Agnostic Process Tomography (2410.11957v1)

Abstract: Characterizing a quantum system by learning its state or evolution is a fundamental problem in quantum physics and learning theory with a myriad of applications. Recently, as a new approach to this problem, the task of agnostic state tomography was defined, in which one aims to approximate an arbitrary quantum state by a simpler one in a given class. Generalizing this notion to quantum processes, we initiate the study of agnostic process tomography: given query access to an unknown quantum channel $\Phi$ and a known concept class $\mathcal{C}$ of channels, output a quantum channel that approximates $\Phi$ as well as any channel in the concept class $\mathcal{C}$, up to some error. In this work, we propose several natural applications for this new task in quantum machine learning, quantum metrology, classical simulation, and error mitigation. In addition, we give efficient agnostic process tomography algorithms for a wide variety of concept classes, including Pauli strings, Pauli channels, quantum junta channels, low-degree channels, and a class of channels produced by $\mathsf{QAC}^0$ circuits. The main technical tool we use is Pauli spectrum analysis of operators and superoperators. We also prove that, using ancilla qubits, any agnostic state tomography algorithm can be extended to one solving agnostic process tomography for a compatible concept class of unitaries, immediately giving us efficient agnostic learning algorithms for Clifford circuits, Clifford circuits with few T gates, and circuits consisting of a tensor product of single-qubit gates. Together, our results provide insight into the conditions and new algorithms necessary to extend the learnability of a concept class from the standard tomographic setting to the agnostic one.

• The paper introduces agnostic process tomography, approximating unknown quantum channels within a defined concept class.
• It employs advanced algorithmic techniques like Fourier and Pauli spectrum analysis alongside classical shadow methods for efficient channel learning.
• The methodology enhances applications in quantum learning, simulation, error mitigation, and metrology, paving the way for scalable quantum technologies.

Agnostic Process Tomography

The paper "Agnostic Process Tomography" by Chirag Wadhwa, Laura Lewis, Elham Kashefi, and Mina Doosti introduces a novel approach to characterizing quantum processes. The work extends agnostic state tomography to quantum processes, aiming to approximate an unknown quantum channel with respect to a defined concept class. The proposed method enhances existing quantum process tomography techniques by offering a framework that accommodates unknown process structures and arbitrary noise, thereby mitigating the limitations of conventional methods reliant on strict known structures.

Contributions and Methodology

The primary contribution of the paper is the definition and development of agnostic process tomography (APT). This framework enables the approximation of an unknown quantum channel by the closest channel within a specified concept class, optimizing the approximation while considering feasible resource constraints. The authors extend efficient algorithms for various concept classes such as Pauli strings, Pauli channels, quantum junta channels, and low-degree channels. Key algorithmic techniques are inspired by Fourier and Pauli spectrum analysis as previously developed for quantum operators and superoperators.

A highlight of this research is the ability to generalize any agnostic state tomography algorithm to agnostic process tomography under certain compatibility conditions. This involves leveraging ancillary systems and classical shadow techniques for sampling, thereby providing guarantees for specific classes like Clifford circuits. The conversion process includes efficient methods to learn the Choi states of quantum processes, a notable theoretical leap in quantum metrology and machine learning contexts.

Practical and Theoretical Implications

The results of this paper have several significant implications:

1. Quantum Learning: The agnostic framework aligns with classical machine learning's general learning theory, thereby offering scalable techniques for quantum machine learning models that use parameterized quantum circuits.
2. Simulation and Implementation: The methodology supports resource-constrained quantum process implementation and classical simulation of quantum processes, providing methods to approximate complex quantum states using simpler or classically simulable sets.
3. Error Mitigation: APT can be incorporated for noise characterization and mitigation, supporting quantum error correction processes that are robust and efficient under operational constraints.
4. Quantum Metrology: The application of agnostic learning to quantum metrology can lead to tracking parameters under noise-laden conditions, advancing precision measurement techniques in quantum technologies.

Future Directions

While the paper provides a comprehensive toolkit for APT, several avenues for further research emerge notably:

• Algorithmic Refinements: Additional work is needed to improve computational complexities, particularly regarding realization of efficient proper learning outputs in APT scenarios.
• Complexity Analysis: Detailed complexity trade-offs between sample efficiency and computational feasibility need refinement in practical settings, particularly for near-term quantum devices.
• New Concept Classes: Exploring physically motivated concept classes beyond the current investigation, such as parameterized quantum circuits with specific ansatz, may yield new insights and methods.
• Reduced Resource Requirements: Future research could focus on lowering resource needs for state preparation and measurement, critical for scaling quantum process tomography to larger systems.

In conclusion, the strategies and methodologies within this paper propose a compelling framework for advancing the paper of quantum processes, laying groundwork for broader applications across quantum computing and information science. The paper effectively bridges gaps between theoretical advancements in quantum learning and practical implementations under realistic constraints.