Self-Verifying Variational Quantum Simulation of the Lattice Schwinger Model

Abstract: Hybrid classical-quantum algorithms aim at variationally solving optimisation problems, using a feedback loop between a classical computer and a quantum co-processor, while benefitting from quantum resources. Here we present experiments demonstrating self-verifying, hybrid, variational quantum simulation of lattice models in condensed matter and high-energy physics. Contrary to analog quantum simulation, this approach forgoes the requirement of realising the targeted Hamiltonian directly in the laboratory, thus allowing the study of a wide variety of previously intractable target models. Here, we focus on the Lattice Schwinger model, a gauge theory of 1D quantum electrodynamics. Our quantum co-processor is a programmable, trapped-ion analog quantum simulator with up to 20 qubits, capable of generating families of entangled trial states respecting symmetries of the target Hamiltonian. We determine ground states, energy gaps and, by measuring variances of the Schwinger Hamiltonian, we provide algorithmic error bars for energies, thus addressing the long-standing challenge of verifying quantum simulation.

• The paper introduces a hybrid quantum-classical simulation that accurately estimates ground state energies of the lattice Schwinger model using up to 20 qubits.
• It leverages trapped-ion analog quantum simulators to optimize trial state preparation while exploiting intrinsic Hamiltonian symmetries for improved scalability.
• A novel self-verification protocol based on algorithmic error bars validates the simulation results, addressing key challenges in quantum simulation accuracy.

Self-Verifying Variational Quantum Simulation of the Lattice Schwinger Model

The paper discusses the implementation of variational quantum simulation (VQS) using hybrid classical-quantum algorithms to study complex quantum systems. The focus of this work is the lattice Schwinger model, which represents 1-dimensional quantum electrodynamics, a fundamental theory in condensed matter and high-energy physics. This study leverages a trapped-ion analog quantum simulator as a quantum co-processor to overcome the limitations of classical simulations.

Key Contributions

• Hybrid Quantum Simulation Approach: The research leverages VQS, which integrates classical and quantum computational resources, to optimize and execute calculations typically intractable for classical systems alone. This methodology sidesteps the necessity to physically realize complex Hamiltonians in laboratories, thus allowing simulations of challenging models like the lattice Schwinger model.
• Trapped-Ion Analog Quantum Simulator: The quantum hardware comprises trapped ions representing up to 20 qubits, used to prepare families of trial states. This setup takes advantage of the intrinsic symmetries of the Schwinger Hamiltonian, enhancing the simulation's efficiency and scalability.
• Self-Verification Protocol: The paper details a novel approach where algorithmic error bars are derived from measuring Hamiltonian variances, providing a mechanism for validating simulation results directly on quantum hardware. This feature addresses long-standing challenges in verifying quantum simulation accuracy.

Numerical Results

• Ground State Energy Estimation: The research successfully determined ground states and energy gaps for the lattice Schwinger model using up to 20 qubits, achieving high fidelity compared to the exact theoretical solutions.
• Scalability of VQS: The experiments demonstrated scalability up to 20 qubits, indicating that the technique is viable for larger systems, with potential for further scaling given improved quantum hardware and algorithmic adaptations.
• Quantum Phase Transitions: The study also explored quantum phase transitions by varying the model's parameters, and accurately measuring order parameters and entanglement entropies near critical points, showcasing the versatility of VQS.

Implications and Future Directions

• Practical Implications: This work highlights the potential for quantum simulators to offer insights into complex quantum systems, previously inaccessible with classical methods. The successful self-verification feature could encourage wider adoption of VQS in experimental settings, assuring researchers of the reliability of their results.
• Theoretical Insights and AI Developments: From a theoretical standpoint, these techniques could pave the way for advancements in our understanding of quantum lattice models and gauge theories, impacting fields ranging from quantum chemistry to materials science. In terms of AI, these computations can contribute to the development of more robust quantum algorithms and inspire novel methods in optimization and simulation.
• Future Prospects: The research exemplifies a significant step towards efficient and reliable quantum simulations. Future work could focus on expanding capabilities to 2D and 3D quantum simulators, exploring different gauge theories, and incorporating more sophisticated optimization techniques to further improve the efficiency and accuracy of quantum simulations.

This paper provides an insightful example of how hybrid quantum-classical systems can be utilized effectively for complex quantum simulations, marking progress towards practical quantum computing applications.