Calculating how much ‘quantum magic’ exists within complex quantum states has long been challenging due to computational limitations. A new method for computing stabilizer Rényi entropies, a measure of non-stabilizerness, now uses real-space renormalization group techniques from the University of Novi Sad in Serbia. The approach yields a closed-form expression applicable to low-energy systems, enabling detailed analysis of how magic evolves as complexity is reduced and across different physical phases.

A new way to measure ‘quantum magic’, an essential resource for advanced computation exceeding classical capabilities, actively creates a means to do so. This method calculates the amount of non-stabilizerness existing within specific quantum systems; stabilizer states can be efficiently simulated on conventional computers but lack computational power beyond that. The team’s approach uses real-space renormalization techniques to provide a precise calculation applicable to low-energy scenarios, revealing how this magical quality changes as complexity increases and transitions between different physical states occur.

The University of Novi Sad devised a new technique for quantifying ‘quantum magic’, a key resource enabling computations beyond classical limits. This measurement focuses on calculating non-stabilizerness within quantum systems; stabilizer states represent simplified types of quantum states easily mimicked by regular computers but lacking advanced computational power.

The team’s method employs real-space renormalization group techniques, akin to simplifying a detailed map by focusing on broad patterns rather than individual details, yielding precise calculations applicable in low-energy scenarios. This allows researchers to track how much ‘magic exists as complexity increases and transitions between different physical phases occur, offering insights into potential advancements in computation; however, determining precisely how this measure relates to the practical capabilities of these complex systems remains an open question.

Real-space renormalisation streamlines calculation of entanglement entropy in large spin systems

A closed-form expression for stabilizer Rényi entropies is now available, enabling calculations on systems with up to N lattice sites where impractical optimisation procedures were once required beyond few qubits. Applying real-space renormalization group techniques progressively simplifies complex spin Hamiltonians by grouping spins into effective units and revealing larger scale behaviour without intensive numerical computation. The researchers successfully computed these entropies to investigate non-stabilizerness, or ‘quantum magic’, in quantum states arising from spin models; they also analysed how it evolves during coarse-graining across different parameter regimes and phases.

Systems containing up to N lattice sites are now amenable to calculation using a closed-form expression for stabilizer Rényi entropies, representing a sharp increase over previous limitations achieved through work. Real-space renormalization group techniques actively analysed non-stabilizerness, often referred to as ‘quantum magic’, within spin models, demonstrating changes in this property as complexity decreases across various parameters and phases.

These quantities were computed for both transverse field Ising models and XY spin chains, revealing insights into quantum phase transitions alongside measures like coherence and discord; analysis of the TFIM showed that the RSRG scheme effectively coarse-grains by grouping spins into larger units, transforming coupling strengths J and h during each step according to specific equations, similar processes also proved effective with the XY model.

Analytical calculation of non-stabilizerness advances for restricted Hamiltonian models

Quantifying ‘quantum magic’, the non-stabilizerness unlocking computational advantages beyond classical simulation, has long been hampered by practical limitations because calculating this property demands immense resources even for modestly sized quantum systems. The new analytical approach developed at University of Novi Sad bypasses some hurdles; however, a significant constraint remains as their current closed-form expression holds true only within low-energy scenarios when analysing spin Hamiltonians. Despite these limitations to specific systems, this represents a valuable step forward in quantifying complex phenomena.

The team has provided a novel analytical tool, a closed-form expression for stabiliser Rényi entropies, which quantifies ‘quantum magic’ or non-stabilizerness and allows exploration of the boundary between classical and quantum computation. Real-space renormalisation group techniques simplify complex systems by focusing on broad patterns rather than individual details, enabling calculations previously limited by computational cost; this approach effectively reduces complexity while preserving key features. By tracking how this ‘magic’ evolves during simplification, scientists can better understand its behaviour across different physical states within spin Hamiltonians.

The researchers developed an analytical method to quantify “quantum magic”, representing properties beyond what classical computers can simulate. This calculation uses stabilizer Rényi entropies and real-space renormalization-group techniques to analyse transverse field Ising models and XY spin chains. The resulting closed-form expression simplifies the analysis of quantum phenomena in low-energy scenarios, allowing for a more efficient understanding of complex systems as they are coarse-grained. Authors suggest further work will explore how this property changes with decreasing complexity under various conditions and phases.

👉 More information
🗞 Real-Space Renormalization of Stabilizer Rényi Entropies in Spin Chains
✍️ Sonja Gombar, Petar Mali, Slobodan Radošević, Milica Rutonjski, Milan Pantić and Milica Pavkov-Hrvojević
🧠 ArXiv: https://arxiv.org/abs/2609.17188

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