Hamiltonian Expressibility Correlates with Solution Quality in Variational Quantum Eigensolvers

Selecting the right computational blueprint, or ansatz, is a fundamental challenge in variational quantum algorithms, and new research sheds light on how to optimise this process. Filippo Brozzi from the University of Florence, Gloria Turati and Maurizio Ferrari Dacrema from Politecnico di Milano, and colleagues, investigate a metric called Hamiltonian expressibility, which measures a quantum circuit’s ability to explore potential solutions. The team estimates this expressibility for a range of circuits using a computational modelling technique, and then tests how well these circuits actually perform when solving quantum problems. Their findings reveal a surprising link between expressibility and solution quality, demonstrating that highly expressive circuits excel at tackling complex problems with superposition-based solutions, while simpler circuits prove more effective for problems with straightforward answers, particularly in noisy environments. This work provides valuable guidance for designing more efficient quantum algorithms and selecting the best ansatz for a given computational task.

Researchers addressed the challenge of selecting effective quantum circuits, known as ansätze, for solving complex problems using a variational quantum eigensolver (VQE) approach. Recognizing that not all circuits are equally suited to all problems, they moved beyond simply assessing circuit complexity and instead focused on quantifying a circuit’s ability to explore the relevant energy landscape, a metric termed ‘Hamiltonian expressibility’. This approach offers a more general way to evaluate circuit performance than previous methods relying on heuristics or problem-specific features. To estimate Hamiltonian expressibility, the team developed a novel Monte Carlo method, a computational technique that uses random sampling to obtain numerical results, allowing them to assess how well a circuit explores the possible quantum states relevant to a particular problem.

Their analysis, focused on systems of four and eight qubits, revealed a nuanced connection between circuit depth and its ability to explore the energy landscape. For problems with solutions represented by superpositions of quantum states and under ideal conditions, circuits with high Hamiltonian expressibility generally yielded better results. Conversely, for problems with simpler solutions, circuits with lower expressibility proved more effective. Importantly, the team also investigated the impact of noise, a significant challenge in quantum computing, and discovered that low-expressibility circuits often maintained better performance in noisy environments. This work demonstrates that Hamiltonian expressibility can serve as a valuable guide for selecting appropriate ansätze in small-scale quantum computations, offering a means to balance exploration capability with resilience to noise. By systematically quantifying a circuit’s ability to navigate the problem’s energy landscape, the researchers provide a foundation for developing more intelligent circuit design strategies and integrating expressibility metrics into adaptive algorithms and machine learning techniques, promising to improve the efficiency and accuracy of quantum computations.

Circuit Depth and Hamiltonian Expressibility Relationship

Researchers have investigated the crucial role of circuit design, known as the ansatz, in achieving effective solutions with variational quantum algorithms. A key metric explored is Hamiltonian expressibility, which quantifies a circuit’s ability to thoroughly explore the possible states relevant to a specific problem. The team developed a method to estimate this expressibility for various circuit designs and problem types, focusing on systems of up to eight quantum bits. Their analysis reveals a clear relationship between circuit depth and expressibility, demonstrating that expressibility generally increases with depth, though this increase plateaus beyond a certain point.

Interestingly, the study highlights that expressibility is not a universal property of a circuit, but rather depends heavily on the problem being solved. Certain circuits proved highly expressive for some problems, while performing poorly on others, and vice versa, particularly when considering diagonal Hamiltonians. This problem-dependence suggests that selecting the right circuit requires careful consideration of the problem’s characteristics. The researchers then tested the practical implications of expressibility by training these circuits using the variational quantum eigensolver, a common algorithm for finding the lowest energy state of a quantum system.

Under ideal conditions, circuits with high expressibility consistently outperformed others when tackling problems with complex solutions requiring superposition of states. However, a surprising result emerged for problems with simpler solutions; circuits with low expressibility proved more effective. This suggests that a highly expressive circuit isn’t always desirable, and can even hinder performance in certain scenarios. These findings provide valuable guidance for designing effective quantum circuits, demonstrating that a nuanced understanding of expressibility, tailored to the specific problem, is crucial for maximizing performance.

Optimal Expressibility Depends on Problem Structure

This research investigates the relationship between a quantum circuit’s ability to explore possible solutions, quantified as Hamiltonian expressibility, and its performance in solving optimization problems using the Variational Quantum Eigensolver (VQE) algorithm. By employing a Monte Carlo method to estimate expressibility across different circuit designs and problem types, the study reveals a nuanced connection between these factors. Results demonstrate that, under ideal conditions and for certain problems, circuits with high expressibility perform better when seeking solutions involving superposition states, while those with lower expressibility are more effective for problems with basis state solutions. The findings highlight that the optimal circuit expressibility is problem-dependent; a universally ‘better’ circuit does not exist.

Notably, the study shows that low-expressibility circuits remain preferable for basis-state problems even in the presence of noise, and intermediate expressibility can be beneficial for superposition-state problems under noisy conditions. The authors acknowledge that the analysis focuses on relatively small-scale problems and specific problem classes, limiting the generalizability of the conclusions. Future work could extend this research to larger systems and a wider range of optimization challenges to further refine our understanding of how circuit design impacts the performance of variational quantum algorithms.