Magic of the Heisenberg Picture (2408.16047v4)

Abstract: We study a non-stabilizerness resource theory for operators, which is dual to that describing states. We identify that the stabilizer R\'enyi 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 R\'enyi 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 non-stabilizerness generation and can inspire Clifford-assisted tensor network methods.

• The paper introduces the Operator Stabilizer Entropy (OSE) in the Heisenberg picture as a novel magic monotone to quantify non-stabilizerness and quantum circuit complexity.
• OSE acts as a robust magic monotone with properties like faithfulness and additivity, offering operational advantages by scaling slower than state-based entropies in local dynamics.
• This research enables potential advancements in classical simulation methods for quantum systems by leveraging operator magic and provides new avenues for studying many-body dynamics.

Insights into Operator Magic in the Heisenberg Picture

The paper under discussion explores the intricate resource theory of quantum magic by focusing on operator-evolved states described in the Heisenberg picture. This research highlights the utility of a novel magic monotone, the Operator Stabilizer Entropy (OSE), which plays a crucial role in assessing quantum circuit complexity and non-stabilizerness, particularly within many-body quantum systems. The paper's central thesis is that operator magic generation, governed by OSE, provides a more nuanced understanding of quantum dynamics compared to state-based measures.

Magic Quantification through Operator Stabilizer Entropy

The OSE emerges as a robust quantifier for non-Clifford operations necessary in quantum computations. This entropy serves as a magic monotone under specific unitary transformations, fulfilling the criteria of faithfulness, stability, additivity, and boundedness. A significant advantage of the OSE is its operational link to the number of Pauli strings required to adequately approximate an operator, akin to the relation between entanglement entropy in tensor networks and circuit complexity.

When analyzing local many-body dynamics, the OSE’s locality manifests through a Lieb-Robinson bound, confining its propagation rate. This constraint contrasts with the often exponential growth observed in other magic measures like the Stabilizer Rényi Entropy (SRE) in chaotic or doped Clifford circuits. Consequently, this outcomes align with current understandings surrounding entanglement spreading and locality in quantum circuits.

Analytical and Numerical Evaluations

Two regimes are analyzed for operator magic growth: one involving random evolution, where the magic almost reaches its peak, and another focusing on integrable systems exemplified by the XXZ circuit. The latter, analyzed through dual unitarity properties and graphical ZX calculus, reveals that magic saturates quickly, illuminating structural insights into the dynamics of many-body quantum systems. The derivation of operator magic reveals it scales significantly slower than state-based entropies, offering computational advantages in systems where locality plays a defining role.

Implications and Future Directions

The introduction of OSE heralds a potential recalibration of magic quantification in quantum computing. It underscores the feasibility of classical simulations by reducing large-scale quantum problems into efficiently computational proxies, leveraging the concept of circuit decompositions in the Heisenberg picture. This could inspire new Clifford-assisted tensor network methods, enhancing the practice of quantum simulations within a classical framework.

The broader implications suggest exciting avenues for exploring many-body dynamics, such as phase transitions and Eigenstate Thermalization Hypothesis (ETH) phenomena, through the lens of operator magic. Moreover, the paper catalyzes future research into the comparison between Schrödinger versus Heisenberg picture stabilizer entropies, potentially informing operational strategies in quantum information processing.

In summary, the examination of OSE within the context of the Heisenberg picture paves the way for further investigations into the resource-theoretic understanding of magic, particularly in local quantum circuits. This research supports the advance of more tractable simulation methodologies that respect the confines of classical computational resources while also contributing to the theoretical mosaic of quantum complexity.