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Towards Quantum Machine Learning with Tensor Networks

William Huggins, Piyush Patel, K. Birgitta Whaley, E. Miles Stoudenmire

arXiv:1803.11537v2quant-phcond-mat.str-elcs.LG

TL;DR

Near-term quantum machine learning needs useful algorithms that operate with limited, noisy qubits on large, high-dimensional data. This paper develops discriminative and generative learning circuits based on tree tensor networks and matrix product states, connecting classical training with quantum optimization. The approach supports qubit-efficient models and demonstrates handwriting recognition with above 95% average test accuracy, while many models retain accuracy close to the noiseless case under tested noise.

  • Problem

    Quantum machine learning needs to perform meaningful tasks on large, high-dimensional datasets despite near-term devices having few noisy qubits.

  • Method

    The paper uses tree tensor network and matrix product state circuits for discriminative and generative learning, with task-specific parameters optimized through a hybrid quantum-classical procedure.

  • Results

    Average test accuracy exceeded 95% for the handwriting-recognition discriminative model, while many models showed only slight accuracy reductions under the tested noise conditions.

  • Takeaways & Limitations

    Classically tractable tensor-network models can initialize more expressive quantum models, and tensor-network architectures can reduce the physical-qubit requirements for processing high-dimensional data.

Abstract

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Machine learning is a promising application of quantum computing, but challenges remain as near-term devices will have a limited number of physical qubits and high error rates. Motivated by the usefulness of tensor networks for machine learning in the classical context, we propose quantum computing approaches to both discriminative and generative learning, with circuits based on tree and matrix product state tensor networks that could have benefits for near-term devices. The result is a unified framework where classical and quantum computing can benefit from the same theoretical and algorithmic developments, and the same model can be trained classically then transferred to the quantum setting for additional optimization. Tensor network circuits can also provide qubit-efficient schemes where, depending on the architecture, the number of physical qubits required scales only logarithmically with, or independently of the input or output data sizes. We demonstrate our proposals with numerical experiments, training a discriminative model to perform handwriting recognition using a optimization procedure that could be carried out on quantum hardware, and testing the noise resilience of the trained model.

I. INTRODUCTION

The paper proposes tensor-network-based quantum circuits for discriminative and generative machine learning on near-term hardware. These circuits aim to address limited qubits and noise while linking classical tensor-network models with quantum optimization.

  • Motivation: Near-term quantum machine learning must handle few noisy qubits while processing large, high-dimensional datasets.The paper identifies datasets with millions of samples and samples containing hundreds or thousands of components as motivating constraints.
  • Proposed framework: The proposed algorithms implement discriminative and generative learning with tree tensor network and matrix product state circuits.These tensor-network architectures were already useful for classical supervised and unsupervised learning.
  • Proposed framework: Only the circuit geometry is fixed initially; unitary-operation parameters are optimized for the specific machine learning task using an auxiliary classical algorithm.This makes the approach conceptually related to variational eigensolver and approximate optimization algorithms.
  • Resource efficiency: Tensor-network models can require physical-qubit counts that scale logarithmically with, or independently of, processed data size.This depends on the tensor-network architecture and whether hardware permits separate measurement of one qubit.
  • Classical–quantum connection: Classically trained tensor-network models can initialize more expressive quantum models, potentially making quantum optimization faster and more likely to succeed.Classical algorithmic improvements can also transfer to the quantum setting.
  • Near-term suitability: Tensor-network circuits may offer inductive biases from local correlations and rigorous noise-resilience bounds, while requiring only product-state preparation, one- and two-qubit gates, and computational-basis measurements.These operations are available on near-term quantum hardware.

II. LEARNING WITH TENSOR NETWORK QUANTUM CIRCUITS

The paper constructs discriminative and generative quantum machine-learning models from tree tensor networks and matrix product states. These architectures map data into quantum states, progressively coarse-grain or entangle qubits, and produce predictions or samples while supporting classical simulation and quantum implementation.

  • Architectures: Tree tensor networks and matrix product states are implemented precisely by quantum circuits and provide increasingly expressive models as bond dimension grows.At sufficiently high bond dimension, quantum implementations can extend beyond classically tractable models.
  • Discriminative Algorithm: Discriminative models map an N-component real vector to an N-qubit product state, then use a tree of parameterized unitaries to coarse-grain it into output qubits.Virtual qubits connect subtrees; some qubits are discarded while others continue to later nodes.
  • Discriminative Algorithm: Tree connectivity can group nearby image pixels, with successive coarse-graining steps alternating between horizontal and vertical directions.This organization is illustrated for a 4x4 image.
  • Discriminative Algorithm: The model predicts the most probable output state by repeatedly evaluating the circuit and estimating the output-qubit distribution.The designated output qubits encode the possible class labels.
  • Architectures: MPS models pass only one set of V qubits after each operation and are suited to data with one-dimensional correlation patterns such as time series, language, or audio.An MPS is described as a maximally unbalanced tree.

B. Generative Algorithm

The generative proposal reverses the discriminative architecture: it prepares and entangles reference-state qubits, then measures them to produce samples. Both tree and MPS variants use tensor-network-specific state-preparation patterns.

  • Generative Algorithm: The generative algorithm prepares a quantum state and measures it in the computational basis, placing it within the family of Born machines.Unlike a completely general state, it uses tree or MPS state-preparation patterns.
  • Generative Algorithm: In the tree version, 2V reference-state qubits are repeatedly entangled with newly prepared qubits through unitary operations across successive layers.The process recursively splits outputs into groups until the desired number of qubits has been processed.
  • Generative Algorithm: Each qubit is measured after all unitaries acting on it have been applied, and the computational-basis outcomes form one sample from the model.Measurements can be performed opportunistically to reduce physical-qubit overhead.
  • Generative Algorithm: An MPS-based generative algorithm provides an alternative architecture for one-dimensional data.The corresponding circuit is shown separately from the tree construction.

III. NUMERICAL EXPERIMENTS

The numerical experiment tests whether a tree tensor-network discriminative circuit can perform supervised image labeling using a near-term-compatible training procedure. The experimental architecture is a low-bond-dimension special case of the proposed model.

  • Numerical Experiments: The experiment trains a tree tensor-network discriminative model for supervised image labeling using operations that could be carried out on a near-term quantum device.The architecture has bond dimension D = 2 because each entangling operation acts on two qubits and only one continues to the next scale.
  • Loss Function: The loss function evaluates the model’s label probabilities and penalizes examples where the true-label probability is insufficiently separated from the most likely incorrect label.The hyperparameters λ and η are selected using validation performance.

B. Optimization

Training uses mini-batch optimization and SPSA, a stochastic procedure that estimates loss gradients from finite differences along random directions. A momentum term mixes previous updates into the current parameter update.

  • Optimization: The training procedure optimizes a stochastic mini-batch estimate of the loss instead of evaluating the full training distribution at every step.Mini-batches are randomly selected from the training data and compared with held-out test data for generalization.
  • Optimization: SPSA estimates the loss gradient through a finite difference along a random direction and updates the circuit parameters accordingly.The method was chosen to emulate optimization feasible on near-term quantum hardware.
  • Optimization: A momentum variable mixes a fraction of previous update steps into the current update during SPSA optimization.The update procedure initializes the parameters randomly and the momentum to zero.
  • Optimization: The optimization schedule randomly partitions training data into mini-batches and uses predefined constants to set step-size and perturbation sequences.The procedure iterates over k from 0 through M.

C. Results

The proposed discriminative tensor-network circuit was tested on MNIST binary digit classification using SPSA optimization, achieving average test accuracy above 95%. Performance varied across digit pairs and depended on hyper-parameter choices.

  • C. Results: The experiment trained tree-network circuits with one output qubit per node to classify 8 × 8 MNIST images from two digit classes.The study evaluated all 45 pairwise classifiers among handwritten digits 0 through 9.
  • C. Results: Test accuracy was plotted against SPSA epochs for binary classification of handwritten 0’s and 7’s.The figure uses M = 30 in the notation of the preceding section.
  • C. Results: The tree-unitary parameters were represented as U = exp(iH), with Hermitian matrices allowed to differ across nodes.This parameterization produced 1008 free parameters for the 8 × 8 image-recognition task.
  • C. Results: Average test accuracy exceeded 95% for the trained pairwise classifiers.The reported hyper-parameter choices included λ = .234, η = 5.59, a = 28.0, b = 33.0, A = 74.1, s = 4.13, t = .658, γ = 0.882, and n = 222.
  • C. Results: The individual pairwise classifiers showed significant performance differences, partly possibly reflecting the difficulty of distinguishing similar digits from 64-pixel images.Different hyper-parameter choices could also significantly change which digit pairs were classified most accurately.

IV. IMPLEMENTATION ON NEAR-TERM DEVICES

The tensor-network circuits are intended for near-term hardware with individual-qubit measurement and reset capabilities. Their proposed advantages include low physical-qubit requirements and potential noise robustness.

  • IV. IMPLEMENTATION ON NEAR-TERM DEVICES: Tensor-network models could be implemented using a small number of physical qubits.The required hardware must permit measurement of individual physical qubits without disturbing the others.
  • IV. IMPLEMENTATION ON NEAR-TERM DEVICES: For tree networks, the required physical-qubit count can scale logarithmically with data dimension and bond dimension.The section then treats matrix product states as a special case with different qubit requirements.
  • IV. IMPLEMENTATION ON NEAR-TERM DEVICES: Noise resilience was investigated by evaluating the trained model with random errors and observing whether it continued to produce useful results.The experiment used the model trained in Section III.

A. Qubit-Efficient Tree Network Models

Tree tensor-network models reduce the physical qubits needed to process many inputs or outputs by measuring and resetting reusable qubits. Their resource requirement scales logarithmically with data size, while larger virtual-qubit numbers increase expressivity and bond dimension.

  • A. Qubit-Efficient Tree Network Models: The virtual-qubit number V counts qubit lines connecting each subtree to higher nodes, and the bond dimension is D = 2^V.At each tree node, two subtrees of V qubits are entangled using a unitary acting on 2V qubits.
  • A. Qubit-Efficient Tree Network Models: Q(N, V) = V lg(2N/V) physical qubits suffice for a discriminative tree model processing N inputs when measurement and reset are available.Without measurement and reset, evaluating the model would require N physical qubits.
  • A. Qubit-Efficient Tree Network Models: The qubit reduction works by measuring and reusing V qubits while retaining the other V entangled qubits, so only V new qubits are needed for each additional 2^V inputs.The construction is illustrated for the discriminative circuit in Fig. 10(a).
  • A. Qubit-Efficient Tree Network Models: Generative tree models require the same physical-qubit count as discriminative models because their unitary pattern is reversed.For V = 2 and N = 16, the generative scheme requires Q = 8 physical qubits.
  • A. Qubit-Efficient Tree Network Models: Q(N, D) ∼ lg(D) lg(N), while increasing V increases expressivity through the bond dimension D = 2^V.The paper notes that V = 16 corresponds to a bond dimension exceeding the largest dimensions used in the cited state-of-the-art classical calculations.

B. Qubit-Efficient Matrix Product Models

Matrix product state circuits are maximally unbalanced tree networks and can process arbitrarily many inputs or outputs with only V + 1 physical qubits when measurement and reset are available.

  • B. Qubit-Efficient Matrix Product Models: An MPS is a maximally unbalanced tree tensor network, suited to one-dimensional correlation patterns without sacrificing expressivity for one-dimensional distributions.Examples include time-series, language, and audio data.
  • B. Qubit-Efficient Matrix Product Models: A discriminative or generative MPS with V virtual qubits can process an arbitrary number of inputs or outputs using V + 1 physical qubits.The circuits implementing this reuse strategy are shown in Fig. 11.
  • B. Qubit-Efficient Matrix Product Models: The discriminative MPS repeatedly measures and resets one qubit to the next input state while entangling all V + 1 qubits.After all inputs are processed, the model output is obtained by sampling one or more physical qubits.
  • B. Qubit-Efficient Matrix Product Models: The generative MPS repeatedly measures one qubit to produce output values, resets it to |0⟩, and reapplies a (V + 1)-qubit unitary.This process continues until N outputs have been generated.
  • B. Qubit-Efficient Matrix Product Models: Mapping the generative circuit to tensor-network notation merges V qubit indices into one index of dimension D = 2^V, yielding a standard MPS diagram.The resulting diagram represents the amplitude for observing a particular set of measured-qubit values.

C. Noise Resilience

The paper evaluates tensor-network classifiers under amplitude-damping and dephasing noise, finding that many retain near-noise-free accuracy and that correctly classified examples remain robust across broad noise ranges.

  • Accuracy under noise: Under T1 = 5 µs, T2 = 7 µs, and a 200 ns gate time, test accuracy is comparable in most pairwise classifiers to training without noise.Successful noisy training required a different set of hyper-parameters.
  • Noise model: The study applies independent amplitude-damping and dephasing channels to pairwise tree-network discriminative models.Noise is composed during each node-contraction step on the left and right child density matrices.
  • Noise model: Tg = 200 ns, T1 = 50 µs, and T2 = 70 µs correspond to pa = 0.004 and pd = 0.003 in the hardware-motivated model.The experiments also consider stronger noise because these realistic values produced almost no observable effects.
  • Evaluation and mitigation: The Kraus-operator formalism directly calculates the labeling qubit’s reduced density matrix under noise without explicitly sampling noise realizations.The reported coherence times are described as achievable on early hardware platforms, and many models show only slightly reduced test accuracy.
  • Evaluation and mitigation: Majority voting over repeated circuit executions can substantially improve classification reliability for images with success probability above .5.The paper gives 500 executions and an individual success probability of .55 as an example yielding almost 99% accuracy.
  • Accuracy under noise: For examples correctly classified in the low-noise limit, success probability remains appreciably above .5 across a wide range of noise levels.Success probabilities saturate at coherence times much shorter than today’s hardware values and drop sharply near T1 ∼1 µs.

V. DISCUSSION

The discussion presents tensor networks as a resource-aware framework for quantum machine learning and connects classical optimization with quantum-model initialization, while noting opportunities for architecture-specific training.

  • Discussion: Tensor networks provide a hierarchy of quantum states whose complexity and resource requirements can be matched to a task.They can also reduce the number of qubits needed for high-dimensional data.
  • Discussion: An optimized, classically tractable tensor network can initialize a more powerful quantum model, avoiding random parameters associated with vanishing-gradient regions.This connects classical tensor-network optimization with subsequent quantum optimization.
  • Discussion: The numerical optimization worked well, but training algorithms specialized to tensor-network architectures could still be devised.The paper suggests separate subtree objectives and use of MPS orthogonality or light-cone properties as examples.
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