Core Concepts

A data-driven approach for the constrained optimization task of quantum process tomography, utilizing advanced stochastic objective optimizers and considering the quantum process as residing in a Stiefel manifold to perform local updates that respect the geometry.

Abstract

This paper introduces a modified version of stochastic gradient descent on a Riemannian manifold for the task of quantum process tomography (QPT). The key contributions are:
Formulating Riemmannian optimization via gradient descent on the Stiefel manifold for QPT.
Adapting the stochastic Adam optimizer for QPT.
Providing an open-source implementation Qutee.jl with native accelerated computation via GPUs.
Demonstrating the effectiveness of QPT with respect to process dimension and characterization on a set of real and simulated experiments.
The authors model the quantum process geometrically, where all possible quantum channels exist on a smooth continuous surface known as a manifold. Gradient-based methods are used to find the point that minimizes the loss function by taking steps along the manifold. The unique properties of quantum channels restrict which channels are physically possible, changing the shape of the manifold. Specifically, quantum channels exist on the Steifel manifold.
The authors utilize a strategy similar to deep learning, where the computational model can be fine-tuned with automatic differentiation using gradient-based techniques. They adapt the stochastic Adam optimizer for QPT, which estimates and updates the moments of the gradient via random subsamples (minibatches) of data points. This helps reduce the oscillations in the loss function arising from randomness and accelerates convergence.
The authors demonstrate the effectiveness of their approach on simulations of quantum processes with varying dimensions, Kraus ranks, and measurement errors. They are able to reconstruct quantum channels in a few seconds, even at maximal rank, outperforming traditional tomography techniques. They also evaluate their method on hardware by characterizing an engineered process on quantum computers.

Stats

The data-driven approach enables accurate, order-of-magnitude faster results, and works with incomplete data.
Reconstruction error saturates as a power curve with respect to noise ϵ.
High fidelity is observed even with the absence of a fraction of data.

Quotes

The key challenge, however, is the exponentially growing size of the process representation, e.g., the Choi representation is a complex-valued 4n×4n matrix for n qubits.
Deep learning involves optimizing complex computational models, such as neural networks, tensor networks, and differential equation solvers, using gradient-based methods.
Automatic differentiation enables the computational process to be differentiable with respect to its constituent elements, such as a hidden layer in deep learning.

Key Insights Distilled From

by Daniel Volya... at **arxiv.org** 04-30-2024

Deeper Inquiries

The Riemannian optimization framework proposed for quantum process tomography can be extended to other representations of quantum processes, such as tensor networks, to characterize non-Markovian noise processes. Tensor networks are powerful tools for representing quantum states and operations, especially in the context of many-body quantum systems. By incorporating Riemannian optimization techniques into tensor network-based quantum process tomography, it is possible to efficiently characterize and model non-Markovian noise processes in quantum systems.
One approach to extending the framework to tensor networks is to represent the quantum processes as tensor networks, where the Kraus operators are encoded in the structure of the network. The optimization task can then be formulated as finding the optimal tensor network structure that best represents the quantum process while adhering to the constraints imposed by the physical properties of the system. By leveraging the geometry of the tensor network manifold, similar to the Stiefel manifold in the current framework, efficient optimization algorithms can be developed to navigate the high-dimensional parameter space of tensor networks.
Furthermore, the use of automatic differentiation techniques can be extended to compute gradients with respect to the parameters of the tensor network representation. By integrating Riemannian optimization methods with tensor network structures, it becomes possible to capture the complex dynamics of non-Markovian noise processes and accurately characterize quantum operations in a variety of quantum systems.

The Riemannian optimization framework proposed for quantum process tomography can be extended to other representations of quantum processes, such as tensor networks, to characterize non-Markovian noise processes. Tensor networks are powerful tools for representing quantum states and operations, especially in the context of many-body quantum systems. By incorporating Riemannian optimization techniques into tensor network-based quantum process tomography, it is possible to efficiently characterize and model non-Markovian noise processes in quantum systems.
One approach to extending the framework to tensor networks is to represent the quantum processes as tensor networks, where the Kraus operators are encoded in the structure of the network. The optimization task can then be formulated as finding the optimal tensor network structure that best represents the quantum process while adhering to the constraints imposed by the physical properties of the system. By leveraging the geometry of the tensor network manifold, similar to the Stiefel manifold in the current framework, efficient optimization algorithms can be developed to navigate the high-dimensional parameter space of tensor networks.
Furthermore, the use of automatic differentiation techniques can be extended to compute gradients with respect to the parameters of the tensor network representation. By integrating Riemannian optimization methods with tensor network structures, it becomes possible to capture the complex dynamics of non-Markovian noise processes and accurately characterize quantum operations in a variety of quantum systems.

The Riemannian optimization approach to quantum process tomography offers several theoretical guarantees and error bounds that ensure the accuracy and reliability of the optimization process.
Convergence: Riemannian optimization guarantees convergence to a local minimum on the Stiefel manifold, where the quantum processes reside. By leveraging the geometry of the manifold, the optimization algorithm can efficiently navigate the parameter space and converge to a solution that minimizes the loss function.
Geometric Constraints: The Riemannian optimization framework ensures that the intermediate solutions obtained during the optimization process satisfy the physical constraints imposed by quantum processes, such as unitarity and complete positivity. This guarantees that the optimized quantum processes remain physically valid.
Error Bounds: The optimization algorithm provides error bounds on the reconstructed quantum processes. These error bounds quantify the accuracy of the reconstructed processes compared to the true underlying quantum processes. By analyzing these error bounds, researchers can assess the quality of the tomographic reconstruction and understand the level of uncertainty in the estimated quantum processes.
Stability: The Riemannian optimization approach offers stability guarantees, ensuring that small perturbations in the input data or parameters do not lead to significant changes in the optimized quantum processes. This stability property is crucial for robust and reliable quantum process tomography.
Overall, the theoretical guarantees and error bounds provided by the Riemannian optimization framework contribute to the trustworthiness and effectiveness of quantum process tomography, enabling researchers to accurately characterize quantum operations and understand the dynamics of quantum systems.

The insights gained from the efficient gradient-based optimization approach proposed for quantum process tomography can indeed be applied to other constrained optimization problems in quantum computing, such as variational quantum algorithms and quantum control. Here are some ways in which these insights can be leveraged:
Variational Quantum Algorithms: Variational quantum algorithms, such as the Variational Quantum Eigensolver (VQE) and Quantum Approximate Optimization Algorithm (QAOA), rely on optimizing a parameterized quantum circuit to solve specific computational problems. By applying the Riemannian optimization techniques developed for quantum process tomography, researchers can enhance the optimization process in variational quantum algorithms. This can lead to faster convergence, improved accuracy, and better exploration of the parameter space, ultimately enhancing the performance of variational quantum algorithms.
Quantum Control: In quantum control tasks, optimizing control parameters to achieve desired quantum operations is essential. By adapting the gradient-based optimization methods from quantum process tomography, quantum control tasks can benefit from more efficient and reliable optimization processes. Whether optimizing pulse sequences in quantum gates or designing control strategies for quantum error correction, the insights from Riemannian optimization can improve the effectiveness of quantum control techniques.
Noise Mitigation: Quantum systems are inherently noisy, and mitigating noise effects is crucial for achieving reliable quantum computations. The optimization insights from quantum process tomography can be applied to develop noise mitigation strategies in quantum algorithms. By optimizing noise models, error correction schemes, or noise-resilient quantum circuits, researchers can enhance the robustness of quantum computations in the presence of noise.
By transferring the efficient gradient-based optimization techniques and geometric insights from quantum process tomography to other areas of quantum computing, researchers can advance the field by improving optimization efficiency, accuracy, and robustness in various quantum applications.

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