Quantum Tomography via Compressed Sensing: Error Bounds, Sample Complexity, and Efficient Estimators (1205.2300v2)

Abstract: Intuitively, if a density operator has small rank, then it should be easier to estimate from experimental data, since in this case only a few eigenvectors need to be learned. We prove two complementary results that confirm this intuition. First, we show that a low-rank density matrix can be estimated using fewer copies of the state, i.e., the sample complexity of tomography decreases with the rank. Second, we show that unknown low-rank states can be reconstructed from an incomplete set of measurements, using techniques from compressed sensing and matrix completion. These techniques use simple Pauli measurements, and their output can be certified without making any assumptions about the unknown state. We give a new theoretical analysis of compressed tomography, based on the restricted isometry property (RIP) for low-rank matrices. Using these tools, we obtain near-optimal error bounds, for the realistic situation where the data contains noise due to finite statistics, and the density matrix is full-rank with decaying eigenvalues. We also obtain upper-bounds on the sample complexity of compressed tomography, and almost-matching lower bounds on the sample complexity of any procedure using adaptive sequences of Pauli measurements. Using numerical simulations, we compare the performance of two compressed sensing estimators with standard maximum-likelihood estimation (MLE). We find that, given comparable experimental resources, the compressed sensing estimators consistently produce higher-fidelity state reconstructions than MLE. In addition, the use of an incomplete set of measurements leads to faster classical processing with no loss of accuracy. Finally, we show how to certify the accuracy of a low rank estimate using direct fidelity estimation and we describe a method for compressed quantum process tomography that works for processes with small Kraus rank.

• The paper demonstrates that compressed sensing significantly reduces sample complexity for estimating low-rank quantum states.
• It introduces efficient estimators, including the matrix Dantzig selector and matrix Lasso, with near-optimal error bounds.
• The study streamlines experimental resource use and improves quantum error correction by ensuring high-fidelity state reconstruction.

Quantum Tomography via Compressed Sensing: Insights and Implications

The paper "Quantum Tomography via Compressed Sensing: Error Bounds, Sample Complexity, and Efficient Estimators" explores the application of compressed sensing techniques to quantum tomography, particularly focusing on estimating low-rank quantum states. The authors present a novel methodology that leverages the structure of quantum states to enhance the efficiency of quantum state estimation.

Overview of Key Contributions

The authors make several critical contributions to advancing quantum tomography using compressed sensing methodology:

1. Sample Complexity Reduction: The paper demonstrates that the sample complexity of estimating low-rank quantum states decreases with the rank. This is a significant improvement over traditional methods, where the number of required samples grows exponentially with the system's size.
2. Use of Compressed Sensing Techniques: Compressed sensing, a method well-known in classical signal processing for its ability to recover sparse signals from a small set of measurements, is adeptly applied here. This approach particularly benefits scenarios involving low-rank density matrices, where compressed sensing helps reconstruct quantum states accurately from partial measurements.
3. Error Bounds and Estimators: The authors introduce theoretical frameworks that provide near-optimal error bounds using the restricted isometry property (RIP). Two estimators—the matrix Dantzig selector and the matrix Lasso—are introduced and benchmarked against the standard maximum likelihood estimation (MLE). Numerical simulations suggest that these compressed sensing estimators yield higher-fidelity state reconstructions compared to MLE.
4. Certification Procedures: The paper extends direct fidelity estimation (DFE) techniques to allow the certification of low-rank state estimates, thus providing a robust means of verifying the accuracy of compressed tomography results.
5. Quantum Process Tomography Extension: The authors also propose a method for applying compressed sensing to quantum process tomography, detailing procedures that minimize experimental complexity by requiring only a limited set of Pauli measurements.

Implications for Quantum Computing

The findings hold substantial implications for both theoretical and practical aspects of quantum computing:

• Theoretical Advancements: The application of compressed sensing to quantum tomography enriches the theoretical toolkit available for quantum state estimation, potentially influencing future research on quantum measurement techniques.
• Experimental Reduction of Resources: By significantly reducing the number of required measurements, compressed sensing enables more efficient use of experimental resources. This is crucial for scaling quantum systems, aligning well with ongoing efforts to realize practical quantum computing.
• Influence on Quantum Error Correction: Improved state reconstruction accuracy facilitates more reliable quantum error correction, enhancing the robustness and fault tolerance of quantum systems.

Future Developments

While the paper sets a solid foundation, it also opens avenues for further exploration:

• Optimization of Parameters: Determining optimal parameter settings for the proposed estimators could further enhance performance, ensuring robust state reconstruction across diverse quantum architectures.
• Exploring Alternative Measurement Bases: Extending the framework to non-Pauli and more experimentally accessible bases could broaden the applicability scope, making compressed sensing-based tomography feasible in a wider range of quantum systems.
• Algorithmic Developments: Refining fast algorithms beyond the field of convex programming to handle larger systems efficiently will be crucial for practical applications in quantum research and industry.

In summary, this paper presents a compelling approach to quantum tomography that significantly optimizes resource use while maintaining high accuracy. It consolidates compressed sensing as a vital component of the quantum information processing toolkit, offering promising directions for future research and application in quantum computing.