Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of Physics A Mathematical and Theoretical

Quantifying magic resource via quantum Jensen–Shannon divergence

Peihua Tian, Yuan Sun

Published Dec 3, 2024• 7 citations
Publisher

Abstract

Abstract Magic is a precious resource necessary for achieving universal fault-tolerant quantum computation. Therefore, it is of vital importance to study the detection and quantification of the magic resource encompassed in quantum states and quantum gates both theoretically and experimentally. In this work, we adopt the quantum Jensen–Shannon divergence to quantify the magic resource of quantum states and quantum gates. On the one hand, we determine the magic resource of a pure state as the minimal and average distance between this state and the set of pure stabilizer states via the quantum Jensen–Shannon divergence, respectively, and extend them to the general mixed states through the method of convex roof construction. We investigate the basic properties of these two magic quantifiers and utilize them to evaluate the magic resource for some typical qubit and qutrit states. By comparing the magic quantifier via the quantum Jensen–Shannon divergence with the min-relative entropy of magic and the stabilizer α-Rényi entropies, we find that the min-relative entropy of magic provides both an upper bound and a lower bound for the magic quantifier via the quantum Jensen-Shannon divergence, and the stabilizer α-Rényi entropies provide a series of lower bounds for the magic quantifier via the quantum Jensen–Shannon divergence. On the other hand, based on the magic quantifier via the quantum Jensen–Shannon divergence for quantum states, we further propose two quantifiers for the magic-resource-generating power of quantum gates and demonstrate that the T-gate is the optimal diagonal unitary gate in creating magic resource for both qubit and qutrit systems in the sense of Clifford equivalence.

DOI
10.1088/1751-8121/ad9a28

Related papers

Quantifying magic via quantum $(α,β)$ Jensen-Shannon divergence

Linmao Wang, Zhaoqi Wu

Magic states play an important role in fault-tolerant quantum computation, and so the quantification of magic for quantum states is of great significance. In this work, we propose two new magic quantifiers by introducing two versions of quantum $(α,β)$ Jensen-Shannon divergence based on the quantum $(α,β)$ entropy and the quantum $(α,β)$-relative entropy, respectively. We derive many desirable properties for our magic quantifiers, and find that they are efficiently computable in low-dimensional Hilbert spaces. We also show that the initial nonstabilizerness in the input state can boost the magic generating power for our magic quantifiers with appropriate parameter ranges for a certain class of quantum gates. Our magic quantifiers may provide new tools for addressing some specific problems in magic resource theory.

quant-pharXiv:2604.06604•Communications in Theoretical Physics•Apr 9, 2026•1 citation

Optimality of T-gate for generating magic resource

Xiaohui Li, Shunlong Luo

Abstract In the stabilizer formalism of fault-tolerant quantum computation, stabilizer states serve as classical objects, while magic states (non-stabilizer states) are a kind of quantum resource (called magic resource) for promoting stabilizer circuits to universal quantum computation. In this framework, the T-gate is widely used as a non-Clifford gate which generates magic resource from stabilizer states. A natural question arises as whether the T-gate is in some sense optimal for generating magic resource. We address this issue by employing an intuitive and computable quantifier of magic based on characteristic functions (Weyl transforms) of quantum states. We demonstrate that the qubit T-gate, as well as its qutrit extension, the qutrit T-gate, are indeed optimal for generating magic resource among the class of diagonal unitary operators. Moreover, up to Clifford equivalence, the T-gate is essentially the only gate having such an optimal property. This reveals some intrinsic optimal features of the T-gate. We further compare the T-gate with general unitary gates for generating magic resource.

Communications in Theoretical Physics•Dec 30, 2022•19 citations

Magic-Resource Generating Power of Quantum Channels

Xiaohui Li

In the stabilizer formalism of quantum error correction and fault-tolerant quantum computation, stabilizer states, as common eigenstates (with the eigenvalue 1) of operators in maximal Abelian subgroups of the Pauli group, play the role of classical states, while magic (non-stabilizer) states are quantum resources enabling universal quantum computation when injected into stabilizer circuits.The natural issue arises as how to characterize and quantify the ability of quantum channels (including quantum gates) in generating magic resource (non-stabilzerness) from stabilizer states.In this work, we introduce two intuitive quantities characterizing magic-resource generating power of quantum channels and reveal their basic properties.We evaluate these quantifies for several prototypical quantum channels and quantum gates.In particular, we demonstrate that the qubit T -gate (i.e., π/8-gate) and its qutrit generalization are optimal in generating magic resource in a class of diagonal unitaries.This highlights the significance of the T -gate and its extensions from a resource-theoretic perspective and provides a theoretical support of their wide applications in stabilizer quantum computation.

Advances in physics research/Advances in Physics Research•Jan 1, 2024•0 citations

Quantifying magic resource for quantum channels via channel-state duality

Xiaohui Li, Shunlong Luo, Zijian Zhang

According to the celebrated Gottesman-Knill theorem, magic resource (nonstabilizerness) is necessary to drive universal quantum computation. Thus characterization and quantification of magic resource for quantum states and quantum channels are significant issues. In this work, we propose a method to construct quantifiers of magic resource for quantum channels from those for quantum states by employing the channel-state duality to convert quantum channels to the corresponding Choi states. We focus on a specific quantifier of magic resource based on characteristic functions (Fourier transforms) of quantum states. This quantifier is well defined for all dimensions and easy to calculate, as opposed to those defined via discrete Wigner functions. It satisfies some desirable properties and provides an upper bound to the mana (a kind of logarithmic negativity of discrete Wigner functions) of quantum channels. Using this quantity, we evaluate the amount of magic resource for some typical quantum channels and compare them with some existing quantifiers of generating-power of magic resource. For the dephasing channels in prime dimensional systems, our quantifier of magic resource essentially characterizes the average generating power of magic resource, while, for general channels, they are quite different and some challenging issues arise.

Physical Review A•May 13, 2025•5 citations

Algorithm for estimating α -stabilizer Rényi entropies via purity

Benjamin Stratton

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.

arXiv:2507.02540•Physical Review A•Jan 23, 2026•0 citations

The relative entropy of magic and its nonadditivity

Carolin Deckers, Justus Neumann, Hermann Kampermann +1 more

In most stabilizer-based quantum computing schemes, so-called magic states are a necessary resource for implementing non-transversal quantum gates. With the resource theory of magic, it is possible to analyze and quantify the generation of the non-stabilizer states. The relative entropy is a measure used in various resource theories. For single qubits, we characterize magic states and their closest stabilizer states by applying analytical results known from the relative entropy of entanglement and show that the magic states and their closest stabilizer states are arranged symmetrically around the states at the centers of the faces of the stabilizer octahedron. For tensor products of single-qubit states, we prove analytically that the relative entropy of magic is nonadditive in almost all cases.

quant-pharXiv:2605.22392•May 22, 2026•0 citations
Abstract Magic is a precious resource necessary for achieving universal fault-tolerant quantum computation. Therefore, it is of vital importance to study the detection and quantification of the magic resource encompassed in quantum states and quantum gates both theoretically and experimentally. In this work, we adopt the quantum Jensen–Shannon divergence to quantify the magic resource of quantum states and quantum gates. On the one hand, we determine the magic resource of a pure state as the minimal and average distance between this state and the set of pure stabilizer states via the quantum Jensen–Shannon divergence, respectively, and extend them to the general mixed states through the method of convex roof construction. We investigate the basic properties of these two magic quantifiers and utilize them to evaluate the magic resource for some typical qubit and qutrit states. By comparing the magic quantifier via the quantum Jensen–Shannon divergence with the min-relative entropy of magic and the stabilizer α-Rényi entropies, we find that the min-relative entropy of magic provides both an upper bound and a lower bound for the magic quantifier via the quantum Jensen-Shannon divergence, and the stabilizer α-Rényi entropies provide a series of lower bounds for the magic quantifier via the quantum Jensen–Shannon divergence. On the other hand, based on the magic quantifier via the quantum Jensen–Shannon divergence for quantum states, we further propose two quantifiers for the magic-resource-generating power of quantum gates and demonstrate that the T-gate is the optimal diagonal unitary gate in creating magic resource for both qubit and qutrit systems in the sense of Clifford equivalence.
Quantifying magic via quantum (α,β)(α,β)(α,β) Jensen-Shannon divergence
Magic states play an important role in fault-tolerant quantum computation, and so the quantification of magic for quantum states is of great significance. In this work, we propose two new magic quantifiers by introducing two versions of quantum (α,β)(α,β)(α,β) Jensen-Shannon divergence based on the quantum (α,β)(α,β)(α,β) entropy and the quantum (α,β)(α,β)(α,β)-relative entropy, respectively. We derive many desirable properties for our magic quantifiers, and find that they are efficiently computable in low-dimensional Hilbert spaces. We also show that the initial nonstabilizerness in the input state can boost the magic generating power for our magic quantifiers with appropriate parameter ranges for a certain class of quantum gates. Our magic quantifiers may provide new tools for addressing some specific problems in magic resource theory.
Abstract In the stabilizer formalism of fault-tolerant quantum computation, stabilizer states serve as classical objects, while magic states (non-stabilizer states) are a kind of quantum resource (called magic resource) for promoting stabilizer circuits to universal quantum computation. In this framework, the T-gate is widely used as a non-Clifford gate which generates magic resource from stabilizer states. A natural question arises as whether the T-gate is in some sense optimal for generating magic resource. We address this issue by employing an intuitive and computable quantifier of magic based on characteristic functions (Weyl transforms) of quantum states. We demonstrate that the qubit T-gate, as well as its qutrit extension, the qutrit T-gate, are indeed optimal for generating magic resource among the class of diagonal unitary operators. Moreover, up to Clifford equivalence, the T-gate is essentially the only gate having such an optimal property. This reveals some intrinsic optimal features of the T-gate. We further compare the T-gate with general unitary gates for generating magic resource.
In the stabilizer formalism of quantum error correction and fault-tolerant quantum computation, stabilizer states, as common eigenstates (with the eigenvalue 1) of operators in maximal Abelian subgroups of the Pauli group, play the role of classical states, while magic (non-stabilizer) states are quantum resources enabling universal quantum computation when injected into stabilizer circuits.The natural issue arises as how to characterize and quantify the ability of quantum channels (including quantum gates) in generating magic resource (non-stabilzerness) from stabilizer states.In this work, we introduce two intuitive quantities characterizing magic-resource generating power of quantum channels and reveal their basic properties.We evaluate these quantifies for several prototypical quantum channels and quantum gates.In particular, we demonstrate that the qubit T -gate (i.e., π/8-gate) and its qutrit generalization are optimal in generating magic resource in a class of diagonal unitaries.This highlights the significance of the T -gate and its extensions from a resource-theoretic perspective and provides a theoretical support of their wide applications in stabilizer quantum computation.
According to the celebrated Gottesman-Knill theorem, magic resource (nonstabilizerness) is necessary to drive universal quantum computation. Thus characterization and quantification of magic resource for quantum states and quantum channels are significant issues. In this work, we propose a method to construct quantifiers of magic resource for quantum channels from those for quantum states by employing the channel-state duality to convert quantum channels to the corresponding Choi states. We focus on a specific quantifier of magic resource based on characteristic functions (Fourier transforms) of quantum states. This quantifier is well defined for all dimensions and easy to calculate, as opposed to those defined via discrete Wigner functions. It satisfies some desirable properties and provides an upper bound to the mana (a kind of logarithmic negativity of discrete Wigner functions) of quantum channels. Using this quantity, we evaluate the amount of magic resource for some typical quantum channels and compare them with some existing quantifiers of generating-power of magic resource. For the dephasing channels in prime dimensional systems, our quantifier of magic resource essentially characterizes the average generating power of magic resource, while, for general channels, they are quite different and some challenging issues arise.
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.
In most stabilizer-based quantum computing schemes, so-called magic states are a necessary resource for implementing non-transversal quantum gates. With the resource theory of magic, it is possible to analyze and quantify the generation of the non-stabilizer states. The relative entropy is a measure used in various resource theories. For single qubits, we characterize magic states and their closest stabilizer states by applying analytical results known from the relative entropy of entanglement and show that the magic states and their closest stabilizer states are arranged symmetrically around the states at the centers of the faces of the stabilizer octahedron. For tensor products of single-qubit states, we prove analytically that the relative entropy of magic is nonadditive in almost all cases.