Non-Unitary Quantum Machine Learning

Abstract

We introduce several probabilistic quantum algorithms that overcome the normal unitary restrictions in quantum machine learning by leveraging the Linear Combination of Unitaries (LCU) method. Among our investigations are quantum native implementations of Residual Networks (ResNet), where we show that residual connections between layers of a variational ansatz can prevent barren plateaus in models which would otherwise contain them. Secondly, we implement a quantum analogue of average pooling layers from convolutional networks using single qubit controlled basic arithmetic operators and show that the LCU success probability remains stable for the MNIST database. This method can be further generalised to convolutional filters, while using exponentially fewer controlled unitaries than previous approaches. Finally, we propose a general framework for applying a linear combination of irreducible subspace projections on quantum encoded data. This enables a quantum state to remain within an exponentially large space, while selectively amplifying specific subspaces relative to others, alleviating simulability concerns that arise when fully projecting to a polynomially sized subspace. We demonstrate improved classification performance for partially amplified permutation invariant encoded point cloud data when compared to non-invariant or fully permutation invariant encodings. We also demonstrate a novel rotationally invariant encoding for point cloud data via Schur-Weyl duality. These quantum computing frameworks are all constructed using the LCU method, suggesting that further novel quantum machine learning algorithms could be created by utilising the LCU technique.

arXiv comment43 pages, 23 figures

References in corpus (29)

• Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer
• Elementary gates for quantum computation
• Supervised learning with quantum enhanced feature spaces
• Quantum Chemistry in the Age of Quantum Computing
• Quantum Convolutional Neural Networks
• Quantum Networks for Elementary Arithmetic Operations
• Simulating Hamiltonian dynamics with a truncated Taylor series
• Power of data in quantum machine learning
• Efficient Learning for Deep Quantum Neural Networks
• Quantum convolutional neural network for classical data classification
• Hybrid Quantum-Classical Convolutional Neural Networks
• Quantum Image Processing and Its Application to Edge Detection: Theory and Experiment
• Efficient Quantum Circuits for Schur and Clebsch-Gordan Transforms
• Group-Invariant Quantum Machine Learning
• Equivalence of quantum barren plateaus to cost concentration and narrow gorges
• A Lie Algebraic Theory of Barren Plateaus for Deep Parameterized Quantum Circuits
• Hamiltonian Simulation Using Linear Combinations of Unitary Operations
• Theoretical Guarantees for Permutation-Equivariant Quantum Neural Networks
• The Adjoint Is All You Need: Characterizing Barren Plateaus in Quantum Ansätze
• Benchmarking Adversarially Robust Quantum Machine Learning at Scale
• Hybrid quantum ResNet for car classification and its hyperparameter optimization
• ResQNets: A Residual Approach for Mitigating Barren Plateaus in Quantum Neural Networks
• An efficient high dimensional quantum Schur transform
• Provably Trainable Rotationally Equivariant Quantum Machine Learning
• Optimal manipulations with qubits: Universal quantum entanglers
• Efficient classical algorithms for simulating symmetric quantum systems
• Drastic Circuit Depth Reductions with Preserved Adversarial Robustness by Approximate Encoding for Quantum Machine Learning
• Permutation Invariant Encodings for Quantum Machine Learning with Point Cloud Data
• Symmetry breaking in geometric quantum machine learning in the presence of noise

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• Classical Autoencoder Distillation of Quantum Adversarial Manipulations