Quantum State Designs with Clifford Enhanced Matrix Product States

TL;DR

The paper quantifies average nonstabilizerness (magic) in random Matrix Product States (RMPS) and introduces Clifford-enhanced RMPS (CMPS) to access Haar-like randomness with bounded entanglement. It derives precise χ-dependent scaling laws for Stabilizer Rényi Entropies under open and periodic boundary conditions, and shows RMPS can approach Haar averages for moments up to the 4th via CMPS, with deviations scaling as $N χ^{-2}$ (and higher orders). The authors compute explicit Haar averages for $m_2$ and $m_3$, analyze transfer-matrix spectra, and connect the 4-design quality to the χ-dependent magic of underlying RMPS through a tight frame-potential relation. This work demonstrates a controllable route from low-entanglement tensor networks to highly nontrivial, Haar-like quantum states, with potential implications for hybrid classical-quantum simulations and design of quantum resources.

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 $χ$. We demonstrate that the $2$-Stabilizer Rényi Entropy converges to that of Haar random states as $N/χ^2$, 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 $χ^{-2}$. Our findings indicate that combining Clifford unitaries with polynomially complex tensor network states can generate highly non-trivial quantum states.

Figures (5)

• Figure 1: $a)$ A random circuit comprising two local random unitary is considered. The initial state evolves as an MPS, with the bond dimension growing up to $\chi$. $b)$ Left: convergence of the magic with the MPS bond dimension ($N=22$ qubits, $500$ trajectories). Deviation of the $n$-SRE ($n=2$) from the averaged Haar value, $\delta_{\chi}^{(n)}$, is plotted over discrete circuit time $t$. (Inset: maximum entanglement entropy $S$ of the evolved MPS). Right: $\delta_{\chi}^{(n)}$ ($n=2,3$) for the final state compared with exact RMPS average obtained for the same size and $\chi$. Black lines represents power laws $\chi^{-2}$ and $\chi^{-3}$.
• Figure 2: Leading eigenvalues of the transfer matrix $\mathcal{T}$ in the bulk for $n=2$ (cirles) and $n=3$ (squares). Straight lines represent leading term for large $\chi$ extracted from analytical expansion ($n=2$) or from a linear fit ($n=3$).
• Figure 3: Magic of RMPS for periodic boundary conditions, see Eq. \ref{['eq:pbc_or_obc']}$a)$. Upper panels: magic deviation from Haar $\delta_{\chi}^{(n)}$ for RMPS at finite size $N$ and $n=2$ ($a$), $n=3$ ($b$). Here, we employ periodic boundary conditions, see Eq. \ref{['eq:pbc_or_obc']}$a)$. Dotted lines represent the linear fit in log log scale (Eq.\ref{['eq:linear_fit_pbc']}). Lower panels: coefficients $a_n$ as a function of $N$ for $n=2$ ($c$), $n=3$ ($d$).
• Figure 4: Magic of RMPS with open boundary conditions, see Eq. \ref{['eq:pbc_or_obc']}$b)$. Upper panels: magic deviation from Haar $\delta_{\chi}^{(n)}$ for RMPS at finite size $N$ and $n=2$ ($a$), $n=3$ ($b$). Dotted lines represent the linear fit in log log scale (Eq.\ref{['eq:linear_fit_obc']}). Lower panels: coefficients $b_n$ as a function of $N$ for $n=2$ ($c$), $n=3$ ($d$).
• Figure 5: Entanglement cooling is applied to doped circuit (see Eq. \ref{['eq:scheme_cooling']}). Continuous lines represent entanglement of the final state $\ket{\psi}$, dotted lines the entanglement of the Clifford disentangled state $\mathcal{U}_c^{\dag}\ket{\psi}$.