Magic of the Heisenberg Picture

• URL: http://arxiv.org/abs/2408.16047v2
• Date: Tue, 24 Sep 2024 05:58:29 GMT
• Title: Magic of the Heisenberg Picture
• Authors: Neil Dowling, Pavel Kos, Xhek Turkeshi,
• Abstract summary: We study a non-stabilizerness resource theory for operators, which is dual to that describing states. We identify that the stabilizer R'enyi entropy analog in operator space is a good magic monotone satisfying the usual conditions. This monotone reveals structural properties of many-body magic generation, and can inspire Clifford-assisted tensor network methods.
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• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Magic quantifies the non-Clifford operations required for preparing a state on quantum processors and sets bounds on the classical computational complexity of simulating quantum dynamics. We study a non-stabilizerness resource theory for operators, which is dual to that describing states. We identify that the stabilizer R\'enyi entropy analog in operator space is a good magic monotone satisfying the usual conditions, while inheriting efficient computability properties and providing a tight lower-bound to the minimum number of non-Clifford gates in a circuit. It is operationally well-defined as quantifying how well one can approximate an operator with one that has only few Pauli strings; analogous to the relation between entanglement entropy and tensor-network truncation. An immediate advantage is that the operator stabilizer entropies exhibit inherent locality through a Lieb-Robinson bound, making them particularly suited for studying local dynamical magic generation in many-body systems. We compute this quantity analytically in two distinct regimes. First, we show that random evolution typically has approximately maximal magic in the Heisenberg picture for all R\'enyi indices, and evaluate the Page correction. Second, harnessing both dual unitarity and ZX graphical calculus, we compute the operator stabilizer entropy evolution for an interacting integrable XXZ circuit. In this case, magic quickly saturates to a constant. This monotone reveals structural properties of many-body magic generation, and can inspire Clifford-assisted tensor network methods.

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