Agnostic Process Tomography

TL;DR

This work introduces agnostic process tomography, formalizing the problem of approximating an unknown quantum channel Φ by a member Φ′ of a tractable concept class C under noise. It builds on Pauli spectrum and Fourier analysis of superoperators to design efficient agnostic learning algorithms for Pauli strings, Pauli channels, junta channels, low-degree channels, QAC0 channels, and bounded-gate circuits, with a key extension from agnostic state tomography via ancilla to obtain unitary/CPTP projections where feasible. The authors also connect agnostic learning to practical scenarios such as resource-constrained implementation, approximate classical simulation, and error mitigation, and discuss converting agnostic state tomography results into agnostic process tomography guarantees for compatible classes. They provide both proper and improper learning guarantees, analyze time versus sample complexities, and outline major open questions, including robust CPTP projections and improved error-bounds, highlighting the framework’s potential to broaden learnability from realizable to agnostic settings in quantum information.

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 $Φ$ and a known concept class $\mathcal{C}$ of channels, output a quantum channel that approximates $Φ$ 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.

Figures (2)

• Figure 1: (Proper) Agnostic process tomography. (Left) Conventional tomography. In the realizable case, the unknown quantum process $\Phi$ is promised to fall in the class $\mathcal{C}$, and the goal is to find some $\Phi' \in \mathcal{C}$ that is close to $\Phi$. (Right) Agnostic tomography. Here, the unknown process $\Phi$ can be any completely positive trace-preserving (CPTP) map. The goal is to find some $\Phi' \in \mathcal{C}$ that is not far from the closest channel $\Phi^*$ to $\Phi$ in $\mathcal{C}$.
• Figure 2: Applications of agnostic process tomography.(a) Resource-constrained implementation. A client with limited computational resources has access to a quantum server, which can implement the client's desired complex quantum process $\mathcal{E}$. The client chooses a suitable concept class based on its available resources and provides it as input to an agnostic process tomography (APT) algorithm. The APT algorithm has access to $\mathcal{E}$ via the quantum server and can (potentially adaptively) query $\mathcal{E}$ on any input state. The APT algorithm then outputs the best approximate implementation $\hat{\mathcal{E}}$ of $\mathcal{E}$, which the client can implement on their local device. (b) Classical simulation. Consider a user with access to a quantum device, which can implement some quantum process $\mathcal{E}$, and the user wishes to classically simulate $\mathcal{E}$. Given the class of Clifford circuits (or any classically simulable class) and after querying $\mathcal{E}$ from the quantum device, the APT algorithm outputs a Clifford circuit $\hat{C}$ that approximates $\mathcal{E}$, which can then be classically simulated. (c) Quantum machine learning and metrology. Consider a class $\mathcal{C}_\theta = \{\mathcal{E}(\theta_i)\}_i$ of parameterized quantum processes, which naturally occur in quantum machine learning (QML), e.g, parameterized quantum circuits (PQC) and linear combination of unitaries (LCU), and quantum metrology. Running APT with respect to $\mathcal{C}_\theta$ and an appropriate access model (e.g., real-world data), one can use APT to find the unknown parameter $\theta^*$ corresponding to the QML model or metrology problem. (d) Error mitigation. Consider the parameterized class of noise channels $\{\mathcal{E}_\theta\}_\theta$ that can be mitigated via classical post-processing when $\theta$ is known. In addition, suppose we obtain noisy data $\hat{o}_{\mathrm{real}}$ by running classically-simulable circuits on our noisy device and ideal data $\hat{o}_{\mathrm{ideal}}$ by classically simulating the same circuits. Passing the concept class and data $\{(\hat{o}_{\mathrm{real}},\hat{o}_{\mathrm{ideal}} )\}$ into an APT algorithm, we learn the parameter $\theta$ that best models the noise. Then, on a new quantum circuit run on our noisy device, we can mitigate the noise via classical post-processing from our knowledge of $\theta$.

Theorems & Definitions (34)

• Definition 1: Agnostic process tomography
• Definition 2: Choi state of a unitary
• Definition 3: Choi state of a quantum channel
• Lemma 1: Properties of Choi representations of quantum channels watrous2018theory
• Definition 4: Pauli spectrum montanaro2008quantum
• Lemma 2: Proposition 10 in montanaro2008quantum
• Definition 5: Pauli weight at level $k$ montanaro2008quantumnadimpalli2024pauli
• Definition 6: Pauli weight on a subset
• Definition 7: Fourier expansion of superoperators; Definition 7 in bao2023testing
• Lemma 3: Properties of Fourier coefficients; Lemma 8 in bao2023testing