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arXiv:2404.18768 · Physical review. B./Physical review. B

Nonstabilizerness versus entanglement in matrix product states

M. Frau, Poetri Sonya Tarabunga, Mario Collura, Marcello Dalmonte, Emanuele Tirrito

Published Jul 1, 2024• 54 citations
arXivHTMLPDF

Abstract

In this paper, we investigate the relationship between entanglement and nonstabilizerness (also known as magic) in matrix product states (MPSs). We study the relation between magic and the bond dimension used to approximate the ground state of a many-body system in two different contexts: full state of magic and mutual magic (the nonstabilizer analog of mutual information, thus free of boundary effects) of spin-1 anisotropic Heisenberg chains. Our results indicate that obtaining converged results for nonstabilizerness is typically considerably easier than entanglement. For full state magic at critical points and at sufficiently large volumes, we observe convergence with $1/{\ensuremath{\chi}}^{2}$, with $\ensuremath{\chi}$ being the MPS bond dimension. At small volumes, magic saturation is so quick that, within error bars, we cannot appreciate any finite-$\ensuremath{\chi}$ correction. Mutual magic also shows a fast convergence with bond dimension, whose specific functional form is however hindered by sampling errors. As a byproduct of our study, we show how Pauli-Markov chains (originally formulated to evaluate magic) resets the state of the art in terms of computing mutual information for MPS. We illustrate this last fact by verifying the logarithmic increase of mutual information between connected partitions at critical points. By comparing mutual information and mutual magic, we observe that, for connected partitions, the latter is typically scaling much slower---if at all---with the partition size, while for disconnected partitions, both are constant in size.

DOI
10.1103/physrevb.110.045101

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In this paper, we investigate the relationship between entanglement and nonstabilizerness (also known as magic) in matrix product states (MPSs). We study the relation between magic and the bond dimension used to approximate the ground state of a many-body system in two different contexts: full state of magic and mutual magic (the nonstabilizer analog of mutual information, thus free of boundary effects) of spin-1 anisotropic Heisenberg chains. Our results indicate that obtaining converged results for nonstabilizerness is typically considerably easier than entanglement. For full state magic at critical points and at sufficiently large volumes, we observe convergence with 1/\ensuremathχ21/{\ensuremath{\chi}}^{2}1/\ensuremathχ2, with \ensuremathχ\ensuremath{\chi}\ensuremathχ being the MPS bond dimension. At small volumes, magic saturation is so quick that, within error bars, we cannot appreciate any finite-\ensuremathχ\ensuremath{\chi}\ensuremathχ correction. Mutual magic also shows a fast convergence with bond dimension, whose specific functional form is however hindered by sampling errors. As a byproduct of our study, we show how Pauli-Markov chains (originally formulated to evaluate magic) resets the state of the art in terms of computing mutual information for MPS. We illustrate this last fact by verifying the logarithmic increase of mutual information between connected partitions at critical points. By comparing mutual information and mutual magic, we observe that, for connected partitions, the latter is typically scaling much slower---if at all---with the partition size, while for disconnected partitions, both are constant in size.
Nonstabilizerness, also known as magic, quantifies the number of non-Clifford operations needed to prepare a quantum state. As typical measures either involve minimization procedures or a computational cost exponential in the number of qubits NNN, it is notoriously hard to characterize for many-body states. In this paper, we show that nonstabilizerness, as quantified by the recently introduced stabilizer R\'enyi entropies (SREs), can be computed efficiently for matrix product states (MPSs). Specifically, given an MPS of bond dimension $\ensuremath{\chi}$ and integer R\'enyi index n>1n>1n>1, we show that the SRE can be expressed in terms of the norm of an MPS with bond dimension ${\ensuremath{\chi}}^{2n}$. For translation-invariant states, this allows us to extract it from a single tensor, the transfer matrix, while for generic MPSs this construction yields a computational cost linear in NNN and polynomial in $\ensuremath{\chi}$. We exploit this observation to revisit the study of ground-state nonstabilizerness in the quantum Ising chain, providing accurate numerical results up to large system sizes. We analyze the SRE near criticality and investigate its dependence on the local computational basis, showing that it is, in general, not maximal at the critical point.
Stabilizer Rényi entropies (SREs) probe the non-stabilizerness (or “magic”) of many-body systems and quantum computers. Here, we introduce the mutual von-Neumann SRE and magic capacity, which can be efficiently computed in time O(N\chi^3) O ( N χ 3 ) for matrix product states (MPSs) of bond dimension \chi χ . We find that mutual SRE characterizes the critical point of ground states of the transverse-field Ising model, independently of the chosen local basis. Then, we relate the magic capacity to the anti-flatness of the Pauli spectrum, which quantifies the complexity of computing SREs. The magic capacity characterizes transitions in the ground state of the Heisenberg and Ising model, randomness of Clifford+T circuits, and distinguishes typical and atypical states. Finally, we make progress on numerical techniques: we design two improved Monte-Carlo algorithms to compute the mutual 2 2 -SRE, overcoming limitations of previous approaches based on local update. We also give improved statevector simulation methods for Bell sampling and SREs with O(8^{N/2}) O ( 8 N / 2 ) time and O(2^N) O ( 2 N ) memory, which we demonstrate for 24 24 qubits. Our work uncovers improved approaches to study the complexity of quantum many-body systems.
Nonstabilizerness, also known as "magic," stands as a crucial resource for achieving a potential advantage in quantum computing. Its connection to many-body physical phenomena is poorly understood at present, mostly due to a lack of practical methods to compute it at large scales. We present a novel approach for the evaluation of nonstabilizerness within the framework of matrix product states (MPSs), based on expressing the MPS directly in the Pauli basis. Our framework provides a powerful tool for efficiently calculating various measures of nonstabilizerness, including stabilizer Rényi entropies, stabilizer nullity, and Bell magic, and enables the learning of the stabilizer group of an MPS. We showcase the efficacy and versatility of our method in the ground states of Ising and XXZ spin chains, as well as in circuits dynamics that has recently been realized in Rydberg atom arrays, where we provide concrete benchmarks for future experiments on logical qubits up to twice the sizes already realized.
Nonstabilizerness, or “magic,” is a critical quantum resource that, together with entanglement, characterizes the nonclassical complexity of quantum states. Here, we address the problem of quantifying the average nonstabilizerness of random matrix product states (RMPSs). RMPSs represent a generalization of random product states featuring bounded entanglement that scales logarithmically with the bond dimension χ . We demonstrate that the stabilizer Rényi entropies converge to that of Haar-random states as N / χ α , where N is the system size and the α are integer exponents. This indicates that MPSs with a modest bond dimension are as magical as generic states. Subsequently, we introduce the ensemble of Clifford-enhanced matrix product states ( C MP Ss ), built by the action of Clifford unitaries on RMPSs. Leveraging our previous result, we show that C MP Ss can approximate quantum state 4-designs with arbitrary accuracy. Specifically, for a constant N , C MP Ss 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 nontrivial quantum states.
We introduce a novel approach to evaluate the nonstabilizerness of an N-qubits matrix product state (MPS) with bond dimension χ. In particular, we consider the recently introduced stabilizer Rényi entropies (SREs). We show that the exponentially hard evaluation of the SREs can be achieved by means of a simple perfect sampling of the many-body wave function over the Pauli string configurations. The sampling is achieved with a novel MPS technique, which enables us to compute each sample in an efficient way with a computational cost O(Nχ^{3}). We benchmark our method over randomly generated magic states, as well as in the ground-state of the quantum Ising chain. Exploiting the extremely favorable scaling, we easily have access to the nonequilibrium dynamics of the SREs after a quantum quench.
We study the behavior of magic as a bipartite correlation in the quantum Ising chain across its quantum phase transition and at finite temperature. To quantify the magic of partitions rigorously, we formulate a hybrid scheme that combines stochastic sampling of reduced density matrices via quantum Monte Carlo, with state-of-the-art estimators for the robustness of magic - a bona fide measure of magic for mixed states. This allows us to compute the mutual robustness of magic for partitions up to eight sites, embedded into a much larger system. We show how mutual robustness is directly related to critical behaviors: at the critical point, it displays a power-law decay as a function of the distance between partitions, whose exponent is related to the partition size. Once finite temperature is included, mutual magic retains its low temperature value up to an effective critical temperature, whose dependence on size is also algebraic. This suggests that magic, differently from entanglement, does not necessarily undergo a sudden death.