.
- Interpretation: Greater noise drives the output state toward the maximally mixed state, bringing the two class probabilities closer to 0.5.More measurements would then be needed to estimate the class reliably.
2.8 Experimental results: Deployment on a quantum computer
The authors deployed a trained Iris TTN classifier on IBM’s ibmqx4 quantum computer. Using repeated circuit runs and majority voting, it correctly classified every example in the 34-example test set.
- Hardware deployment: The deployed TTN circuit contained three CNOT gates and seven rotations in the Y direction.The classifier was trained classically before deployment.
- Hardware deployment: 100% of the 34 unseen Iris test examples were correctly classified on the ibmqx4 quantum computer.Each example was evaluated using 401 circuit runs, with the most likely class selected from the samples.
- Hardware deployment: The hardware evaluation achieved a test cost function value of 0.0811.The cost was computed from the sampled class predictions.
3 Discussion
The discussion positions hierarchical quantum circuits as a route for combining tensor-network structure with quantum computation for classification. It emphasizes more expressive MERA circuits, quantum-data classification, hardware-compatible gates, and successful real-device deployment.
- Discussion: The work explores hierarchical quantum circuits for supervised classification of both classical and quantum data.Tensor networks are presented as a natural connection between neural-network structure and quantum circuits.
- Discussion: MERA circuits are studied as more complex hierarchical networks than TTNs, and their superiority relative to simpler networks is demonstrated.The paper frames this comparison within broader efforts to bring machine-learning capabilities to quantum computation.
- Discussion: The approach uses networks constructed from simple two-qubit gates that can be natively implemented on available hardware.This hardware-oriented design supports the paper’s quantum-computing context rather than relying only on universal idealized computers.
- Discussion: A trained tensor network was successfully deployed on the ibmqx4 quantum device.This provides a real-device demonstration alongside the paper’s classical and simulated experiments.
4 Conclusion
The report finds that hierarchical quantum classifiers can classify classical and quantum data, with MERA outperforming TTN and showing resistance to depolarizing noise. It also identifies unresolved architectural, scaling, training-noise, and weight-sharing questions, while highlighting quantum-classification settings that may be classically intractable.
- Conclusion: MERA circuits outperform simpler TTN circuits, use a gate set compatible with existing quantum computers, resist depolarizing noise, and were deployed on ibmqx4.The report presents MERA as the more expressive hierarchical circuit family and demonstrates hardware deployment for an Iris classifier.
- Limitations and future directions: The ideal architecture remains unclear because TTN circuits may trade off entanglement needed to share correlations against optimization difficulty, especially at larger input sizes.The report calls for further examination of entanglement's role in classification circuits.
- Limitations and future directions: The study did not simulate circuit noise during training, although it showed high resistance to depolarizing noise during prediction.The authors identify additional regularization methods and training-time noise as future considerations.
- Limitations and future directions: The role of weight sharing in hierarchical quantum classifiers remains an open question, despite its use for translation and scale invariance in related architectures.The report describes restricting unitaries within or between layers as possible ways to enforce these invariances.
- Significance: Entangled quantum inputs may make classical classification require expensive tomography and become intractable as system size grows, motivating quantum methods for quantum data.The report identifies this as one of two settings where classical simulation is thought to be exponentially harder than quantum computation.