Nonstabilizerness is a fundamental resource for quantum advantage, as it quantifies the extent to which a quantum state diverges from those states that can be efficiently simulated on a classical computer, the stabilizer states. The stabilizer Rényi entropy (SRE) is one of the most investigated measures of nonstabilizerness because of its computational properties and suitability for experimental measurements on quantum processors. Because computing the SRE for arbitrary quantum states is a computationally hard problem, we propose a supervised machine-learning approach to estimate it. In this work, we frame SRE estimation as a regression task and train a Random Forest Regressor and a Support Vector Regressor (SVR) on a comprehensive dataset, including both unstructured random quantum circuits and structured circuits derived from the physics-motivated one-dimensional transverse Ising model (TIM). We compare the machine-learning models using two different quantum circuit representations: one based on classical shadows and the other on circuit-level features. Furthermore, we assess the generalization capabilities of the models on out-of-distribution instances. Experimental results show that an SVR trained on circuit-level features achieves the best overall performance. On the random circuits dataset, our approach converges to accurate SRE estimations, but struggles to generalize out of distribution. In contrast, it generalizes well on the structured TIM dataset, even to deeper and larger circuits. In line with previous work, our experiments suggest that machine learning offers a viable path for efficient nonstabilizerness estimation.
Nonstabilizerness is a fundamental resource for quantum advantage, as it quantifies the extent to which a quantum state diverges from those states that can be efficiently simulated on a classical computer, the stabilizer states. The stabilizer Rényi entropy (SRE) is one of the most investigated measures of nonstabilizerness because of its computational properties and suitability for experimental measurements on quantum processors. Because computing the SRE for arbitrary quantum states is a computationally hard problem, we propose a supervised machine-learning approach to estimate it. In this work, we frame SRE estimation as a regression task and train a Random Forest Regressor and a Support Vector Regressor (SVR) on a comprehensive dataset, including both unstructured random quantum circuits and structured circuits derived from the physics-motivated one-dimensional transverse Ising model (TIM). We compare the machine-learning models using two different quantum circuit representations: one based on classical shadows and the other on circuit-level features. Furthermore, we assess the generalization capabilities of the models on out-of-distribution instances. Experimental results show that an SVR trained on circuit-level features achieves the best overall performance. On the random circuits dataset, our approach converges to accurate SRE estimations, but struggles to generalize out of distribution. In contrast, it generalizes well on the structured TIM dataset, even to deeper and larger circuits. In line with previous work, our experiments suggest that machine learning offers a viable path for efficient nonstabilizerness estimation.
This article proposes a Graph Neural Network (GNN) approach to estimate nonstabilizerness in quantum circuits, measured by the stabilizer Rényi entropy (SRE). Nonstabilizerness is a fundamental resource for quantum advantage, and efficient SRE estimations are highly beneficial in practical applications. We address the nonstabilizerness estimation problem through three supervised learning formulations starting from easier classification tasks to the more challenging regression task. Experimental results show that the proposed GNN manages to capture meaningful features from the graph-based circuit representation, resulting in robust generalization performances achieved across diverse scenarios. In classification tasks, the GNN is trained on product states and generalizes on circuits evolved under Clifford operations, entangled states, and circuits with higher number of qubits. In the regression task, the GNN significantly improves the SRE estimation on out-of-distribution circuits with higher number of qubits and gate counts compared to previous work, for both unstructured random quantum circuits and structured circuits derived from the transverse-field Ising model. Moreover, the graph representation of quantum circuits naturally integrates hardware-specific information. Simulations on noisy quantum hardware highlight the potential of the proposed GNN to predict the SRE measured on quantum devices.
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' 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.
Nonstabilizerness, also known as magic, plays a central role in universal quantum computation. Hypergraph states are nonstabilizer generalizations of graph states and constitute a key class of quantum states in various areas of quantum physics, such as the demonstration of quantum advantage, measurement-based quantum computation, and the study of topological phases. In this work, we investigate nonstabilizerness of 3-uniform hypergraph states, which are solely generated by controlled-controlled-Z gates, in terms of the stabilizer Rényi entropy (SRE). We find that the SRE of 3-uniform hypergraph states can be expressed using the matrix rank, which reduces computational cost from O(23N) to O(N32N) for N-qubit states. Based on this result, we exactly evaluate SREs of one-dimensional hypergraph states. We also present numerical results of SREs of several large-scale 3-uniform hypergraph states. Our results would contribute to an understanding of the role of nonstabilizerness in a wide range of physical settings where hypergraph states are employed.
Quantum magic, or nonstabilizerness, provides a crucial characterization of quantum systems, regarding the classical simulability with stabilizer states. In this work, we propose an alternative and efficient algorithm for computing stabilizer R\'enyi entropy, one of the measures for quantum magic, in spin systems with sign-problem free Hamiltonians. This algorithm is based on the quantum Monte Carlo simulation of the path integral of the work between two partition function ensembles and it applies to all spatial dimensions and temperatures. We demonstrate this algorithm on the one- and two-dimensional transverse field Ising model at both finite and zero temperatures and show the quantitative agreements with tensor-network based algorithms. We analyze the computational cost and provide analytical and numerical evidences for it to be polynomial in system size. This work also suggests a unifying framework for calculating various types of entropy quantities including entanglement R\'enyi entropy and entanglement R\'enyi negativity.
We introduce a methodology to estimate nonstabilizerness or “magic,” a key resource for quantum complexity, with neural quantum states (NQS). Our framework relies on two schemes based on Monte Carlo sampling to quantify nonstabilizerness via stabilizer Rényi entropy (SRE) in arbitrary variational wave functions. When combined with NQS, this approach is effective for systems with strong correlations and in dimensions larger than one, unlike tensor network methods. First, we study the magic content in an ensemble of random NQS, demonstrating that neural network parametrizations of the wave function capture finite nonstabilizerness besides large entanglement. Second, we investigate the nonstabilizerness in the ground state of the J 1 − J 2 Heisenberg model. In one dimension, we find that the SRE vanishes at the Majumdar-Ghosh point J 2 = J 1 / 2 , consistent with a stabilizer ground state. In two dimensions, a dip in the SRE is observed near maximum frustration around J 2 / J 1 ≈ 0.6 , suggesting a valence bond solid between the two antiferromagnetic phases.