Quantum State Designs with Clifford Enhanced Matrix Product States

TL;DR

This paper investigates the nonstabilizerness of random Matrix Product States and introduces Clifford enhanced MPS, showing they can approximate complex quantum state designs with modest bond dimensions, thus generating highly non-trivial states.

Contribution

The paper demonstrates that Clifford enhanced MPS can efficiently approximate 4-designs, revealing a new method to generate complex quantum states with bounded entanglement.

Findings

RMPS with modest bond dimension are as magical as Haar random states.

Clifford enhanced MPS can approximate 4-designs with polynomial bond dimension scaling.

Combining Clifford unitaries with tensor networks produces highly non-trivial quantum states.

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.