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 (``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, also known as "magic," stands as a crucial resource for achieving a potential advantage in quantum computing. Its connection to many-body physical phenomena is poorly understood at present, mostly due to a lack of practical methods to compute it at large scales. We present a novel approach for the evaluation of nonstabilizerness within the framework of matrix product states (MPSs), based on expressing the MPS directly in the Pauli basis. Our framework provides a powerful tool for efficiently calculating various measures of nonstabilizerness, including stabilizer Rényi entropies, stabilizer nullity, and Bell magic, and enables the learning of the stabilizer group of an MPS. We showcase the efficacy and versatility of our method in the ground states of Ising and XXZ spin chains, as well as in circuits dynamics that has recently been realized in Rydberg atom arrays, where we provide concrete benchmarks for future experiments on logical qubits up to twice the sizes already realized.
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.
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.
Unraveling the secrets of how much nonstabilizerness a quantum dynamic can generate is crucial for harnessing the power of magic states, the essential resources for achieving quantum advantage and realizing fault-tolerant quantum computation. In this work, we introduce the amortized α-stabilizer Rényi entropy, a magic monotone for unitary operations that quantifies the nonstabilizerness generation capability of quantum dynamics. Amortization is key in quantifying the magic of quantum dynamics, as we reveal that nonstabilizerness generation can be enhanced by prior nonstabilizerness in input states when considering the α-stabilizer Rényi entropy, while this is not the case for robustness of magic or stabilizer extent. We demonstrate the versatility of the amortized α-stabilizer Rényi entropy in investigating the nonstabilizerness resources of quantum dynamics of computational and fundamental interest. In particular, we establish improved lower bounds on the T-count of quantum Fourier transforms and the quantum evolutions of one-dimensional Heisenberg Hamiltonians, showcasing the power of this tool in studying quantum advantages and the corresponding cost in fault-tolerant quantum computation.
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.
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.