Quantum State Designs with Clifford Enhanced Matrix Product States

• URL: http://arxiv.org/abs/2404.18751v2
• Date: Tue, 08 Oct 2024 11:37:40 GMT
• Title: Quantum State Designs with Clifford Enhanced Matrix Product States
• Authors: Guglielmo Lami, Tobias Haug, Jacopo De Nardis,
• Abstract summary: Nonstabilizerness, or magic', is a critical quantum resource that characterizes the non-trivial complexity of quantum states. We show that Clifford enhanced Matrix Product States ($mathcalC$MPS) can approximate $4$-spherical designs with arbitrary accuracy.
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• Abstract: Nonstabilizerness, or `magic', is a critical quantum resource that, together with entanglement, characterizes the non-classical complexity of quantum states. Here, we address the problem of quantifying the average nonstabilizerness of random Matrix Product States (RMPS). RMPS represent a generalization of random product states featuring bounded entanglement that scales logarithmically with the bond dimension $\chi$. We demonstrate that the $2$-Stabilizer R\'enyi Entropy converges to that of Haar random states as $N/\chi^2$, where $N$ is the system size. This indicates that MPS with a modest bond dimension are as magical as generic states. Subsequently, we introduce the ensemble of Clifford enhanced Matrix Product States ($\mathcal{C}$MPS), built by the action of Clifford unitaries on RMPS. Leveraging our previous result, we show that $\mathcal{C}$MPS can approximate $4$-spherical designs with arbitrary accuracy. Specifically, for a constant $N$, $\mathcal{C}$MPS become close to $4$-designs with a scaling as $\chi^{-2}$. Our findings indicate that combining Clifford unitaries with polynomially complex tensor network states can generate highly non-trivial quantum states.

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