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.
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.
We study different aspects of the stabilizer entropies (SEs) and compare them against known nonstabilizerness monotones such as the min-relative entropy and the robustness of magic. First, by means of explicit examples, we show that, for Rényi index 0≤n≤2 , the SEs are not monotones with respect to stabilizer protocols which include computational-basis measurements, not even when restricting to pure states (while the question remains open for n≥2 ). Next, we show that, for any Rényi index, the SEs do not satisfy a strong monotonicity condition with respect to computational-basis measurements. We further study SEs in different classes of many-body states. We compare the SEs with other measures, either proving or providing numerical evidence for inequalities between them.Finally, we discuss exact or efficient tensor-network numerical methods to compute SEs of matrix-product states (MPSs) for large numbers of qubits. In addition to previously developed exact methods to compute the Rényi SEs, we also put forward a scheme based on perfect MPS sampling, allowing us to compute efficiently the von Neumann SE for large bond dimensions.
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.
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 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.
The advent of quantum technologies brought forward much attention to the theoretical characterization of the computational resources they provide. A method to quantify quantum resources is to use a class of functions called magic monotones and stabilizer entropies, which are, however, notoriously hard and impractical to evaluate for large system sizes. In recent studies, a fundamental connection between information scrambling, the magic monotone mana and 2-Renyi stabilizer entropy was established. This connection simplified magic monotone calculation, but this class of methods still suffers from exponential scaling with respect to the number of qubits. In this work, we establish a way to sample an out-of-time-order correlator that approximates magic monotones and 2-Renyi stabilizer entropy. We numerically show the relation of these sampled correlators to different non-stabilizerness measures for both qubit and qutrit systems and provide an analytical relation to 2-Renyi stabilizer entropy. Furthermore, we put forward and simulate a protocol to measure the monotonic behaviour of magic for the time evolution of local Hamiltonians.
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 wave functions. 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(2^{N}) 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.