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arXiv:2606.12035

Shadow Engineering of Quantum Processes

Tian-Ci Tian, De-Tao Jiang, Wei-Ming Zhu, Wei-You Liao, Hong-Wei Li, He-Liang Huang

quant-phSubmitted Jun 10, 2026• Announced Jun 11, 2026• 0 citations
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Abstract

Characterizing quantum processes is essential for hardware benchmarking, error diagnosis, and algorithm verification. While recent work [PRX QUANTUM \textbf{4}, 040337 (2023)] extended classical shadows from quantum state to quantum process, enabling efficient single-channel $\mathcal{E}$ property prediction, its applicability to composite processes $f(\mathcal{E}_1, \mathcal{E}_2,\cdots, \mathcal{E}_k)$ remains unexplored. We introduce shadow engineering, a framework encoding the classical shadows of processes into sparse transfer matrices to predict $f(\mathcal{E}_1, \mathcal{E}_2,\cdots, \mathcal{E}_k)$ properties with proven polynomial sample complexity, matching single-channel efficiency while exponentially lower than quantum process tomography. Crucially, this approach repurposes existing $\mathcal{E}_m$-shadow data without physical execution of $f(\mathcal{E}_1, \mathcal{E}_2,\cdots, \mathcal{E}_k)$, enabling flexible quantum process characterization with minimal hardware overhead. We demonstrate the framework's effectiveness and practicality on a superconducting quantum processor for typical applications such as error mitigation and Hamiltonian dynamical simulation. This framework unlocks new capabilities for predicting complex quantum behaviors without physical re-execution, with immediate applications in near-term device calibration and quantum simulation.

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This thesis presents developments in the theory and application of quantum learning.The first two chapters fall under the head of classical shadow tomography, a randomized measurement technique which estimates a large number of expectation values effectively in parallel. In the first chapter on this topic, we extend classical shadow state tomography to the characterization of quantum channels. We prove that with a simple protocol involving randomized measurement of randomly generated input states evolved through channel, one can estimate many channel properties in parallel, with rigorous guarantees on the sample complexity required. In the next chapter, we bring together finite-depth classical shadows, derandomization, and tensor network simulation to present an algorithm for tailoring a sequence of measurements to a set of observables of interest, demonstrating the effectiveness of our technique for a number of tasks in quantum chemistry and simulation. The second part of this thesis presents a step toward applying ideas from quantum learning to quantum error correction. We investigate the use of prior information on qubit error rates in the decoding task, showing that prior information can significantly change decoder performance when a small number of qubits have high error rates. We present a protocol to learn the location of error-prone sites from decoding data in real time, and then apply this in turn to develop in situ protocol for calibration of gates while processing logical information.

•Jan 1, 2025•0 citations
Characterizing quantum processes is essential for hardware benchmarking, error diagnosis, and algorithm verification. While recent work [PRX QUANTUM \textbf{4}, 040337 (2023)] extended classical shadows from quantum state to quantum process, enabling efficient single-channel E\mathcal{E}E property prediction, its applicability to composite processes f(E1,E2,⋯ ,Ek)f(\mathcal{E}_1, \mathcal{E}_2,\cdots, \mathcal{E}_k)f(E1​,E2​,⋯,Ek​) remains unexplored. We introduce shadow engineering, a framework encoding the classical shadows of processes into sparse transfer matrices to predict f(E1,E2,⋯ ,Ek)f(\mathcal{E}_1, \mathcal{E}_2,\cdots, \mathcal{E}_k)f(E1​,E2​,⋯,Ek​) properties with proven polynomial sample complexity, matching single-channel efficiency while exponentially lower than quantum process tomography. Crucially, this approach repurposes existing Em\mathcal{E}_mEm​-shadow data without physical execution of f(E1,E2,⋯ ,Ek)f(\mathcal{E}_1, \mathcal{E}_2,\cdots, \mathcal{E}_k)f(E1​,E2​,⋯,Ek​), enabling flexible quantum process characterization with minimal hardware overhead. We demonstrate the framework's effectiveness and practicality on a superconducting quantum processor for typical applications such as error mitigation and Hamiltonian dynamical simulation. This framework unlocks new capabilities for predicting complex quantum behaviors without physical re-execution, with immediate applications in near-term device calibration and quantum simulation.
Classical shadows enable us to learn many properties of a quantum state ρ with very few measurements. However, near-term and early fault-tolerant quantum computers will only be able to prepare noisy quantum states ρ and it is thus a considerable challenge to efficiently learn properties of an ideal, noise-free state ρ id . We consider error mitigation techniques, such as probabilistic error cancelation (PEC), zero noise extrapolation (ZNE), and symmetry verification (SV), which have been developed for mitigating errors in single expected value measurements and generalize them for mitigating errors in classical shadows. We find that PEC is the most natural candidate and thus develop a thorough theoretical framework for PEC shadows with the following rigorous theoretical guarantees: PEC shadows are an unbiased estimator for the ideal quantum state ρ id ; the sample complexity for simultaneously predicting many linear properties of ρ id is identical to that of the conventional shadows approach up to a multiplicative factor, which is the sample overhead due to error mitigation. Due to efficient postprocessing of shadows, this overhead does not depend directly on the number of qubits but rather grows exponentially with the number of noisy gates. The broad set of tools introduced in this work may be instrumental in exploiting near-term and early fault-tolerant quantum computers: we demonstrate in detailed numerical simulations a range of practical applications of quantum computers that will significantly benefit from our techniques. Published by the American Physical Society 2024
Extracting information efficiently from quantum systems is crucial for quantum information processing. Classical shadows enable predicting many properties of arbitrary quantum states using few measurements. While random single-qubit measurements are experimentally friendly and suitable for learning low-weight Pauli observables, they perform poorly for nonlocal observables. Introducing a shallow random quantum circuit before measurements improves sample efficiency for high-weight Pauli observables and low-rank properties. However, in practice, these circuits can be noisy and bias the measurement results. Here, we propose the robust shallow shadows, which employs Bayesian inference to learn and mitigate noise in postprocessing. We analyze noise effects on sample complexity and the optimal circuit depth. We provide theoretical guarantees for the success of error mitigation under a wide class of noise processes. Experimental validation on a superconducting quantum processor confirms the advantage of our method, even in the presence of realistic noise, over single-qubit measurements for predicting diverse state properties, such as fidelity and entanglement entropy. Our protocol thus offers a scalable, robust, and sample-efficient method for quantum state characterization on near-term quantum devices. Fast and reliable characterisation of quantum systems is a key part of quantum technologies development. Here, the authors propose and demonstrate a way to embed noise characterisation in classical shadow estimation of nonlocal properties, enabling an efficient way to extract information from noisy quantum systems.
Quantum process tomography is a critical capability for building quantum computers, enabling quantum networks, and understanding quantum sensors. Like quantum state tomography, the process tomography of an arbitrary quantum channel requires a number of measurements that scales exponentially in the number of quantum bits affected. However, the recent field of shadow tomography, applied to quantum states, has demonstrated the ability to extract key information about a state with only polynomially many measurements. In this work, we apply the concepts of shadow state tomography to the challenge of characterizing quantum processes. We make use of the Choi isomorphism to directly apply rigorous bounds from shadow state tomography to shadow process tomography, and we find additional bounds on the number of measurements that are unique to process tomography. Our results, which include algorithms for implementing shadow process tomography, enable new techniques including evaluation of channel concatenation and the application of channels to shadows of quantum states. This provides a dramatic improvement for understanding large-scale quantum systems.
Accurate noise classification is essential for operating near-term quantum processors, yet existing approaches, such as quantum process tomography, scale exponentially with system size, limiting their practicality for routine calibration. We propose a measurement-efficient noise fingerprinting pipeline that combines structured classical shadow tomography with physics-informed feature engineering to identify noise channels from a fixed set of 3-qubit probe circuits. Each sample is represented by a feature vector constructed from randomized Pauli measurements and derived observables designed to resolve physically similar noise channels that produce overlapping signatures under generic measurement sets. We evaluate random forest, extra trees, and a multilayer perceptron on 10,000 labeled samples spanning ten noise models. The three classifiers achieve comparable performance. In the reported runs, random forest and extra trees perform similarly, achieving approximately 0.736 test accuracy and 0.729-0.730 macro F1, compared with 0.715 accuracy and 0.699 macro F1 for the multilayer perceptron. We further analyze the effect of the noise-strength sampling range and conduct a limited sensitivity check using analogous 2- and 4-qubit probes. Confusion analysis shows that readout error, phase flip, thermal relaxation, and bit flip are classified with high reliability, while most remaining errors occur among channels with similar physical effects.
Quantum process tomography is a powerful tool for understanding quantum channels and characterizing the properties of quantum devices. Inspired by recent advances using classical shadows in quantum state tomography [H.-Y. Huang, R. Kueng, and J. Preskill, .], we have developed ShadowQPT, a classical shadow method for quantum process tomography. We introduce two related formulations with and without ancilla qubits. ShadowQPT stochastically reconstructs the Choi matrix of the device allowing for an classical evaluation of the device on arbitrary inputs with respect to arbitrary outputs. Using shadows, we then show how to compute overlaps, generate all k -weight reduced processes, and perform reconstruction via Hamiltonian learning. These latter two tasks are efficient for large systems as the number of quantum measurements needed scales only logarithmically with the number of qubits. A number of additional approximations and improvements are developed, including the use of a pair-factorized Clifford shadow and a series of postprocessing techniques that significantly enhance the accuracy for recovering the quantum channel. We have implemented ShadowQPT using both Pauli and Clifford measurements on the IonQ trapped ion quantum computer for quantum processes up to n=4 qubits, and we achieved good performance. Published by the American Physical Society 2024
This thesis presents developments in the theory and application of quantum learning.The first two chapters fall under the head of classical shadow tomography, a randomized measurement technique which estimates a large number of expectation values effectively in parallel. In the first chapter on this topic, we extend classical shadow state tomography to the characterization of quantum channels. We prove that with a simple protocol involving randomized measurement of randomly generated input states evolved through channel, one can estimate many channel properties in parallel, with rigorous guarantees on the sample complexity required. In the next chapter, we bring together finite-depth classical shadows, derandomization, and tensor network simulation to present an algorithm for tailoring a sequence of measurements to a set of observables of interest, demonstrating the effectiveness of our technique for a number of tasks in quantum chemistry and simulation. The second part of this thesis presents a step toward applying ideas from quantum learning to quantum error correction. We investigate the use of prior information on qubit error rates in the decoding task, showing that prior information can significantly change decoder performance when a small number of qubits have high error rates. We present a protocol to learn the location of error-prone sites from decoding data in real time, and then apply this in turn to develop in situ protocol for calibration of gates while processing logical information.