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 R & eacute,.nyi 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.

We introduce a nonstabilizerness monotone which we name basis-minimized stabilizerness asymmetry (BMSA). It is based on the notion of G-asymmetry, a measure of how much a certain state deviates from being symmetric with respect to a symmetry group G. For pure states, we show that the BMSA is a strong monotone for magic-state resource theory, while it can be extended to mixed states via the convex roof construction. We discuss its relation with other magic monotones, first showing that the BMSA coincides with the recently introduced basis-minimized measurement entropy, thereby establishing the strong monotonicity of the latter. Next, we provide inequalities between the BMSA and other nonstabilizerness measures such as the robustness of magic, stabilizer extent, stabilizer rank, stabilizer fidelity and stabilizer R & eacute,.nyi entropy. We also prove that the stabilizer fidelity, stabilizer R & eacute,.nyi entropy and BMSA with index alpha >= 2 have the same asymptotic scaling with qubit number. Finally, we present numerical methods to compute the BMSA, highlighting its advantages and drawbacks compared to other nonstabilizerness measures in the context of pure many-body quantum states. We also discuss the importance of additivity and strong monotonicity for measures of nonstabilizerness in many-body physics, motivating the search for additional computable nonstabilizerness monotones.

Quantum Monte Carlo (QMC) methods are essential for the numerical study of large-scale quantum many-body systems, yet their utility has been significantly hampered by the difficulty in computing key quantities such as off-diagonal operators and entanglement. This Letter introduces Bell-QMC, a novel QMC framework leveraging Bell sampling, a two-copy measurement protocol in the transversal Bell basis. We demonstrate that Bell-QMC enables an efficient and unbiased estimation of both challenging classes of observables, offering a significant advantage over previous QMC approaches. Notably, the entanglement across all system partitions can be computed in a single Bell-QMC simulation. We implement this method within the stochastic series expansion, where we design an efficient update scheme for sampling the configurations in the Bell basis. We demonstrate our algorithm in the one-dimensional transverse-field Ising model and the two-dimensional Z2 lattice gauge theory, extracting universal quantum features using only simple diagonal measurements. This Letter establishes Bell-QMC as a powerful framework that significantly expands the accessible quantum properties in QMC simulations, providing a substantial advantage over conventional QMC.

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.

"Stabilizer R & eacute,.nyi entropies (SREs) probe the non-stabilizerness (or ""magic"") of manybody 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) for matrix product states (MPSs) of bond dimension chi. 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-SRE, overcoming limitations of previous approaches based on local update. We also give improved statevector simulation methods for Bell sampling and SREs with O(8N/2) time and O(2N) memory, which we demonstrate for 24 qubits. Our work uncovers improved approaches to study the complexity of quantum many-body systems."

Understanding how entanglement can be reduced through simple operations is crucial for both classical and quantum algorithms. We investigate the entanglement properties of lattice models hosting conformal field theories cooled via local Clifford operations, a procedure we refer to as stabilizer disentangling. We uncover two distinct regimes: a constant gain regime, where disentangling is volume-independent, and a log-gain regime, where disentanglement increases with volume, characterized by a reduced effective central charge. In both cases, disentangling efficiency correlates with the target state magic, with larger magic leading to more effective cooling. The dichotomy between the two cases stems from mutual stabilizer R & eacute,.nyi entropy, which influences the entanglement cooling process. We provide an analytical understanding of such effect in the context of cluster Ising models, that feature disentangling global Clifford operations. Our findings indicate that matrix product states possess subclasses based on the relationship between entanglement and magic, and clarifying the potential of new classes of variational states embedding Clifford dynamics within matrix product states.

"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 R & eacute,.nyi 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."