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arXiv:2506.23806 · Physical Review A

Reducing complexity of shadow process tomography with generalized measurements

Haitao Wang, Kan He

Published Jun 23, 2025• 0 citations
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Abstract

Quantum process tomography (QPT) is crucial for advancing quantum technologies, including quantum computers, quantum networks and quantum sensors. Shadow process tomography (SPT) utilizes the Choi isomorphism to map QPT to shadow state tomography (SST), significantly reducing the sample complexity for extracting information from quantum processes. However, SPT relies on random unitary operators and complicates the determination of the optimal unitary operator that minimizes the shadow norm, which is the key factor influencing the sample complexity. In this work, we propose a generalized SPT framework that minimizes the shadow norm by replacing unitary operators with generalized measurements (POVMs). This approach, termed shadow process tomography with POVMs (POVM-SPT), uses convex optimization to identify the optimal POVM for minimizing the shadow norm, thereby further reducing sample complexity. We demonstrate the identification of the optimal POVM through numerical simulations and provide the corresponding optimization algorithms. Our numerical experiments demonstrate that POVM-SPT achieves a substantial reduction in shadow norm compared to conventional SPT, with an approximate 7-fold improvement for single-qubit input states and a remarkable $2^{180}$-fold enhancement for 64-qubit input states. These results reveal that POVM-SPT offers significant advantages in simplifying SPT tasks, particularly for large-scale quantum systems.

DOI
10.1103/13ct-8vtv

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Quantum process tomography (QPT) is crucial for advancing quantum technologies, including quantum computers, quantum networks and quantum sensors. Shadow process tomography (SPT) utilizes the Choi isomorphism to map QPT to shadow state tomography (SST), significantly reducing the sample complexity for extracting information from quantum processes. However, SPT relies on random unitary operators and complicates the determination of the optimal unitary operator that minimizes the shadow norm, which is the key factor influencing the sample complexity. In this work, we propose a generalized SPT framework that minimizes the shadow norm by replacing unitary operators with generalized measurements (POVMs). This approach, termed shadow process tomography with POVMs (POVM-SPT), uses convex optimization to identify the optimal POVM for minimizing the shadow norm, thereby further reducing sample complexity. We demonstrate the identification of the optimal POVM through numerical simulations and provide the corresponding optimization algorithms. Our numerical experiments demonstrate that POVM-SPT achieves a substantial reduction in shadow norm compared to conventional SPT, with an approximate 7-fold improvement for single-qubit input states and a remarkable 21802^{180}2180-fold enhancement for 64-qubit input states. These results reveal that POVM-SPT offers significant advantages in simplifying SPT tasks, particularly for large-scale 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.
Advances in quantum technology require scalable techniques to efficiently extract information from a quantum system. Traditional tomography is limited to a handful of qubits, and shadow tomography has been suggested as a scalable replacement for larger systems. Shadow tomography is conventionally analyzed based on outcomes of ideal projective measurements on the system upon application of randomized unitaries. Here, we suggest that shadow tomography can be much more straightforwardly formulated for generalized measurements, or positive operator valued measures. Based on the idea of the least-square estimator shadow tomography with generalized measurements is both more general and simpler than the traditional formulation with randomization of unitaries. In particular, this formulation allows us to analyze theoretical aspects of shadow tomography in detail. For example, we provide a detailed study of the implication of symmetries in shadow tomography. Moreover, with this generalization we also demonstrate how the optimization of measurements for shadow tomography tailored toward a particular set of observables can be carried out.
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
The rapid development of quantum technology demands efficient characterization of complex quantum many-body states. However, full quantum state tomography requires an exponential number of measurements in system size, preventing its practical use in large-scale quantum devices. A major recent breakthrough in this direction, called classical shadow tomography, significantly reduces the sample complexity, the number of samples needed to estimate properties of a state, by implementing random Clifford rotations before measurements. Despite many recent efforts, reducing the sample complexity below 2 k for extracting any non-successive local operators with a size ~ k remains a challenge. In this work, we achieve a significantly smaller sample complexity of ~ 1.8 k using a protocol that hybridizes locally random and globally deterministic unitary operations. The key insight is the discovery of a deterministic global unitary, termed as contractive unitary , which is more efficient in reducing the operator size to enhance tomography efficiency. The contractive unitary perfectly matches the advantages of the atom array quantum computation platform and is readily realized in the atom array quantum processor. More importantly, it highlights a new strategy in classical shadow tomography, demonstrating that a random-deterministic hybridized protocol can be more efficient than fully random measurements.
Informationally overcomplete POVMs are known to outperform minimally complete measurements in many tomography and estimation tasks, and they also leave a purely classical freedom in shadow tomography: the same observable admits infinitely many unbiased linear reconstructions from identical measurement data. We formulate the choice of reconstruction coefficients as a convex minimax problem and give an algorithm with guaranteed convergence that returns the tightest state-independent variance bound achievable by post-processing for a fixed POVM and observable. Numerical examples show that the resulting estimators can dramatically reduce sampling complexity relative to standard (canonical) reconstructions, and can even improve the qualitative scaling with system size for structured noncommuting targets.
Shadow tomography is a promising way for quantum system characterization, enabling the extraction of multiple properties of unknown quantum states using only single-copy randomized measurements. While single-copy measurements are experimentally friendly and efficient for estimating linear functions, they show limited efficiency for nonlinear ones. In contrast, multicopy measurements enabled by joint entangling operations can remarkably reduce the sample complexity for nonlinear functions. Here, we design and realize a deterministic optical three-qubit Fredkin gate by comprehensive control over multiple degrees of freedom of a single photon, achieving controlled-swap operation with high process fidelity of $0.935\ifmmode\pm\else\textpm\fi{}0.001$. Utilizing this Fredkin gate, we demonstrate hybrid shadow tomography, in which the higher-order functions like the moments can be efficiently estimated from the two-copy swap operation in conjunction with the single-copy randomized measurement. Our demonstration indicates a significant reduction in sample complexity compared with the single-copy setting. Furthermore, we show the potential advantage of our scheme in a proof-of-principle quantum metrology experiment, wherein the accuracy of parameter estimation is enhanced with the assistance of virtual distillation. Our results suggest that hybrid shadow tomography is an efficient and powerful tool for shadow-estimation-enabled quantum information processing.