Connecting geometry and performance of two-qubit parameterized quantum circuits

Abstract

Parameterized quantum circuits (PQCs) are a central component of many variational quantum algorithms, yet there is a lack of understanding of how their parameterization impacts algorithm performance. We initiate this discussion by using principal bundles to geometrically characterize two-qubit PQCs. On the base manifold, we use the Mannoury-Fubini-Study metric to find a simple equation relating the Ricci scalar (geometry) and concurrence (entanglement). By calculating the Ricci scalar during a variational quantum eigensolver (VQE) optimization process, this offers us a new perspective to how and why Quantum Natural Gradient outperforms the standard gradient descent. We argue that the key to the Quantum Natural Gradient's superior performance is its ability to find regions of high negative curvature early in the optimization process. These regions of high negative curvature appear to be important in accelerating the optimization process.

References in corpus (15)

• Supplementary information for "Quantum supremacy using a programmable superconducting processor"
• Variational Quantum Algorithms
• Noisy intermediate-scale quantum (NISQ) algorithms
• Connecting ansatz expressibility to gradient magnitudes and barren plateaus
• A Quantum Computing View on Unitary Coupled Cluster Theory
• Equivalence of quantum barren plateaus to cost concentration and narrow gorges
• Capacity and quantum geometry of parametrized quantum circuits
• Measurement cost of metric-aware variational quantum algorithms
• Adaptive pruning-based optimization of parameterized quantum circuits
• Dimensional Expressivity Analysis of Parametric Quantum Circuits
• Two-Qubit Bloch Sphere
• An application benchmark for fermionic quantum simulations
• Optimal training of variational quantum algorithms without barren plateaus
• Reducing the amount of single-qubit rotations in VQE and related algorithms
• Efficient classical calculation of the Quantum Natural Gradient

Cited by in corpus (9)

• Optimizing quantum circuits with Riemannian gradient flow
• Natural parameterized quantum circuit
• Quantum variational learning for entanglement witnessing
• Solving Fractional Differential Equations on a Quantum Computer: A Variational Approach
• Quantum Natural Gradient with Geodesic Corrections for Small Shallow Quantum Circuits
• Learning Fourier series with parametrized quantum circuits
• Hybrid Quantum Singular Spectrum Decomposition for Time Series Analysis
• Modified Conjugate Quantum Natural Gradient
• A Brillouin torus decomposition for two-dimensional topological insulators