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Physical review. D/Physical review. D.

Probing chaos by magic monotones

Kanato Goto, Tomoki Nosaka, Masahiro Nozaki A

Published Dec 19, 2022• 27 citations
Publisher

Abstract

There is a property of a quantum state called ``magic.'' As shown by the Gottesman-Knill theorem, so-called stabilizer states, which are composed of only Clifford gates, can be efficiently computed on a classical computer, and thus quantum computation gives no advantage. Nonstabilizer states are called magic states, which are necessary to achieve the universal quantum computation. Magic (monotone) is the measure of the amount of nonstabilizer resource, and it measures how difficult it is for a classical computer to simulate the state. We study magic of states in the integrable and chaotic regimes of the higher-spin generalization of the Ising model through two quantities: ``mana'' and ``robustness of magic'' (RoM). We find that in the chaotic regime, mana increases monotonically in time in the early-time region, and at late times these quantities oscillate around some nonzero value that increases linearly with respect to the system size. Our result also suggests that under chaotic dynamics, any state evolves to a state whose mana almost saturates the optimal upper bound; i.e., the state becomes ``maximally magical.'' We find that RoM also shows similar behaviors. On the other hand, in the integrable regime, mana and RoM behave periodically in time in contrast to the chaotic case. In addition to mana and RoM, for the early-time behavior of magic, we study the stabilizer R\'enyi entropy, which can be numerically computed for larger systems than mana and RoM. In the anti-de Sitter/conformal field theory correspondence, classical spacetime emerges from the chaotic nature of the dual quantum system. Our results suggest that magic of quantum states is strongly involved in the emergence of spacetime geometry.

DOI
10.1103/physrevd.106.126009

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There is a property of a quantum state called magic.'' As shown by the Gottesman-Knill theorem, so-called stabilizer states, which are composed of only Clifford gates, can be efficiently computed on a classical computer, and thus quantum computation gives no advantage. Nonstabilizer states are called magic states, which are necessary to achieve the universal quantum computation. Magic (monotone) is the measure of the amount of nonstabilizer resource, and it measures how difficult it is for a classical computer to simulate the state. We study magic of states in the integrable and chaotic regimes of the higher-spin generalization of the Ising model through two quantities: mana'' and robustness of magic'' (RoM). We find that in the chaotic regime, mana increases monotonically in time in the early-time region, and at late times these quantities oscillate around some nonzero value that increases linearly with respect to the system size. Our result also suggests that under chaotic dynamics, any state evolves to a state whose mana almost saturates the optimal upper bound; i.e., the state becomes maximally magical.'' We find that RoM also shows similar behaviors. On the other hand, in the integrable regime, mana and RoM behave periodically in time in contrast to the chaotic case. In addition to mana and RoM, for the early-time behavior of magic, we study the stabilizer R'enyi entropy, which can be numerically computed for larger systems than mana and RoM. In the anti-de Sitter/conformal field theory correspondence, classical spacetime emerges from the chaotic nature of the dual quantum system. Our results suggest that magic of quantum states is strongly involved in the emergence of spacetime geometry.
The advent of quantum technologies brought forward much attention to the theoretical characterization of the computational resources they provide. A method to quantify quantum resources is to use a class of functions called magic monotones and stabilizer entropies, which are, however, notoriously hard and impractical to evaluate for large system sizes. In recent studies, a fundamental connection between information scrambling, the magic monotone mana and 2-Renyi stabilizer entropy was established. This connection simplified magic monotone calculation, but this class of methods still suffers from exponential scaling with respect to the number of qubits. In this work, we establish a way to sample an out-of-time-order correlator that approximates magic monotones and 2-Renyi stabilizer entropy. We numerically show the relation of these sampled correlators to different non-stabilizerness measures for both qubit and qutrit systems and provide an analytical relation to 2-Renyi stabilizer entropy. Furthermore, we put forward and simulate a protocol to measure the monotonic behaviour of magic for the time evolution of local Hamiltonians.
Magic refers to the degree of “quantumness” in a system that cannot be fully described by stabilizer states and Clifford operations alone. In quantum computing, stabilizer states and Clifford operations can be efficiently simulated on a classical computer, even though they may appear complicated from the perspective of entanglement. In this sense, magic is a crucial resource for unlocking the unique computational power of quantum computers to address problems that are classically intractable. Magic can be quantified by measures such as Wigner negativity and mana that satisfy fundamental properties such as monotonicity under Clifford operations. In this paper, we generalize the statistical mechanical mapping methods of large- q random circuits to the calculation of Rényi Wigner negativity and mana. Based on this, we find: (1) a precise formula describing the competition between magic and entanglement in many-body states prepared under Haar random circuits; (2) a formula describing the spreading and scrambling of magic in states evolved under random Clifford circuits; (3) a quantitative description of magic “squeezing” and “teleportation” under measurements. Finally, we comment on the relation between coherent information and magic.
Identifying the boundary between classical and quantum computation is a central challenge in quantum information. In multi-qubit systems, entanglement and magic are the key resources underlying genuinely quantum behaviour. While entanglement is well understood, magic - essential for universal quantum computation - remains relatively poorly characterised. Here we show that determining membership in the stabilizer polytope, which defines the free states of magic-state resource theory, requires super-exponential time $\class{exp} ( n^2)$ in the number of qubits nnn, even approximately. We reduce the problem to solving a 333-\class{SAT} instance on n2n^2n2 variables and, by invoking the exponential time hypothesis, the result follows. As a consequence, both quantifying and certifying magic are fundamentally intractable: any magic monotone for general states must be super-exponentially hard to compute, and deciding whether an operator is a valid magic witness is equally difficult. As a corollary, we establish the robustness of magic as computationally optimal among monotones. This barrier extends even to classically simulable regimes: deciding whether a state lies in the convex hull of states generated by a logarithmic number of non-Clifford gates is also super-exponentially hard. Together, these results reveal intrinsic computational limits on assessing classical simulability, distilling pathological magic states, and ultimately probing and exploiting magic as a quantum resource.
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.
Non-stabilizerness – commonly known as magic – measures the extent to which a quantum state deviates from stabilizer states and is a fundamental resource for achieving universal quantum computation. In this work, we investigate the behavior of non-stabilizerness around criticality in quantum spin chains. To quantify non-stabilizerness, we employ a monotone called mana, based on the negativity of the discrete Wigner function. This measure captures non-stabilizerness for both pure and mixed states. We introduce Rényi generalizations of mana, which are also measures of non-stabilizerness for pure states, and utilize it to compute mana in large quantum systems. We consider the three-state Potts model and its non-integrable extension and we provide strong evidence that the mutual mana exhibits universal logarithmic scaling with distance in conformal field theory, as is the case for entanglement.
Nonstabilizerness, or magic, is an essential quantum resource for performing universal quantum computation. Specifically, Robustness of Magic (RoM) characterizes the usefulness of a given quantum state for non-Clifford operations. While the mathematical formalism of RoM is concise, determining RoM in practice is highly challenging due to the necessity of dealing with a super-exponential number of stabilizer states. In this work, we present novel algorithms to compute RoM. The key technique is a subroutine for calculating overlaps between all stabilizer states and a target state, with the following remarkable features: (i) the time complexity per state is reduced {exponentially}, and (ii) the total space complexity is reduced {super-exponentially}. Based on this subroutine, we present algorithms to compute RoM for arbitrary states up to n = 8 qubits, whereas the naive method requires a memory size of at least 86 PiB , which is infeasible for any current classical computer. Additionally, the proposed subroutine enables the computation of stabilizer fidelity for a mixed state up to n = 8 qubits. We further propose novel algorithms that exploit prior knowledge of the structure of the target quantum state, such as permutation symmetry or disentanglement, and numerically demonstrate our results for copies of magic states and partially disentangled quantum states. This series of algorithms constitutes a comprehensive ``handbook'' for scaling up the computation of RoM. We envision that the proposed technique will also apply to the computation of other quantum resource measures.