Quantum magic, or non-stabilizerness, is an important quantum resource that characterizes computational power beyond classically simulable Clifford operations and is therefore essential for achieving quantum advantage. While previous studies have explored non-stabilizerness dynamics in random circuits and under time-independent generators, here we extend the study of its universal dynamics to time-dependent driving across quantum phase transitions. In particular, we show that the stabilizer Rényi entropies and the cumulants of the Pauli spectrum exhibit universal power-law scaling with the driving rate in slow processes. Moreover, we show that the logarithmic Pauli spectrum is asymptotically Gaussian, implying a lognormal distribution for the Pauli spectrum values. Our results are explicitly demonstrated by exact results in the transverse-field Ising model and by analytical approximations in long-range Kitaev models.
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5 figures in main text, 8 figures in supplementary
Quantum magic, or non-stabilizerness, is an important quantum resource that characterizes computational power beyond classically simulable Clifford operations and is therefore essential for achieving quantum advantage. While previous studies have explored non-stabilizerness dynamics in random circuits and under time-independent generators, here we extend the study of its universal dynamics to time-dependent driving across quantum phase transitions. In particular, we show that the stabilizer Rényi entropies and the cumulants of the Pauli spectrum exhibit universal power-law scaling with the driving rate in slow processes. Moreover, we show that the logarithmic Pauli spectrum is asymptotically Gaussian, implying a lognormal distribution for the Pauli spectrum values. Our results are explicitly demonstrated by exact results in the transverse-field Ising model and by analytical approximations in long-range Kitaev models.
Stabilizer states constitute an important class of quantum states that can be generated from computational-basis states using Pauli operators and Clifford gates. Although they may exhibit substantial multipartite entanglement, quantum circuits restricted to stabilizer operations can be efficiently simulated classically and therefore cannot, by themselves, provide a quantum computational advantage. Such an advantage requires non-stabilizer resources, commonly referred to as quantum magic. In this paper, we investigate the non-stabilizerness of quantum states arising in a class of spin Hamiltonians by computing their stabilizer Rényi entropies. Using real-space renormalization-group techniques, we obtain a closed-form expression valid in the low-energy, large-distance regime. We analyze how quantum magic evolves under coarse-graining and explore its behavior across different parameter regimes and quantum phases.
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.
The quantum-trajectory description of a noisy quantum circuit is not unique: every Kraus representation of the noise channel defines a different unraveling, i.e., a different ensemble of pure states that averages to the same density matrix. While all unravelings share the same expectation values of observables, they generally differ in the amount of quantum resources carried by the individual trajectories. Here we study non-stabilizerness, or magic, quantified by the stabilizer Rényi entropy, in quantum-trajectory unravelings of Haar-random and T-doped random Clifford brickwork circuits subject to local dephasing noise of strength p. The canonical unraveling in terms of Pauli jumps is Clifford and leaves magic untouched, so that trajectories retain the Haar-random value of non-stabilizerness. Exploring the full two-parameter family of minimal unravelings, we find that the trajectory-averaged magic can be substantially reduced by a state-independent choice of Kraus operators, and that the optimal choice is not in general the projective-measurement unraveling, but a two-outcome generalized measurement in which the Kraus operators correspond to a weak measurement of the noisy qubit. The optimal unraveling reduces the extensive magic density of the trajectories by a p-dependent factor without altering its linear scaling with system size. At the same time, it produces a very broad distribution of magic over trajectories, with a sizable fraction of trajectories retaining Haar-random magic. Since the cost of stabilizer-based classical simulation grows exponentially with the magic of the state, these high-magic trajectories, rather than the average, control the simulation cost. Our results delineate the potential and the limitations of unraveling optimization as a route to the classical simulation of noisy quantum dynamics with stabilizer methods.
An important question of quantum information is to characterize genuinely quantum (beyond-Clifford) resources necessary for universal quantum computing. Here, we use the Pauli spectrum to quantify how ``magic,'' beyond Clifford, typical many-qubit states are. We first present a phenomenological picture of the Pauli spectrum based on quantum typicality, and then we confirm it for Haar random states. We then introduce filtered stabilizer entropy, a magic measure that can resolve the difference between typical and atypical states. We proceed with the numerical study of the Pauli spectrum of states created by random circuits as well as for eigenstates of chaotic Hamiltonians. We find that in both cases, the Pauli spectrum approaches the one of Haar random states, up to exponentially suppressed tails. We discuss how the Pauli spectrum changes when ergodicity is broken due to disorder. Our results underscore the difference between typical and atypical states from the point of view of quantum information.
Nonstabilizerness, also known as "magic," quantifies the deviation of quantum states from stabilizer states, capturing the complexity necessary for quantum computational advantage. In this Letter, we investigate the dynamics of nonstabilizerness in disordered many-body localized (MBL) systems using the stabilizer Rényi entropy (SRE). Leveraging a phenomenological description based on the ℓ-bit model, we analytically and numerically demonstrate that interactions profoundly influence nonstabilizerness spreading, inducing a power-law growth of SRE that markedly contrasts with the rapid saturation observed in ergodic systems. We validate our theoretical predictions through numerical simulations of the disordered transverse-field Ising model, showing excellent agreement across various disorder strengths, system sizes, and initial states. Additionally, we uncover a universal relationship between SRE and entanglement entropy, revealing their common scaling in the MBL regime independent of disorder strength and system size. Our results offer critical insights into the interplay of disorder, interactions, and complexity in quantum many-body systems.
Quantum spin liquids (QSLs) are long-range entangled phases of matter and a natural setting for exploring how quantum correlations generate quantum complexity. Motivated by the emergence of magic, or nonstabilizerness, as a diagnostic of many-body complexity beyond entanglement, we study magic of QSLs by computing the stabilizer Rényi entropy (SRE) of the Kitaev honeycomb model. We derive a correspondence between Pauli strings and products of itinerant Majorana operators in the model's free-fermion description; this enables the sampling of SRE using an optimized algorithm for Gaussian states in systems with thousands of spins. Our results show that magic is largest in the gapless phase, where subleading volume-law corrections indicate nonlocal contributions, while in the gapped phases, SRE decreases in quantitative agreement with a perturbative expansion that we develop in the anisotropic limit. At the topological phase transition, SRE exhibits universal scaling dictated by critical exponents.