Understanding quantum magic (i.e., nonstabilizerness) in many-body quantum systems is challenging but essential to the study of quantum computation and many-body physics. We investigate how noise affects magic properties in entangled many-body quantum states by quantitatively examining the magic decay under noise, with a primary aim being to understand the stability of magic associated with different kinds of entanglement structures. As a standard model, we study hypergraph states, a representative class of many-body magic states, subject to depolarizing noise. First, we show that interactions facilitated by high-degree gates are fragile to noise. In particular, the $\mathrm{C}^{n-1}Z$ state family exhibits a vanishing magic threshold of $Θ(1/n)$. Furthermore, we demonstrate efficiently preparable families of hypergraph states without local magic but with a non-vanishing magic threshold which signifies robust magic that is entirely embedded in global entanglement. We also discuss the qudit case based on the discrete Wigner formalism.
Understanding quantum magic (i.e., nonstabilizerness) in many-body quantum systems is challenging but essential to the study of quantum computation and many-body physics. We investigate how noise affects magic properties in entangled many-body quantum states by quantitatively examining the magic decay under noise, with a primary aim being to understand the stability of magic associated with different kinds of entanglement structures. As a standard model, we study hypergraph states, a representative class of many-body magic states, subject to depolarizing noise. First, we show that interactions facilitated by high-degree gates are fragile to noise. In particular, the Cn−1Z state family exhibits a vanishing magic threshold of Θ(1/n). Furthermore, we demonstrate efficiently preparable families of hypergraph states without local magic but with a non-vanishing magic threshold which signifies robust magic that is entirely embedded in global entanglement. We also discuss the qudit case based on the discrete Wigner formalism.
Magic (nonstabilizerness) is a necessary but "expensive" kind of "fuel" to drive universal fault-tolerant quantum computation.To properly study and characterize the origin of quantum "complexity" in computation as well as physics, it is crucial to develop a rigorous understanding of the quantification of magic.Previous studies of magic mostly focused on small systems and largely relied on the discrete Wigner formalism (which is only well behaved in odd prime power dimensions).Here we present an initiatory study of the magic of genuinely many-body quantum states that may be strongly entangled, with focus on the important case of many qubits, at a quantitative level.We first address the basic question of how "magical" a many-body state can be, and show that the maximum magic of an n-qubit state is essentially n, simultaneously for a range of "good" magic measures.As a corollary, the resource theory of magic has asymptotic golden currency states.We then show that, in fact, almost all n-qubit pure states have magic of nearly n.In the quest for explicit, scalable cases of highly entangled states whose magic can be understood, we connect the magic of hypergraph states with the second-order nonlinearity of their underlying Boolean functions.Next, we go on and investigate many-body magic in practical and physical contexts.We first consider a variant of measurement-based quantum computation where the client is restricted to Pauli measurements, in which magic is a necessary feature of the initial "resource" state.We show that n-qubit states with nearly n magic, or indeed almost all states, cannot supply nontrivial speedups over classical computers.We then present an example of analyzing the magic of "natural" condensed matter systems of physical interest.We apply the Boolean function techniques to derive explicit bounds on the magic of certain representative two-dimensional symmetry-protected topological states, and comment on possible further connections between magic and the quantum complexity of phases of matter.
Quantum hypergraph states extend the well-studied class of graph states by taking into account multi-qubit interactions through hyperedges. They provide a powerful framework to represent a family of quantum states with genuine multipartite entanglement. In this review, we provide a compact overview of the formal structure, entanglement characteristics, and operational relevance of hypergraph states in quantum information theory. We begin by introducing their mathematical foundations and generalizations of the stabilizer formalism. A central focus is placed on their entanglement properties, including the classification under local unitary (LU) and stochastic local operations with classical communication (SLOCC), the quantification of multipartite entanglement, and detection techniques via entanglement witnesses. We also explore other nonclassical features of hypergraph states, such as contextuality and genuine multipartite nonlocality, derived from stabilizer-based Bell-type inequalities. Additional attention is given to the role of hypergraph states in error correction, and as a computational resource in measurement-based quantum computation (MBQC), and to their non-stabilizer character - quantified via resource-theoretic measures of quantum magic. Finally we review their generalization to higher dimensions, i.e. to qudits and continuous variables.
Magic, or nonstabilizerness, characterizes the deviation of a quantum state from the set of stabilizer states, playing a fundamental role in quantum state complexity and universal fault-tolerant quantum computing. However, analytical and numerical characterizations of magic are very challenging, especially for multi-qubit systems, even with a moderate qubit number. Here, we systemically and analytically investigate the magic resource of archetypal multipartite quantum states—quantum hypergraph states, which can be generated by multi-qubit controlled-phase gates encoded by hypergraphs. We first derive the magic formula in terms of the stabilizer Rényi- α entropies for general quantum hypergraph states. If the average degree of the corresponding hypergraph is constant, we show that magic cannot reach the maximal value, i.e., the number of qubits n . Then, we investigate the statistical behaviors of random hypergraph states' magic and prove a concentration result, indicating that random hypergraph states typically reach the maximum magic. This also suggests an efficient way to generate maximal magic states with random diagonal circuits. Finally, we study hypergraph states with permutation symmetry, such as 3 -complete hypergraph states, where any three vertices are connected by a hyperedge. Counterintuitively, such states can only possess constant or even exponentially small magic for α≥2 . Our study advances the understanding of multipartite quantum magic and could lead to applications in quantum computing and quantum many-body physics.
Nonstabilizerness, or ‘magic’, is a crucial resource for quantum computation, but quantifying the magic of mixed states has been a notoriously difficult task. We introduce efficient magic witnesses based on stabilizer Rényi entropy that both robustly indicate magic and quantitatively estimate magic monotones. Building on these witnesses, we design testing algorithms that distinguish high- and low-magic states under entropy constraints and apply them to certify the number of noisy T-gates for a broad class of noise models. Using the IonQ quantum computer, we experimentally verify magic in noisy random circuits and find that magic remains robust, persisting even under depolarizing noise with probability exponentially close to one. Our witnesses are efficiently computable for matrix product states, showing that subsystems of many-body states can host extensive magic even when the system is entangled. Finally, we show that mimicking high-magic states with minimal magic requires an extensive amount of entropy, implying that entropy is a necessary cryptographic resource for hiding magic from eavesdroppers. Our results provide practical tools for characterizing the complexity of noisy quantum systems.
Magic, or nonstabilizerness, is the resource that lifts Clifford circuits to universal quantum computation and has become a standard diagnostic of many-body states. For a state shared between two parties, however, a basic question has remained open: how much of the magic resides in the correlations between the parties rather than in their local bases? Isolating this nonlocal magic requires minimizing over all local bases, an optimization that has so far resisted exact solution. Here we solve it for the stabilizer fidelity: the nonlocal magic of every pure multiqubit state is the distance of its entanglement spectrum from the closest spectrum of Bell pairs. The same quantity governs an apparently unrelated task: a family of states universally embezzles entanglement under local operations and classical communication if and only if its nonlocal magic diverges. The deciding property is not the amount of entanglement but the way the entanglement spectrum spreads its weight across factor-of-two windows of rank, so that critical chains and random-singlet states, with identical logarithmic entanglement scaling, carry unbounded and vanishing nonlocal magic, respectively. Nonlocal magic thereby becomes an operationally meaningful property of quantum correlations, directly accessible to tensor-network simulations and, through entanglement spectroscopy, to experiments.
Magic, a quantum resource distinct from entanglement, measures departure from states stabilized by products of Pauli operators. Such stabilizer states can be maximally entangled yet admit efficient classical description: quantum advantage requires magic. We develop a powerful framework to evaluate the degree of magic in generic many-body states. We use it to study the fate of magic under thermal and dynamical evolution. In the transverse field Ising model, we find a thermal sudden death of magic that saturates with system size, and occurs after the death of entanglement. We reach similar conclusions in generic spin chains, and in the Kitaev quantum spin liquid on the honeycomb lattice, where the threshold shows an intricate dependence on subregion shape. We then study how magic equilibrates after a quantum quench. In quenches of generic spin chains, the magic of a subregion typically dies before entanglement, contrary to thermal trajectories. An explanation is given in terms of the geometry of non-magical states. As a third type of trajectory, we examine the fate of magic with separation with an appropriate notion of non-local magic and establish a sudden death. Finally, we show how to experimentally witness a volume law of frame-robust magic in quantum materials by using neutron scattering.