Quantum State Tomography via Nonconvex Riemannian Gradient Descent

Area of Science:

• Quantum Information Science
• Quantum Computing
• Quantum State Tomography

Background:

• Reconstructing large quantum states (density matrices) demands substantial computational power.
• Factored Gradient Descent (FGD) algorithms mitigate dimensionality but suffer slow convergence due to state condition number dependency.
• Current FGD methods require O(sqrt[κ]ln(1/ϵ)) iterations for error ϵ, hindering practical applications.

Purpose of the Study:

• To develop a quantum state tomography scheme with improved convergence rates.
• To overcome the limitations of existing methods by reducing dependency on the state's condition number.
• To achieve faster and more resource-efficient quantum state reconstruction.

Main Methods:

• Introduced a novel quantum state tomography scheme.
• Employed nonconvex Riemannian Gradient Descent (RGD) for state reconstruction.
• Derived theoretical convergence bounds independent of the ground truth state's condition number.

Main Results:

• Achieved O(ln(1/κϵ)) convergence, a significant improvement over O(sqrt[κ]ln(1/ϵ)).
• The new algorithm's convergence rate is independent of the condition number κ.
• Numerical simulations corroborate theoretical findings of fast convergence and near-optimal error bounds.

Conclusions:

• The proposed RGD-based quantum state tomography offers substantially faster convergence.
• This method significantly reduces computational resources needed for accurate density matrix recovery.
• The findings pave the way for more efficient quantum information processing and analysis.