Quantum Hamiltonian-Based Models and the Variational Quantum Thermalizer Algorithm (1910.02071v1)

Abstract: We introduce a new class of generative quantum-neural-network-based models called Quantum Hamiltonian-Based Models (QHBMs). In doing so, we establish a paradigmatic approach for quantum-probabilistic hybrid variational learning, where we efficiently decompose the tasks of learning classical and quantum correlations in a way which maximizes the utility of both classical and quantum processors. In addition, we introduce the Variational Quantum Thermalizer (VQT) for generating the thermal state of a given Hamiltonian and target temperature, a task for which QHBMs are naturally well-suited. The VQT can be seen as a generalization of the Variational Quantum Eigensolver (VQE) to thermal states: we show that the VQT converges to the VQE in the zero temperature limit. We provide numerical results demonstrating the efficacy of these techniques in illustrative examples. We use QHBMs and the VQT on Heisenberg spin systems, we apply QHBMs to learn entanglement Hamiltonians and compression codes in simulated free Bosonic systems, and finally we use the VQT to prepare thermal Fermionic Gaussian states for quantum simulation.

• The paper proposes hybrid quantum neural network models to learn effective modular Hamiltonians for simulating mixed quantum states.
• It introduces the Variational Quantum Thermalizer algorithm, generalizing VQE to approximate thermal states via free energy minimization.
• It demonstrates efficacy on Heisenberg, Bosonic, and Fermionic systems, suggesting promising avenues for future research.

Quantum Hamiltonian-Based Models and the Variational Quantum Thermalizer Algorithm

The paper in discussion presents a novel approach to generative modeling in quantum systems through the introduction of Quantum Hamiltonian-Based Models (QHBMs) and the Variational Quantum Thermalizer (VQT) algorithm. These methodologies are designed to address the challenges posed by near-term quantum devices, focusing on efficiently leveraging both quantum and classical processors to model complex quantum systems.

QHBMs represent a new class of hybrid quantum-neural-network-based models that are particularly well-suited for tasks involving mixed quantum states. They seek to decompose learning tasks into components that can separately address classical and quantum correlations, thereby optimizing the use of computational resources in noisy intermediate-scale quantum (NISQ) devices. By employing a variational approach, these models aim to learn the effective modular Hamiltonian of a given quantum system, which can then be used to replicate the statistical properties of an unknown target mixed state.

A key innovation of QHBMs is the coupling of quantum neural network unitaries with parameterized latent distributions to create an exponentially parameterized ansatz. This ansatz facilitates the modeling of quantum distributions as thermal states of a parameterized Hamiltonian, which is beneficial for capturing the quantum statistical mechanics inherent to mixed states. The learning process involves minimizing quantum relative entropy, a natural cost function in this context, which leads to optimizing the variational parameters until the model's distribution closely aligns with the target distribution.

The Variational Quantum Thermalizer (VQT) algorithm is a specific application within this framework aimed at simulating thermal states of quantum systems. It generalizes the widely known Variational Quantum Eigensolver (VQE) to finite-temperature scenarios, wherein the task is to approximate thermal or mixed quantum states rather than pure ground states. VQT employs a free energy minimization approach, where the goal is to find a variational state minimizing a combination of the Hamiltonian expectation value and state entropy, thereby approximating the thermal state at a specific temperature.

Illustrative examples on Heisenberg spin systems, Bosonic systems, and Fermionic systems with Gaussian statistics demonstrate the efficacy of these methods. The paper details the application of QHBMs to learn effective Hamiltonians that represent complex quantum distributions efficiently. These applications include thermal state preparation, modular Hamiltonian learning for entanglement spectra, and compression code learning in simulated quantum systems.

The implications of this work are significant, suggesting potential advancements in quantum simulation, machine learning, and quantum communication. QHBMs and the VQT algorithm offer a pathway to bridge the capabilities of classical and quantum computation, potentially expanding the problem-solving capacity of quantum-inspired techniques in practical applications.

While the paper provides strong numerical evidence supporting the effectiveness of the proposed models, several avenues for future research are apparent. These include exploring more generalized latent space distributions through the integration of classical probabilistic modeling techniques (such as energy-based models or normalizing flows) and expanding the application scope to non-Gaussian and high-dimensional quantum systems beyond reach via classical methods. As the field progresses, QHBMs might offer new insights into complex quantum phenomena, providing both theoretical and practical advancements in harnessing quantum computational power.