Group-Invariant Quantum Machine Learning

Abstract

Quantum Machine Learning (QML) models are aimed at learning from data encoded in quantum states. Recently, it has been shown that models with little to no inductive biases (i.e., with no assumptions about the problem embedded in the model) are likely to have trainability and generalization issues, especially for large problem sizes. As such, it is fundamental to develop schemes that encode as much information as available about the problem at hand. In this work we present a simple, yet powerful, framework where the underlying invariances in the data are used to build QML models that, by construction, respect those symmetries. These so-called group-invariant models produce outputs that remain invariant under the action of any element of the symmetry group $\mathfrak{G}$ associated to the dataset. We present theoretical results underpinning the design of $\mathfrak{G}$-invariant models, and exemplify their application through several paradigmatic QML classification tasks including cases when $\mathfrak{G}$ is a continuous Lie group and also when it is a discrete symmetry group. Notably, our framework allows us to recover, in an elegant way, several well known algorithms for the literature, as well as to discover new ones. Taken together, we expect that our results will help pave the way towards a more geometric and group-theoretic approach to QML model design.

19 + 10 pages, 9 + 1 figures, updated to published version

References in corpus (2)

• Generalized concurrence measure for faithful quantification of multiparticle pure state entanglement using Lagrange's identity and wedge product
• Discriminating Non-Isomorphic Graphs with an Experimental Quantum Annealer

Cited by in corpus (86)

• Challenges and Opportunities in Quantum Machine Learning
• Barren Plateaus in Variational Quantum Computing
• Quantum Computing for High-Energy Physics: State of the Art and Challenges. Summary of the QC4HEP Working Group
• Introduction to Haar Measure Tools in Quantum Information: A Beginner's Tutorial
• Theory for Equivariant Quantum Neural Networks
• Theoretical Guarantees for Permutation-Equivariant Quantum Neural Networks
• Does provable absence of barren plateaus imply classical simulability?
• Understanding quantum machine learning also requires rethinking generalization
• On the practical usefulness of the Hardware Efficient Ansatz
• The Adjoint Is All You Need: Characterizing Barren Plateaus in Quantum Ansätze
• Generalization despite overfitting in quantum machine learning models
• Building spatial symmetries into parameterized quantum circuits for faster training
• Reflection Equivariant Quantum Neural Networks for Enhanced Image Classification
• Trainability barriers and opportunities in quantum generative modeling
• Provably Trainable Rotationally Equivariant Quantum Machine Learning
• Model-Independent Learning of Quantum Phases of Matter with Quantum Convolutional Neural Networks
• Quantum Feature Maps for Graph Machine Learning on a Neutral Atom Quantum Processor
• Quantum Deep Hedging
• Efficient classical algorithms for simulating symmetric quantum systems
• Quantum Mixed State Compiling
• Quantum Phase Processing and its Applications in Estimating Phase and Entropies
• Hierarchical quantum circuit representations for neural architecture search
• Generalization of Quantum Machine Learning Models Using Quantum Fisher Information Metric
• Numerical evidence against advantage with quantum fidelity kernels on classical data
• Quantum Convolutional Neural Networks are Effectively Classically Simulable
• Lie-algebraic classical simulations for quantum computing
• Potential and limitations of random Fourier features for dequantizing quantum machine learning
• Neural Quantum Embedding: Pushing the Limits of Quantum Supervised Learning
• The advantage of quantum control in many-body Hamiltonian learning
• Quantum neural networks form Gaussian processes
• Symmetry breaking in geometric quantum machine learning in the presence of noise
• Quantum algorithms for scientific computing
• Approximately Equivariant Quantum Neural Network for $p4m$ Group Symmetries in Images
• Trainability and Expressivity of Hamming-Weight Preserving Quantum Circuits for Machine Learning
• What can we learn from quantum convolutional neural networks?
• On the universality of $S_n$-equivariant $k$-body gates
• Symmetric Tensor Networks for Generative Modeling and Constrained Combinatorial Optimization
• Neural quantum kernels: training quantum kernels with quantum neural networks
• Non-Unitary Quantum Machine Learning
• Inferring physical properties of symmetric states from the fewest copies
• Reinforcement learning-based architecture search for quantum machine learning
• The role of data embedding in equivariant quantum convolutional neural networks
• Enabling High Performance Debugging for Variational Quantum Algorithms using Compressed Sensing
• Symmetry-invariant quantum machine learning force fields
• Splitting and Parallelizing of Quantum Convolutional Neural Networks for Learning Translationally Symmetric Data
• Characterization of variational quantum algorithms using free fermions
• Unitary Designs of Symmetric Local Random Circuits
• Barren plateaus are swamped with traps
• Image Classification with Rotation-Invariant Variational Quantum Circuits
• Toward Super-polynomial Quantum Speedup of Equivariant Quantum Algorithms with SU($d$) Symmetry
• Quantum Algorithms for Geologic Fracture Networks
• Analyzing the quantum approximate optimization algorithm: ansätze, symmetries, and Lie algebras
• Framework for Learning and Control in the Classical and Quantum Domains
• Geometric Quantum Machine Learning with Horizontal Quantum Gates
• Characterizing randomness in parameterized quantum circuits through expressibility and average entanglement
• Linear programming with unitary-equivariant constraints
• All you need is spin: SU(2) equivariant variational quantum circuits based on spin networks
• Trade-off between Gradient Measurement Efficiency and Expressivity in Deep Quantum Neural Networks
• High-fidelity dimer excitations using quantum hardware
• Efficient quantum algorithms for testing symmetries of open quantum systems
• Quantum subspace expansion approach for simulating dynamical response functions of Kitaev spin liquids
• Ground state-based quantum feature maps
• Permutation-equivariant quantum convolutional neural networks
• On the role of symmetry and geometry in global quantum sensing
• Imaginary Hamiltonian variational ansatz for combinatorial optimization problems
• Improving the efficiency of learning-based error mitigation
• Parallel-in-time quantum simulation via Page and Wootters quantum time
• Quantum enhanced stratification of Breast Cancer: exploring quantum expressivity for real omics data
• Probing many-body Bell correlation depth with superconducting qubits
• Quantum Computational Complexity and Symmetry
• Learning Equivariant Maps with Variational Quantum Circuits
• Learning Fourier series with parametrized quantum circuits
• Pitfalls when tackling the exponential concentration of parameterized quantum models
• Introduction to Quantum Error Correction with Stabilizer Codes
• Barren plateaus are amplified by the dimension of qudits
• Quantum Active Learning
• A quantum tug of war between randomness and symmetries on homogeneous spaces
• Expressibility, entangling power and quantum average causal effect for causally indefinite circuits
• Automated detection of symmetry-protected subspaces in quantum simulations
• Perspectives on Utilization of Measurements in Quantum Algorithms
• SU(d)-Symmetric Random Unitaries: Quantum Scrambling, Error Correction, and Machine Learning
• Harnessing a 256-qubit Neutral Atom Simulator for Graph Classification
• Learning quantum symmetries with interactive quantum-classical variational algorithms
• Scaling of symmetry-restricted quantum circuits
• Estimation of multivariate traces of states given partial classical information
• Learning complexity gradually in quantum machine learning models