Classical shadows with Pauli-invariant unitary ensembles

Abstract

The classical shadow estimation protocol is a noise-resilient and sample-efficient quantum algorithm for learning the properties of quantum systems. Its performance depends on the choice of a unitary ensemble, which must be chosen by a user in advance. What is the weakest assumption that can be made on the chosen unitary ensemble that would still yield meaningful and interesting results? To address this question, we consider the class of Pauli-invariant unitary ensembles, i.e. unitary ensembles that are invariant under multiplication by a Pauli operator. This class includes many previously studied ensembles like the local and global Clifford ensembles as well as locally scrambled unitary ensembles. For this class of ensembles, we provide an explicit formula for the reconstruction map corresponding to the shadow channel and give explicit sample complexity bounds. In addition, we provide two applications of our results. Our first application is to locally scrambled unitary ensembles, where we give explicit formulas for the reconstruction map and sample complexity bounds that circumvent the need to solve an exponential-sized linear system. Our second application is to the classical shadow tomography of quantum channels with Pauli-invariant unitary ensembles. Our results pave the way for more efficient or robust protocols for predicting important properties of quantum states, such as their fidelity, entanglement entropy, and quantum Fisher information.

8+9 pages

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• Error-mitigated fermionic classical shadows on noisy quantum devices
• Demonstration of Robust and Efficient Quantum Property Learning with Shallow Shadows
• Machine learning on quantum experimental data toward solving quantum many-body problems
• Magic Resource Can Enhance the Quantum Capacity of Channels
• Classical shadows based on locally-entangled measurements
• Magic of Random Matrix Product States
• Optimising quantum tomography via shadow inversion
• Solving Fractional Differential Equations on a Quantum Computer: A Variational Approach
• Efficient Classical Shadow Tomography through Many-body Localization Dynamics
• Simulating quantum circuits with arbitrary local noise using Pauli Propagation
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• Classical shadows meet quantum optimal mass transport
• On the connection between least squares, regularization, and classical shadows
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• Holographic Classical Shadow Tomography
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• Decoded Quantum Interferometry Under Noise
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