Magic states and magic gates are crucial for achieving universal quantum computation, but important questions about how magic resources should be implemented to attain maximal quantum advantage have remained unexplored, especially in the context of measurement-based quantum computation (MQC). This work bridges the gap between MQC and the resource theory of magic by introducing the key concepts of "invested" and "potential" magic resources. The former quantifies the magic cost associated with MQC, serving as both a resource witness and a feasible upper bound for the practical realization, and is gate-order independent; The latter represents the maximal achievable magic resource in a given graph structure defining MQC. We utilize both concepts to analyze the quantum Fourier transform (QFT) and provide a fresh perspective on the universality of MQC, highlighting the crucial role of non-Pauli measurements in injecting magic. In particular, we theoretically prove that high-dimensional graphs can generate an exponential advantage of MQC compared to classical computing. We demonstrate experimentally our theoretical findings in a high-fidelity four-photon setup, surpassing conventional magic state injection (MSI) methods in both qubit efficiency and resource utilization. Our findings pave the way for future research exploring magic resource optimization and novel distillation schemes within the MQC framework, advancing fault-tolerant universal quantum computation.
Magic states and magic gates are crucial for achieving universal quantum computation, but important questions about how magic resources should be implemented to attain maximal quantum advantage have remained unexplored, especially in the context of measurement-based quantum computation (MQC). This work bridges the gap between MQC and the resource theory of magic by introducing the key concepts of "invested" and "potential" magic resources. The former quantifies the magic cost associated with MQC, serving as both a resource witness and a feasible upper bound for the practical realization, and is gate-order independent; The latter represents the maximal achievable magic resource in a given graph structure defining MQC. We utilize both concepts to analyze the quantum Fourier transform (QFT) and provide a fresh perspective on the universality of MQC, highlighting the crucial role of non-Pauli measurements in injecting magic. In particular, we theoretically prove that high-dimensional graphs can generate an exponential advantage of MQC compared to classical computing. We demonstrate experimentally our theoretical findings in a high-fidelity four-photon setup, surpassing conventional magic state injection (MSI) methods in both qubit efficiency and resource utilization. Our findings pave the way for future research exploring magic resource optimization and novel distillation schemes within the MQC framework, advancing fault-tolerant universal quantum computation.
Magic states and magic gates are crucial for achieving universal quantum computation, but important questions about how magic resources should be implemented to attain maximal quantum advantage have remained unexplored, especially in the context of measurement-based quantum computation (MQC). This Letter bridges the gap between MQC and the resource theory of magic by introducing the key concepts of "invested" and "potential" magic resources. The former quantifies the magic cost associated with MQC, serving as both a resource witness and a feasible upper bound for the practical realization, and is gate-order independent; the latter represents the maximal achievable magic resource in a given graph structure defining MQC. We utilize both concepts to analyze the quantum Fourier transform (QFT) and provide a fresh perspective on the universality of MQC, highlighting the crucial role of non-Pauli measurements in injecting magic. In particular, we theoretically prove that high-dimensional graphs can generate an exponential advantage of MQC compared to classical computing. We demonstrate experimentally our theoretical findings in a high-fidelity four-photon setup, surpassing conventional magic state injection methods in both qubit efficiency and resource utilization. Our findings pave the way for future research exploring magic resource optimization and novel distillation schemes within the MQC framework, advancing fault-tolerant universal quantum computation.
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.
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.
Abstract Magic states have been widely studied in recent years as resource states that help quantum computers achieve fault-tolerant universal quantum computing. The fault-tolerant quantum computing requires fault-tolerant implementation of a set of universal logical gates. Stabilizer code, as a commonly used error correcting code with good properties, can apply the Clifford gates transversally which is fault tolerant. But only Clifford gates cannot realize universal computation. Magic states are introduced to construct non-Clifford gates that combine with Clifford operations to achieve universal quantum computing. Since the preparation of quantum states is inevitably accompanied by noise, preparing the magic state with high fidelity and low overhead is the crucial problem to realizing universal quantum computation. In this paper, we survey the related literature in the past 20 years and introduce the common types of magic states, the protocols to obtain high-fidelity magic states, and overhead analysis for these protocols. Finally, we discuss the future directions of this field.
Magic (nonstabilizerness) is a necessary but "expensive" kind of "fuel" to drive universal fault-tolerant quantum computation.To properly study and characterize the origin of quantum "complexity" in computation as well as physics, it is crucial to develop a rigorous understanding of the quantification of magic.Previous studies of magic mostly focused on small systems and largely relied on the discrete Wigner formalism (which is only well behaved in odd prime power dimensions).Here we present an initiatory study of the magic of genuinely many-body quantum states that may be strongly entangled, with focus on the important case of many qubits, at a quantitative level.We first address the basic question of how "magical" a many-body state can be, and show that the maximum magic of an n-qubit state is essentially n, simultaneously for a range of "good" magic measures.As a corollary, the resource theory of magic has asymptotic golden currency states.We then show that, in fact, almost all n-qubit pure states have magic of nearly n.In the quest for explicit, scalable cases of highly entangled states whose magic can be understood, we connect the magic of hypergraph states with the second-order nonlinearity of their underlying Boolean functions.Next, we go on and investigate many-body magic in practical and physical contexts.We first consider a variant of measurement-based quantum computation where the client is restricted to Pauli measurements, in which magic is a necessary feature of the initial "resource" state.We show that n-qubit states with nearly n magic, or indeed almost all states, cannot supply nontrivial speedups over classical computers.We then present an example of analyzing the magic of "natural" condensed matter systems of physical interest.We apply the Boolean function techniques to derive explicit bounds on the magic of certain representative two-dimensional symmetry-protected topological states, and comment on possible further connections between magic and the quantum complexity of phases of matter.
Compared with the quantum gate circuit model, the measurement-based quantum computing model provides an alternative way to realize universal quantum computation, and relevant contents have been greatly enriched after nearly two decades of research and exploration. In this article, we review the research history and status of the measurement-based quantum computing model. First, we briefly introduce the basic theories of this model, including the concept and working principles of quantum graph states as resource states, the model’s computational universality and classical simulation methods, and relevant applications in the field of quantum information processing such as designing quantum algorithms and fault-tolerant error correction schemes. Then, from the perspective of quantum physical properties, which include the specific roles of quantum entanglement, contextuality, quantum correlations, symmetry-protected topological order, and quantum phases of matter as computing resources, the close relationship between measurement-based quantum computing model and quantum many-body system is presented. For example, a type of measurement-based computing model for exploiting quantum correlations can show a quantum advantage over the classical local hidden variable models, or certain symmetry-protected topological order states enable the universal quantum computation to be conducted by using only the measurements of single-qubit Pauli operators. Next, a variety of different technical routes and experimental progress of realizing the measurement-based quantum computing model are summarized, such as photonic systems, ion traps, superconducting circuits, etc. These achievements in various physical areas lay the foundation for future scalable and fault-tolerant quantum computers. Finally, we discuss and prospect the future research directions in this field thereby inspiring readers to further study and explore the relevant subjects.