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.
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.
Abstract To achieve universal quantum computation via general fault-tolerant schemes, stabilizer operations must be supplemented with other non-stabilizer quantum resources. Motivated by this necessity, we develop a resource theory for magic quantum channels to characterize and quantify the quantum ‘magic’ or non-stabilizerness of noisy quantum circuits. For qudit quantum computing with odd dimension d , it is known that quantum states with non-negative Wigner function can be efficiently simulated classically. First, inspired by this observation, we introduce a resource theory based on completely positive-Wigner-preserving quantum operations as free operations, and we show that they can be efficiently simulated via a classical algorithm. Second, we introduce two efficiently computable magic measures for quantum channels, called the mana and thauma of a quantum channel. As applications, we show that these measures not only provide fundamental limits on the distillable magic of quantum channels, but they also lead to lower bounds for the task of synthesizing non-Clifford gates. Third, we propose a classical algorithm for simulating noisy quantum circuits, whose sample complexity can be quantified by the mana of a quantum channel. We further show that this algorithm can outperform another approach for simulating noisy quantum circuits, based on channel robustness. Finally, we explore the threshold of non-stabilizerness for basic quantum circuits under depolarizing noise.
To achieve universal quantum computation via general fault-tolerant schemes, stabilizer operations must be supplemented with other non-stabilizer quantum resources. Motivated by this necessity, we develop a resource theory for magic quantum channels to characterize and quantify the quantum 'magic' or non-stabilizerness of noisy quantum circuits. For qudit quantum computing with odd dimension d, it is known that quantum states with non-negative Wigner function can be efficiently simulated classically. First, inspired by this observation, we introduce a resource theory based on completely positive-Wigner-preserving quantum operations as free operations, and we show that they can be efficiently simulated via a classical algorithm. Second, we introduce two efficiently computable magic measures for quantum channels, called the mana and thauma of a quantum channel. As applications, we show that these measures not only provide fundamental limits on the distillable magic of quantum channels, but they also lead to lower bounds for the task of synthesizing non-Clifford gates. Third, we propose a classical algorithm for simulating noisy quantum circuits, whose sample complexity can be quantified by the mana of a quantum channel. We further show that this algorithm can outperform another approach for simulating noisy quantum circuits, based on channel robustness. Finally, we explore the threshold of non-stabilizerness for basic quantum circuits under depolarizing noise.
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.
Magic (non-stabilizerness) is a key resource for achieving universal fault-tolerant quantum computation beyond classical computation. While previous studies have primarily focused on magic in single systems, its interactions and distribution in multipartite settings remain largely unexplored. In this work, we introduce mutual mana as a measure of magic correlations defined in close analogy with quantum mutual information. Our definition builds upon mana, which is the established quantifier of magic based on discrete Wigner function negativity. We characterize magic correlations generated by discrete beamsplitters, whose Gaussian counterparts are fundamental components in quantum optics and quantum technologies. We show that coupling a magic state with a stabilizer vacuum state via a discrete beamsplitter will induce a full conversion of local magic into mutual mana, thereby establishing a mechanism for redistributing magic resources as magic correlations. We reveal the fundamental properties of mutual mana and derive its explicit expressions for several prototypical qutrit states subject to a discrete beamsplitter. We make a comparative study of mutual mana with several established quantifiers of correlations generated by the qutrit beamsplitter, including quantum mutual information, mutual L1-norm magic, and mutual stabilizer 2-Rényi entropy.
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) 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) 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.