Evolved Quantum Boltzmann Machines

Abstract

We introduce evolved quantum Boltzmann machines as a variational ansatz for quantum optimization and learning tasks. Given two parameterized Hamiltonians $G(\theta )$ and $H(\varphi )$, an evolved quantum Boltzmann machine consists of preparing a thermal state of the first Hamiltonian $G(\theta )$ followed by unitary evolution according to the second Hamiltonian $H(\varphi )$. Alternatively, one can think of it as first realizing imaginary time evolution according to $G(\theta )$ followed by real time evolution according to $H(\varphi )$. After defining this ansatz, we provide analytical expressions for the gradient vector and illustrate their application in ground-state energy estimation and generative modeling, showing how the gradient for these tasks can be estimated by means of quantum algorithms that involve classical sampling, Hamiltonian simulation, and the Hadamard test. We also establish analytical expressions for the Fisher-Bures, Wigner-Yanase, and Kubo-Mori information matrix elements of evolved quantum Boltzmann machines, as well as quantum algorithms for estimating each of them, which leads to at least three different general natural gradient descent algorithms based on this ansatz. Along the way, we establish a broad generalization of the main result of [Luo, Proc. Am. Math. Soc. 132, 885 (2004)], proving that the Fisher-Bures and Wigner-Yanase information matrices of general parameterized families of states differ by no more than a factor of two in the matrix (Loewner) order, making them essentially interchangeable for training when using natural gradient descent.

v3: 21 pages of main text, 42 pages of appendices, 6 figures

References in corpus (61)

• Quantum Machine Learning
• A variational eigenvalue solver on a quantum processor
• Variational Quantum Algorithms
• Supervised learning with quantum enhanced feature spaces
• Barren plateaus in quantum neural network training landscapes
• The theory of variational hybrid quantum-classical algorithms
• Quantum support vector machine for big data classification
• Quantum Circuit Learning
• Rényi Divergence and Kullback-Leibler Divergence
• Parameterized quantum circuits as machine learning models
• Cost Function Dependent Barren Plateaus in Shallow Parametrized Quantum Circuits
• Quantum algorithms: an overview
• Circuit-centric quantum classifiers
• Hamiltonian Simulation by Qubitization
• From the Quantum Approximate Optimization Algorithm to a Quantum Alternating Operator Ansatz
• Quantum Fisher information matrix and multiparameter estimation
• On quantum Renyi entropies: a new generalization and some properties
• Towards Practical Quantum Variational Algorithms
• Connecting ansatz expressibility to gradient magnitudes and barren plateaus
• Toward the first quantum simulation with quantum speedup
• Strong converse for the classical capacity of entanglement-breaking and Hadamard channels via a sandwiched Renyi relative entropy
• Quantum critical scaling of the geometric tensors
• Quantum Natural Gradient
• Measuring thermodynamic length
• Quantum Boltzmann Machine
• Quantum Approximate Optimization Algorithm for MaxCut: A Fermionic View
• Asymptotic Error Rates in Quantum Hypothesis Testing
• Variational quantum algorithms for discovering Hamiltonian spectra
• Quantum enhanced estimation of a multi-dimensional field
• A Geometric Perspective on Quantum Parameter Estimation
• Monotonicity of a relative Rényi entropy
• Geometry of quantum phase transitions
• Tomography and Generative Data Modeling via Quantum Boltzmann Training
• Metric adjusted skew information
• Variational Quantum State Eigensolver
• Sample-efficient learning of quantum many-body systems
• Fisher Information in Noisy Intermediate-Scale Quantum Applications
• Quantum-Assisted Learning of Hardware-Embedded Probabilistic Graphical Models
• Quantum Belief Propagation
• Wigner-Yanase information on quantum state space:the geometric approach
• Measuring Analytic Gradients of General Quantum Evolution with the Stochastic Parameter Shift Rule
• Canonical Energy is Quantum Fisher Information
• How to Build the Thermofield Double State
• A measurement-based variational quantum eigensolver
• Quantum natural gradient generalised to noisy and non-unitary circuits
• Holographic spacetime, black holes and quantum error correcting codes: A review
• Optimized quantum f-divergences and data processing
• Stability of topologically invariant order parameters at finite temperature
• Perturbative analysis of topological entanglement entropy from conditional independence
• New Quantum Algorithms for Computing Quantum Entropies and Distances
• On the Sample Complexity of Quantum Boltzmann Machine Learning
• Fundamental limits of metrology at thermal equilibrium
• Fast Quantum Algorithms for Trace Distance Estimation
• Here comes the SU(N): multivariate quantum gates and gradients
• Geometric approach to quantum statistical inference
• Estimation of Hamiltonian parameters from thermal states
• F-Divergences and Cost Function Locality in Generative Modelling with Quantum Circuits
• Stochastic approach for quantum metrology with generic Hamiltonians
• Quantum algorithms for matrix geometric means
• QSlack: A slack-variable approach for variational quantum semi-definite programming
• Gibbs Sampling gives Quantum Advantage at Constant Temperatures with O(1)-Local Hamiltonians