Scientists are increasingly recognising nonstabilizerness, a key resource for universal computation, as crucial to understanding complex quantum systems. Andrew Hallam, Ryan Smith, and Zlatko Papić, all from the School of Physics and Astronomy at the University of Leeds, have developed a novel spectral transfer-matrix framework to investigate this phenomenon within infinite matrix product states. Their work reveals universal subleading information within the spectrum of the stabilizer Rényi entropy, identifying a distinct SRE correlation length that diverges at continuous phase transitions and dictates the spatial response to local perturbations. By deriving exact SRE expressions for the cluster-Ising model and performing numerical analysis, the researchers demonstrate that nonstabilizerness provides a new and powerful means of detecting criticality and understanding the impact of local disturbances in quantum systems.
Understanding the limits of computation demands fresh ways to characterise complex systems. New work offers a powerful technique for detecting critical points, moments of dramatic change, within these systems, revealing how a system’s inherent computational power relates to its response to even minor disturbances. Scientists have uncovered a new way to measure “magic”, a key resource for advanced quantum computation, within complex quantum systems.
This work details how nonstabilizerness, the property enabling universal quantum gates beyond standard operations, behaves particularly close to critical points where materials undergo phase transitions. Specifically, they identified a unique SRE correlation length, differing from conventional measures, that expands dramatically at these critical transitions and dictates how the SRE responds to localised disturbances.
Exact SRE calculations for a simplified “skeleton” of the cluster-Ising model, alongside numerical investigations, demonstrate that nonstabilizerness provides a novel perspective on the connection between computational power and the emergence of complex behaviour in quantum materials. This discovery offers a new tool for characterising criticality and understanding how local changes propagate through a system.
Assessing nonstabilizerness in large systems is generally computationally demanding, prompting investigations into whether it mirrors entanglement’s ability to characterise universal properties. Quantum entanglement is now central to understanding exotic phases of matter and the dynamics of interacting quantum systems. This research addresses computational challenges by focusing on the SRE, a recently proposed measure for many-qubit wave functions, which can be approximated using Monte Carlo methods or computed directly for matrix product states with limited bond dimensions.
These tools have already provided insights into nonstabilizerness across various many-body settings, including critical systems, scrambling models, and systems of identical particles. Ground states of many spin-chain models exhibit varying degrees of nonstabilizerness but often fail to saturate the SRE bound, even at criticality. This highlights the need for analytically solvable models where the SRE enhancement can be rigorously established, mirroring the understanding of entanglement in condensed matter physics and identifying universal properties encoded within nonstabilizerness.
Numerical studies have reported universal SRE scaling in certain models, supported by conformal field theory and exact calculations for non-interacting systems. However, the reliability of the SRE as a diagnostic of criticality remains unclear, as the critical Ising model displays non-analytic features in the SRE, while a Rydberg-atom model exhibiting the same critical point shows smooth behaviour.
In this work, a spectral framework for nonstabilizerness in infinite matrix product states (iMPS) is developed, utilising the eigenspectrum of their SRE replica transfer matrices. The research demonstrates that the SRE of a subsystem embedded in an infinite chain generally decomposes into three contributions: an extensive, model-dependent term capturing global nonstabilizerness; a boundary term corresponding to the mutual SRE between two semi-infinite subsystems; and subleading terms leading to exponentially-decaying SRE correlations.
This decomposition allows for the definition of an SRE correlation length that diverges at continuous phase transitions and governs the spatial response to local perturbations. For the exactly solvable bond dimension χ = 2 MPS skeleton of the cluster, Ising model, these quantities can be obtained analytically, offering microscopic insight into the behaviour of nonstabilizerness.
Calculating entanglement via the dominant eigenvalue of a transfer matrix
A spectral transfer-matrix framework underpinned this work, designed to examine the stabilizer Rényi entropy (SRE) within infinite matrix product states. SRE, a measure of quantum entanglement, was chosen because it offers a way to quantify nonstabilizerness, a property linked to computational power in quantum systems. This approach allowed researchers to probe the subtle behaviour of nonstabilizerness, particularly as systems approach critical points where properties change dramatically.
By analysing the spectrum of the SRE transfer matrix, the study aimed to uncover universal information about the system’s response to local disturbances. Researchers constructed a modified transfer matrix, denoted as E, with dimensions of χ4n×χ4n, central to determining the SRE density, calculated as (1 −n)−1 log(μ1), where μ1 represents the dominant eigenvalue of E.
While this replica trick is exact for low bond dimensions, its computational cost scales as χ6n, limiting its application to smaller systems. To overcome this, an equivalent method based on Pauli-basis conversion was implemented, enabling calculations with larger bond dimensions, albeit with a slight approximation introduced through bond-dimension truncation. Researchers constructed a modified transfer matrix, denoted as E, with dimensions of χ4n×χ4n, central to determining the SRE density, calculated as (1 −n)−1 log(μ1), where μ1 represents the dominant eigenvalue of E. To overcome this, an equivalent method based on Pauli-basis conversion was implemented, enabling calculations with larger bond dimensions, albeit with a slight approximation introduced through bond-dimension truncation.
Employing a truncated Pauli-MPS, reducing the matrix dimension to χ2n t, where χt is less than or equal to χ2, managed this complexity. Once the SRE transfer matrix was established, its spectral decomposition revealed key insights into the system’s behaviour, examining not only the dominant eigenvalue but also the subleading eigenvalues and their corresponding eigenvectors.
These components hold information about correlation lengths and mutual SRE between adjacent subsystems. The decomposition of the SRE transfer matrix, E, into its eigenvalues and eigenvectors, expressed as E = B∗ B Λ, allowed the researchers to define a nonstabilizerness correlation length, analogous to the standard correlation length used in conventional MPS analysis, which diverges at continuous phase transitions.
Furthermore, the dominant eigenvector was found to encode the mutual SRE shared between neighbouring subsystems, providing a direct measure of entanglement. By focusing on finite subsystems embedded within an infinite chain, the study could then analyse how these subleading eigenvalues influence the SRE and reveal signatures of criticality and local perturbations.
Stabilizer Rényi entropy decomposition reveals scaling behaviour at critical points
Initial analysis of the stabilizer Rényi entropy (SRE) within infinite matrix product states reveals a decomposition into three distinct contributions. An extensive term, proportional to system size and denoted as Nm(n), arises from the dominant eigenvalue of the replica transfer matrix, exhibiting model dependence. Alongside this, a boundary term defines the mutual SRE between adjacent semi-infinite subsystems, diverging logarithmically with a correlation length, L(n) ∞.
Finally, exponentially-decaying contributions define an SRE correlation length, ξ(n) SRE, which demonstrates a power-law divergence near continuous phase transitions. For the bond dimension χ = 2 MPS “skeleton” of the cluster-Ising model, these quantities can be obtained analytically, offering microscopic insight into the behaviour of nonstabilizerness.
Still, the mutual SRE, L(n) ∞, exhibited a power-law divergence as the system approached criticality. Once established, this correlation length provides a measure of how far perturbations propagate within the quantum system. By examining the eigenspectrum of the SRE replica transfer matrices, researchers were able to isolate these contributions and define the SRE correlation length.
Now, this length offers a new lens through which to understand the interplay between computational resources and emergent phenomena in quantum many-body systems. The divergence of ξ(n) SRE signals an increased sensitivity to local perturbations as the system nears a critical point, where even small changes can have widespread effects, as reflected in the diverging correlation length.
Quantum magic unlocks a new detector of system-wide phase transitions
Scientists have long sought to understand how complex systems transition between ordered and disordered states, a quest now receiving a surprising boost from the study of quantum computation. This work reveals a new way to detect these critical points, moments of dramatic change, by examining the subtle “magic” within quantum systems. Researchers have identified a unique correlation length linked to this ‘magic’, a resource vital for universal quantum computing, that signals impending shifts in a system’s behaviour.
It’s a clever approach, using the tools of quantum information to probe the fundamental physics of phase transitions. Pinpointing this correlation length proved challenging because it differs from traditional measures used to characterise critical phenomena. Previous attempts often relied on approximations or specific models, limiting their broad applicability.
Now, a spectral transfer-matrix framework offers a more general method for analysing the behaviour of these systems, demonstrated through detailed calculations on the cluster-Ising model, and confirmed with numerical simulations. This framework could allow physicists to identify previously hidden critical behaviours in a wider range of materials and quantum systems.
While the research focuses on theoretical models, the implications extend to areas like materials science and condensed matter physics, potentially aiding the design of new materials with tailored properties. Scaling these calculations to larger, more complex systems presents a significant hurdle. A key question arises: can this method be adapted to detect critical transitions in systems where the underlying quantum ‘magic’ is less pronounced or more difficult to quantify. For now, this work offers a fresh perspective, but further investigation is needed to fully unlock its potential and determine its limits.
👉 More information
🗞 Spectral signatures of nonstabilizerness and criticality in infinite matrix product states
🧠 ArXiv: https://arxiv.org/abs/2602.15116