Reducing Complexity of Shadow Process Tomography with Generalized Measurements

TL;DR

This paper introduces a generalized shadow process tomography framework using POVMs and convex optimization to minimize the shadow norm, significantly reducing sample complexity in quantum process characterization, especially for large systems.

Contribution

The paper proposes POVM-based shadow process tomography with convex optimization to identify optimal measurements, reducing complexity beyond traditional unitary-based methods.

Findings

Achieves approximately 7-fold reduction in shadow norm for single-qubit states.

Demonstrates a $2^{180}$-fold improvement for 64-qubit states.

Provides algorithms for optimal POVM identification via numerical simulations.

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.