We introduce a novel measure for the quantum property of "nonstabilizerness"-commonly known as "magic"-by considering the Rényi entropy of the probability distribution associated to a pure quantum state given by the square of the expectation value of Pauli strings in that state. We show that this is a good measure of nonstabilizerness from the point of view of resource theory and show bounds with other known measures. The stabilizer Rényi entropy has the advantage of being easily computable because it does not need a minimization procedure. We present a protocol for an experimental measurement by randomized measurements. We show that the nonstabilizerness is intimately connected to out-of-time-order correlation functions and that maximal levels of nonstabilizerness are necessary for quantum chaos.
We introduce a novel measure for the quantum property of "nonstabilizerness"-commonly known as "magic"-by considering the Rényi entropy of the probability distribution associated to a pure quantum state given by the square of the expectation value of Pauli strings in that state. We show that this is a good measure of nonstabilizerness from the point of view of resource theory and show bounds with other known measures. The stabilizer Rényi entropy has the advantage of being easily computable because it does not need a minimization procedure. We present a protocol for an experimental measurement by randomized measurements. We show that the nonstabilizerness is intimately connected to out-of-time-order correlation functions and that maximal levels of nonstabilizerness are necessary for quantum chaos.
We study a nonstabilizerness resource theory for operators, which is dual to that describing states. We identify that the stabilizer Rényi entropy analog in operator space is a good magic monotone satisfying the usual conditions while inheriting efficient computability properties and providing a tight lower bound to the minimum number of non-Clifford gates in a circuit. Operationally, this measure quantifies how well an operator can be approximated by one with only a few Pauli strings-analogous to how entanglement entropy relates to tensor-network truncation. A notable advantage of operator stabilizer entropies is their inherent locality, as captured by a Lieb-Robinson bound. This feature makes them particularly suited for studying local dynamical magic resource generation in many-body systems. We compute this quantity analytically in two distinct regimes. First, we show that under random evolution, operator magic typically reaches near-maximal value for all Rényi indices, and we evaluate the Page correction. Second, harnessing both dual unitarity and ZX graphical calculus, we solve the operator stabilizer entropy for interacting integrable XXZ circuit, finding that it quickly saturates to a constant value. Overall, this measure sheds light on the structural properties of many-body nonstabilizerness generation and can inspire Clifford-assisted tensor network methods.
Nonstabilizerness (``magic'') is an essential part of quantum computation, so the quantitative study of nonstabilizerness has become a very important issue. Since researchers have defined the Pauli spectrum of quantum states, they can use it to calculate other nonstabilizerness measures such as stabilizer R\'enyi entropy and stabilizer nullity. At the same time, the study of the magic-channel resource theory has become a main direction in recent years. In this article we extend the definition of the Pauli spectrum to quantum channels and define the definitions of stabilizer R\'enyi entropy and stabilizer nullity for quantum channels based on the Pauli spectrum. Meanwhile, we study the properties of stabilizer R\'enyi entropy and stabilizer nullity of quantum channels and give the corresponding proofs. Additionally, this article gives the necessary and sufficient condition that the depolarizing channel is a completely stabilizer-preserving operation (CSPO), and we can calculate other channels. Furthermore, we verify that stabilizer R\'enyi entropy of quantum channels satisfies the amortization inequality so that the maximum value of the distillation bounds of the magic channels provided by the stabilizer R\'enyi entropy can be established. Therefore, we can give an alternative method to calculate the magic-channel distillation bounds. Moreover, we compare the channel distillation rate provided by stabilizer R\'enyi entropy with that provided by thauma and find that the channel distillation rate proposed in this article has higher distillation efficiency. Finally, the nonstabilizerness cost of the gate synthesis is given.
Nonstabilizerness, or “magic,” is a resource for universal quantum computation in most fault-tolerant architectures; access to states with nonstabilizerness allows for nonclassically simulable quantum computation to be performed. Quantifying this resource for unknown states is therefore essential to assessing their utility in quantum computation. The stabilizer Rényi entropies have emerged as a leading tool for achieving this, having already enabled one efficient algorithm for measuring nonstabilizerness. In addition, the stabilizer Rényi entropies have proven useful in developing connections between nonstabilizerness and other quantum phenomena. In this work, we introduce an alternative algorithm for measuring the stabilizer Rényi entropies of an unknown quantum state. First, we show the existence of a state, produced from the action of a channel on α copies of some state ρ , that encodes the α -stabilizer Rényi entropy of ρ into its purity. We detail several methods of applying this channel, and then, by employing existing purity-measuring algorithms, provide an algorithm for measuring the α -stabilizer Rényi entropies for all integers α > 1 . This algorithm is benchmarked for qubits and the resource requirements compared to other known algorithms. Finally, a nonstabilizerness-entanglement relationship is shown to exist in the algorithm, demonstrating a relationship between the two resources, before an instance of resource hiding is found.
Nonstabilizerness or magic resource characterizes the amount of non-Clifford operations needed to prepare quantum states.It is a crucial resource for quantum computing and a necessary condition for quantum advantage.However, quantifying magic resource beyond a few qubits has been a major challenge.Here, we introduce efficient measures of magic resource for pure quantum states with a sampling cost that is independent of the number of qubits.Our method uses Bell measurements over two copies of a state, which we implement in experiment together with a cost-free error-mitigation scheme.We show the transition of classically simulable stabilizer states into intractable quantum states on the IonQ quantum computer.For applications, we efficiently distinguish stabilizer and nonstabilizer states with low measurement cost even in the presence of experimental noise.Further, we propose a variational quantum algorithm to maximize our measure via the shift rule.Our algorithm can be free of barren plateaus even for highly expressible variational circuits.Finally, we experimentally demonstrate a Bell-measurement protocol for the stabilizer Rényi entropy as well as the Wallach-Meyer entanglement measure.Our results pave the way to understanding the nonclassical power of quantum computers, quantum simulators, and quantum many-body systems.
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.
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.