Variational Neural Networks Probe Non-Stabilizerness in Qubit-Regularised Field Theory

Understanding the behaviour of fundamental forces presents a major challenge for physicists, particularly when dealing with complex systems like those described by quantum field theory. Samuel Crew and Hsueh Hao Lu, from National Tsing Hua University, along with their colleagues, now investigate this challenge by employing a novel approach using artificial intelligence. They demonstrate how variational neural networks can quantify the ‘magic’, or non-stabilizerness, within the ground states of the Schwinger model, a simplified theory of quantum electrodynamics. This work provides a new tool for exploring the difficulty of simulating gauge theories, offering insights into the classical hardness of these fundamental interactions and potentially paving the way for improved computational methods.

Entanglement has long been recognised as a key resource distinguishing quantum and classical physics. However, it is clear that entanglement alone does not fully capture the quantum nature of a state. For instance, stabilizer states, those generated by Clifford operations, can exhibit significant entanglement while remaining efficiently classically simulable. Understanding the properties that differentiate between efficiently simulable and genuinely quantum states remains a central challenge in quantum information theory. This research investigates how the ‘magic content’ of these states depends on the separation between external probe charges, providing insight into the classical hardness of simulating gauge theories with non-trivial infrared structure.

Quantum Magic Diagnoses Schwinger Model Phases

This research explores the use of neural networks to simulate quantum systems, specifically focusing on the Schwinger model, a simplified model of quantum electrodynamics. The authors investigate quantum magic, a measure of how non-classical a quantum state is, as a potential diagnostic tool for phase transitions, such as confinement and screening, in quantum field theory. The study proposes that the stabilizer Rényi entropy can identify and characterise different phases of matter. Researchers utilise neural networks, specifically neural quantum states, to represent and simulate the quantum states of the Schwinger model, allowing them to explore systems difficult to simulate with traditional methods.

The Schwinger model serves as a test case due to its relative simplicity while still exhibiting interesting quantum phenomena. The stabilizer Rényi entropy quantifies the degree to which a quantum state deviates from being a simple stabilizer state, a classical-like state. Neural networks are employed to represent the wavefunctions of the quantum system, with the network’s parameters optimised to minimise the energy of the system, effectively finding the ground state. The study demonstrates that the stabilizer Rényi entropy does differ between the confinement and screening phases of the Schwinger model, supporting the idea that quantum magic can be used as a diagnostic tool for phase transitions. The authors observe distinct behaviours in the entropy as a function of system parameters, with the confinement phase showing a monotonic decrease and the screening phase exhibiting a local minimum. This approach could be extended to other quantum field theories and potentially used to identify new phases of matter.

Neural Networks Quantify Quantum State Complexity

Researchers have demonstrated a novel approach to quantifying the complexity of quantum states using neural networks, offering insights into the potential for quantum computation. The team applied a neural network quantum state method to model the ground state of the Schwinger model and then measured its “magic content”, a property indicating how far the state deviates from being easily simulated by classical computers. This research establishes that neural network quantum state methods can not only determine the energy of these complex states but also assess their computational non-classicality, a crucial step in evaluating their potential for quantum advantage. The core of this work lies in representing the quantum state as a complex neural network, specifically a restricted Boltzmann machine.

This network architecture allows researchers to encode the probabilities of different quantum configurations, effectively creating a variational approximation of the true ground state. By training the network to minimise the energy of the system, the team generated a state that closely resembles the actual ground state of the Schwinger model, enabling the subsequent calculation of its non-stabilizerness. The method involves expressing the complex quantum properties as classical calculations, allowing for efficient evaluation using standard computational techniques. Researchers then employed the stabilizer Rényi entropy to quantify the “magic” within the generated quantum state.

This entropy, calculated using the neural network representation, provides a direct indication of how difficult it would be to simulate the state using classical algorithms. The results demonstrate that the neural network quantum state method accurately captures the non-classical features of the quantum state, revealing a significant degree of “magic” present within it. Specifically, the team showed that the calculated entropy provides a quantifiable measure of how much quantum advantage the state might offer, potentially exceeding the capabilities of classical simulation. This represents a significant advancement in the field of quantum simulation, as it provides a practical method for assessing the complexity of quantum states and their potential for computational advantage. By combining neural networks with quantum field theory, the researchers have opened new avenues for exploring the boundary between classical and quantum computation.

Non-Stabilizerness Reveals Quantum Field Theory Phases

This work demonstrates the effective use of neural network quantum states as a variational method for investigating non-stabilizerness in qubit-regularised quantum field theory. Researchers computed the stabilizer Rényi entropy of ground states within the Schwinger model, revealing how the ‘magic content’ of these states changes with the separation of external charges. The results accurately approximate the energy of lattice gauge theory ground states and capture features linked to the computational difficulty of simulating these systems. The study shows that non-stabilizerness increases as external charges are brought closer together, and that qualitatively different behaviours exist between the screening and confinement phases of the model, particularly at strong coupling. This provides an initial step towards utilising quantum magic as a diagnostic tool for identifying phase transitions in quantum field theory. Future work should focus on developing an analytical understanding of the observed phenomena and extending the approach to larger, more complex systems.