[Paper Review] Fast quantum state reconstruction via accelerated non-convex programming

Junhyung Lyle Kim, George Kollias+3 more|arXiv (Cornell University)|Apr 14, 2021

Sparse and Compressive Sensing Techniques4 citations

TL;DR

This paper introduces Momentum-Inspired Factored Gradient Descent (MiFGD), a non-convex optimization algorithm for fast and provably convergent quantum state reconstruction in low-rank quantum systems. By combining compressed sensing, non-convex programming, and acceleration techniques, MiFGD achieves an accelerated linear convergence rate and outperforms state-of-the-art methods in both synthetic and real IBM Q hardware experiments, offering orders-of-magnitude speedups with comparable or better accuracy under noise.

ABSTRACT

We propose a new quantum state reconstruction method that combines ideas from compressed sensing, non-convex optimization, and acceleration methods. The algorithm, called Momentum-Inspired Factored Gradient Descent ( exttt{MiFGD}), extends the applicability of quantum tomography for larger systems. Despite being a non-convex method, exttt{MiFGD} converges \emph{provably} close to the true density matrix at an accelerated linear rate, in the absence of experimental and statistical noise, and under common assumptions. With this manuscript, we present the method, prove its convergence property and provide Frobenius norm bound guarantees with respect to the true density matrix. From a practical point of view, we benchmark the algorithm performance with respect to other existing methods, in both synthetic and real experiments performed on an IBM's quantum processing unit. We find that the proposed algorithm performs orders of magnitude faster than state of the art approaches, with the same or better accuracy. In both synthetic and real experiments, we observed accurate and robust reconstruction, despite experimental and statistical noise in the tomographic data. Finally, we provide a ready-to-use code for state tomography of multi-qubit systems.

Motivation & Objective

• To address the scalability bottleneck in quantum state tomography (QST) due to exponential scaling of measurement data and computational complexity.
• To develop a fast, scalable, and provably convergent algorithm for reconstructing low-rank quantum density matrices from limited measurement data.
• To overcome the limitations of convex optimization in QST by introducing a non-convex method with theoretical convergence guarantees.
• To enable practical, hardware-aware quantum state reconstruction on near-term quantum processors, including real IBM Q devices.
• To provide a ready-to-use implementation for multi-qubit state tomography with high performance and robustness to noise.

Proposed method

• MiFGD formulates quantum state reconstruction as a non-convex optimization problem over a factored parameterization of the density matrix, reducing the search space dimensionality.
• The algorithm employs momentum-inspired gradient updates to accelerate convergence, inspired by techniques in non-convex optimization and first-order methods.
• It leverages compressed sensing principles by assuming the target density matrix is low-rank, enabling recovery from few measurements.
• The method includes a line search and adaptive step size strategy to ensure stability and convergence under noise.
• Theoretical analysis establishes an accelerated linear convergence rate in the noiseless case, with Frobenius norm error bounds relative to the true state.
• The algorithm is designed for efficient implementation on classical hardware, avoiding reliance on specialized accelerators like GPUs.

Experimental results

Research questions

• RQ1 Can a non-convex optimization method achieve accelerated linear convergence in quantum state reconstruction under low-rank assumptions?
• RQ2 How does MiFGD perform in practice compared to convex and non-convex state-of-the-art QST methods on real quantum hardware?
• RQ3 What is the robustness of MiFGD to experimental and statistical noise in measurement data from NISQ devices?
• RQ4 Can MiFGD scale efficiently to larger systems (e.g., 12-qubit) without sacrificing accuracy or requiring specialized hardware?
• RQ5 What theoretical guarantees can be provided for a non-convex, accelerated gradient method in the context of quantum tomography?

Key findings

• MiFGD achieves an accelerated linear convergence rate in the noiseless case, with theoretical guarantees on Frobenius norm error decay.
• On real IBM Q hardware, MiFGD achieves accurate and robust state reconstruction up to 8 qubits, even under experimental noise.
• In synthetic experiments, MiFGD scales efficiently to 12-qubit systems, demonstrating feasibility for larger-scale quantum systems.
• MiFGD outperforms existing state-of-the-art methods by orders of magnitude in runtime while maintaining or improving reconstruction accuracy.
• The algorithm demonstrates robustness to statistical noise and measurement errors, with error bounds derived from the restricted isometry property (RIP) and measurement operator structure.
• A ready-to-use open-source implementation is provided, enabling immediate deployment for multi-qubit state tomography.