#nonstabilizerness

Renyi Entropy SRE stabilizer

Simple Metric Links Reveal Quantum ‘Magic’ Metrology Protocol Entanglement and Precision

Quantifying the specific resources that give quantum metrology its increased capabilities has long been an aim of the science, which seeks unprecedented sensing and measurement precision. These resources contain “non-stabilizerness,” or “magic.” This complexity is needed to construct quantum states with a computational quantum advantage.

Piotr Sierant, Marcin Płodzień, Tanausú Hernández-Yanes, and Jakub Zakrzewski from Uniwersytet Jagielloński have found a means to simplify the measuring of this complex resource. They demonstrate that the Stabilizer Rényi Entropy (SRE), a measure of non-stabilizerness for many-particle systems with permutationally symmetric states, can be estimated with a few measured parameters.

Define Non-Stabilizerness and SRE

The Stabilizer The Rényi Entropy quantifies non-stabilizerness by measuring a state's quantum separation from classically simulable stabiliser states. Classical computers may express stabilizer states with great entanglement and coherence. Thus, a quantum device must be non-stabilizery to get a computational quantum advantage.

Due to its mathematical properties, the SRE is a dependable resource indicator. This quantum entanglement metric is useful for expressing almost separable states. Importantly, Stabilizer Renyi Entropy SRE is faithful (zero) only if the state is a stabilizer. It is Clifford invariant and additive on product states, meeting the monotone pure state non-stabilizerness conditions.

Despite its value, non-stabilizerness has been difficult to compute. However, the SRE has been used to study numerous quantum phases and dynamics and may be evaluated computationally and empirically.

The Breakthrough: Simplifying Large System Calculation

The study team developed a new approach to assess this “magic” resource. Historically, permutationally symmetric systems' Stabilizer Renyi Entropy SRE required determining the predicted values of different Pauli-string representatives. Researchers use the symmetry of these states at the big system size limit to simplify computing.

A closed-form expression for the SRE using only a set number of expectation values from collective spin operators is the key achievement. In the limit, the SRE approximation only utilizes six single-axis projections (overlaps) of the studied state for permutation-invariant states.

This condensed formula represents a “drastic simplification” of full quantum state tomography, which needs measurement and postprocessing. An interaction-based readout (twisting-echo) system on platforms with traditional one-axis twisting (OAT) controllers can experimentally achieve the reduced formula's complicated overlaps. This is useful for assessing non-stabilizerness in precision sensing experiments.

Quantum Metrology Protocol SRE Dynamics

This novel analytical framework was utilized to analyze spin-squeezing procedures, which improve quantum metrology measurement accuracy and reduce quantum noise. The work focused on One-Axis Twisting (OAT).

The study showed that optimum spin squeezing generates a considerable increase in non-stabilizerness. The Stabilizer Renyi Entropy SRE diverges logarithmically with system size for states that are more severely spin-squeezed (approaching the minimum parameter). This scale shows that strong non-stabilizerness and severely compressed states are closely related. States with continual spin squeezing have an independent SRE saturation value. Dicke states with zero magnetization and the two-axis countertwisting (TACT) technique corroborated these SRE scaling conclusions in squeezed states.

The Kitten States Paradox

The study also analyzed “kitten” states, macroscopic superpositions of coherent states created later in OAT dynamics. These states have strong many-body Bell correlations, as evaluated by the body correlator, confirming real multipartite entanglement.

Even though kitten states have substantial quantum correlations, their Stabilizer Rényi Entropy is independent of system size. Thus, these states showed an anticorrelation between many-body Bell correlations (E) and SRE (non-stabilizerness).

Greenberger-Horne-Zeilinger (GHZ) is a stabilizer state with diminishing SRE that maximizes many-body Bell correlations. In contrast, the best-squeezed states have reduced Bell correlations, sub-linear Q growth, and logarithmic SRE rise.

Introducing Quantum Monte Carlo Methods

An innovative quantum Monte Carlo method reveals the'magic' of quantum materials

By revealing a powerful tool to investigate non-stabilizerness, researchers gain new insights into critical behaviour and nonlocal quantum correlations.

Many-body quantum systems are difficult to characterise, even though quantum mechanics governs matter and energy. Quantum entanglement, a key component of quantum information, is not enough to realise quantum computers' potential.

The actual driver of quantum advantage is ‘magic’, or ‘non-stabilizerness’. Although extremely entangled, classical computers can imitate stabiliser states using Clifford protocols. This particular property measures the degree to which a quantum state deviates from them. Magic in complex, many-body systems has always been difficult to calculate, especially in higher dimensions or at finite temperatures.

Researchers have developed a quantum Monte Carlo (QMC) scheme to properly measure magic. This unique method can compute the alpha-stabilizer Rényi entropy (SRE), a key indication of magic, and its derivatives in large-scale and high-dimensional quantum systems. The approach is QMC-based, so no prior knowledge of tensor networks is needed, and it can be applied to any Hamiltonian without the "sign problem," which often adds negative weights to QMC simulations, making probabilistic interpretation impossible.

This innovative method interprets alpha-SRE as a ratio of generalised partition functions. The researchers demonstrated that sampling "reduced Pauli strings" limits simulation to a "reduced configuration space." Ingenious solution avoids sign difficulty for efficient classical magic computation.

To simplify SRE value and derivative calculations for various system parameters, the technique uses strong Monte Carlo methods including thermodynamic integration (TI) and reweight-annealing (ReAn). Carefully prepared nonlocal updates that minimise autocorrelations boost performance and ensure accurate and timely results. This is much better than earlier hybrid algorithms that could only calculate alpha-SRE once and extract derivative information, offering very limited physical information.

In one and two dimensions, the researchers demonstrated the strength and adaptability of their unique method using the transverse field Ising (TFI) model, a key component of condensed matter physics. They present a complex and compelling picture of magic at quantum critical points, the temperatures where quantum systems undergo sudden phase shifts.

Researchers separated the 2-SRE's characteristic function (Q-part), which is associated to magic, and free energy (Z-part) for the first time. They found that magic and criticality have a non-trivial link because their derivatives have singularities at critical points.

Magic's behaviour at these vital occasions was more complicated than expected. The 2D TFI model's magic density increased monotonically across the critical point, peaking in the ferromagnetic (FM) phase before decaying, while the 1D model's magic density peaked at the critical point, in line with previous studies. In broad many-body systems, quantum entanglement often peaks around quantum critical points, although alpha-SRE does not always do so. The degree of magic does not always indicate a phase's features.

In addition to magic's magnitude, the study stressed volume-law adjustments to SRE. The non-zero values of these modifications reveal nonlocal magic in correlations that cannot be eradicated by local procedures, making them crucial. The scientists identified discontinuities in these adjustments at quantum crucial places in 1D and 2D TFI models. This sudden change reflects a swift ground-state magical structural shift over the phase transition. They propose that volume-law adjustments can diagnose criticalities better than full-state magic and may even be universal signatures of the boundary conformal field theory's 'g factor'.

As a magic metric, alpha-SRE failed, according to the study. The 2-SRE produced nonphysical results in mixed states (such as finite-temperature Gibbs states) in the 2D TFI model, with singularities appearing at points unrelated to the system's key features. This proves alpha-SRE is unsuitable for mixed-state magic.

Despite this limitation for mixed states, the innovative QMC algorithm opens many research avenues. Because of its versatility, bipartite mutual magic (mSRE) calculation is easy to add. This should help characterise quantum phases and solve difficult problems in finite-temperature phase transitions and open quantum systems.