We investigate the dynamics of nonstabilizerness—also known as “magic”—in monitored quantum circuits composed of random Clifford unitaries and local measurements. For measurements in the computational basis, we derive an analytical model for dynamics of the stabilizer nullity, showing that it decays in quantized steps and requires exponentially many measurements to vanish, which reveals the strong protection through Clifford scrambling. On the other hand, for measurements performed in rotated non-Clifford bases, measurements can both create and destroy nonstabilizerness. Here, the dynamics leads to a steady state with nontrivial nonstabilizerness, independent of the initial state. We find that Haar-random states equilibrate in constant time, whereas stabilizer states exhibit linear-in-size relaxation time. While the stabilizer nullity is insensitive to the rotation angle, stabilizer Rényi entropies expose a richer structure in their dynamics. Our results uncover sharp distinctions between coarse and fine-grained nonstabilizerness diagnostics and demonstrate how measurements can both suppress and sustain quantum computational resources.
We investigate the dynamics of nonstabilizerness—also known as “magic”—in monitored quantum circuits composed of random Clifford unitaries and local measurements. For measurements in the computational basis, we derive an analytical model for dynamics of the stabilizer nullity, showing that it decays in quantized steps and requires exponentially many measurements to vanish, which reveals the strong protection through Clifford scrambling. On the other hand, for measurements performed in rotated non-Clifford bases, measurements can both create and destroy nonstabilizerness. Here, the dynamics leads to a steady state with nontrivial nonstabilizerness, independent of the initial state. We find that Haar-random states equilibrate in constant time, whereas stabilizer states exhibit linear-in-size relaxation time. While the stabilizer nullity is insensitive to the rotation angle, stabilizer Rényi entropies expose a richer structure in their dynamics. Our results uncover sharp distinctions between coarse and fine-grained nonstabilizerness diagnostics and demonstrate how measurements can both suppress and sustain quantum computational resources.
Quantum magic, quantified by nonstabilizerness, measures departures from stabilizer structure and underlies potential quantum speedups. We introduce an efficient classical framework for computing stabilizer Rényi entropies and stabilizer nullity of generic N-qubit wavefunctions. The method combines the fast Walsh-Hadamard transform with an exact partition of Pauli operators, reducing the average cost per sampled Pauli string from O(2N) to O(N). We further develop a Monte Carlo estimator with Clifford preconditioning and find that the required number of samples shows no visible growth with N in our benchmarks. Applying the method to T-doped random Clifford circuits, we identify the scrambling ratio η (Clifford gates per T gate) as the key parameter governing magic growth. Each T gate approaches its dilute-limit nonstabilizerness power with only modest Clifford scrambling. Our approach enables quantitative studies of magic in highly entangled states and long-time nonequilibrium dynamics.
Nonstabilizerness or `magic' is a key resource for quantum computing and a necessary condition for quantum advantage. Non-Clifford operations turn stabilizer states into resourceful states, where the amount of nonstabilizerness is quantified by resource measures such as stabilizer Rényi entropies (SREs). Here, we show that SREs saturate their maximum value at a critical number of non-Clifford operations. Close to the critical point SREs show universal behavior. Remarkably, the derivative of the SRE crosses at the same point independent of the number of qubits and can be rescaled onto a single curve. We find that the critical point depends non-trivially on Rényi index α . For random Clifford circuits doped with T-gates, the critical T-gate density scales independently of α . In contrast, for random Hamiltonian evolution, the critical time scales linearly with qubit number for α > 1 , while it is a constant for α < 1 . This highlights that α -SREs reveal fundamentally different aspects of nonstabilizerness depending on α : α -SREs with α < 1 relate to Clifford simulation complexity, while α > 1 probe the distance to the closest stabilizer state and approximate state certification cost via Pauli measurements. As technical contributions, we observe that the Pauli spectrum of random evolution can be approximated by two highly concentrated peaks which allows us to compute its SRE. Further, we introduce a class of random evolution that can be expressed as random Clifford circuits and rotations, where we provide its exact SRE. Our results opens up new approaches to characterize the complexity of quantum systems.
Monitored quantum circuits combine scrambling with measurement-conditioned state updates, while non-Clifford perturbations inject magic into otherwise stabilizer-compatible dynamics. Rigorous results on stationary magic and its weak-injection asymptotics remain limited even for finite-dimensional Clifford-based monitored models.
We study an N-qudit process of prime local dimension d. Each cycle draws a fresh uniform global Clifford unitary, applies a local weak non-Clifford rotation and a projective measurement on one qudit, and then returns to the inverse Clifford frame. For every fixed injection strength, we prove that the induced pure-state Markov chain has a unique invariant probability measure and attracts every initial law geometrically in Wasserstein distance. At zero injection, the invariant law is supported on the finite stabilizer layer. After rescaling transverse deviations from this layer, the resulting blown-up stationary laws converge weakly to the invariant law of an affine tangent recursion. Combining this tangent law with Poisson representations and the first nonzero local resource germs determines the sharp vanishing rates of stationary magic. For every fixed N≥2, odd-prime Gross--Wigner mana admits a linear expansion with a strictly positive coefficient, whereas qubit 2-stabilizer Rényi entropy admits a quadratic expansion with a strictly positive coefficient. For N=1, both stationary resource averages vanish identically for all injection strengths. The distinct orders arise from the different local resource geometries, together with quadratic-order zero-reference branch contributions in qubit case.
Quantum magic resources, or nonstabilizerness, are a central ingredient for universal quantum computation. In noisy many-body systems, the interplay between these resources and errors leads to sharp magic phase transitions. However, the microscopic mechanism behind these critical phenomena is still an open question, especially since early empirical evidence showed conflicting results regarding their universality classes. In this work, we provide a comprehensive picture of magic phase transitions for the class of encoding-decoding quantum circuits to resolve these ambiguities. We analytically show that the behavior of magic resources is fundamentally dictated by the chosen measurement protocol. When we fix, or post-select, the class of measurement syndromes, the magic transition inherits the universal features of the error-resilience phase transition in the circuits. Interestingly, this clean transition survives even for fully random Haar encoders showing that it is a consequence of initial's state retrieval, and not an artifact of the Clifford encoders. On the other hand, if we consider realistic Born-rule sampling, the intrinsic statistical fluctuations of a given syndrome measurement act as a relevant perturbation. This brings in strong finite-size drifts and an apparent multifractality, which end up altering the critical behavior of the system. We reveal that magic phase transitions are actually direct manifestations of error-resilience thresholds, rather than independent critical phenomena, reconciling conflicting observations from the earlier literature. Ultimately, our framework clarifies how the quantum computational power can survive, or be irreversibly destroyed, due to the competition between scrambling, measurements, and errors.
Stabilizer entropies (SEs) are measures of nonstabilizerness or "magic" that quantify the degree to which a state is described by stabilizers. SEs are especially interesting due to their connections to scrambling, localization and property testing. However, applications have been limited so far as previously known measurement protocols for SEs scale exponentially with the number of qubits. Here, we efficiently measure SEs for integer Rényi index n>1 via Bell measurements. The SE of N-qubit quantum states can be measured with O(n) copies and O(nN) classical computational time, where for even n we additionally require the complex conjugate of the state. We provide efficient bounds of various nonstabilizerness monotones that are intractable to compute beyond a few qubits. Using the IonQ quantum computer, we measure SEs of random Clifford circuits doped with non-Clifford gates and give bounds for the stabilizer fidelity, stabilizer extent, and robustness of magic. We provide efficient algorithms to measure Clifford-averaged 4n-point out-of-time-order correlators and multifractal flatness. With these measures we study the scrambling time of doped Clifford circuits and random Hamiltonian evolution depending on nonstabilizerness. Counterintuitively, random Hamiltonian evolution becomes less scrambled at long times, which we reveal with the multifractal flatness. Our results open up the exploration of nonstabilizerness with quantum computers.
Nonstabilizerness, or magic, constitutes a fundamental resource for quantum computation and a crucial ingredient for quantum advantage. Recent progress has substantially advanced the characterization of magic in many-body quantum systems, with stabilizer Rényi entropy (SRE) emerging as a computable and experimentally accessible measure. In this work, we investigate the spreading of SRE in terms of single-qubit reduced density matrices, where an initial product state that contains magic in a local region evolves under brickwork random Clifford circuits. For the case with Haar-random local Clifford gates, we find that the spreading profile exhibits a diffusive structure within a ballistic light cone when viewed through a normalized version of single-qubit SRE, despite the absence of explicit conserved charges. We further examine the robustness of this non-ballistic behavior of the normalized single-qubit SRE spreading by extending the analysis to a restricted Clifford circuit, where we unveil a superdiffusive spreading. Finally, we discuss that a similar non-ballistic spreading within the light cone is found for another indicator of the magic, i.e., the robustness of magic.